Geometrical Methods for Kinematics and Dynamics in Relativistic Theories of Gravity with Applications to Cosmology and Space Physics

Home , Arthur Geoffrey Walker, Raychaudhuri equation

Geometrical methods for kinematics and dynamics in relativistic theories of gravity with applications to cosmology and space physics

Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert–Ludwigs–Universität Freiburg im Breisgau vorgelegt von

Matteo Carrera

aus Tenero–Contra, Schweiz

August 16, 2010 Dekan: Prof. Dr. Kay Königsmann Betreuer der Dissertation: Prof. Dr. Domenico Giulini Referent: Prof. Dr. Domenico Giulini Korreferent: Prof. Dr. Hartmann Römer Datum der mündlichen Prüfung: October 6, 2010 to my family

Il semble que tout l’effort industriel de l’homme, tous ses calculs, toutes ses nuits de veille sur les épures, n’aboutissent, comme signes visibles, qu’à la seule simplicité, comme s’il fallait l’expérience de plusieurs générations pour dégager peu àpeu la courbe d’une colonne, d’une carène, ou d’un fuselage d’avion, jusqu’àleur rendre la pureté élémentaire de la courbe d’un sein ou d’une épaule. Il semble que le tra- vail des ingénieurs, des dessinateurs, des calculateurs du bureau d’études ne soit ainsi en apparence, que de polir et d’effacer, d’alléger ce raccord, d’équilibrer cette aile, jusqu’àce qu’on ne la remarque plus, jusqu’àce qu’il n’y ait plus une aile accrochée àun fuselage, mais une forme par- faitement épanouie, enfin dégagée de sa gangue, une sorte d’ensemble spontané, mystérieusement lié, et de la même qualitéque celle du poème. Il semble que la perfection soit atteinte non quand il n’y a plus rien à ajouter, mais quand il n’y a plus rien àretrancher.

Antoine de Saint Exup´ery, Terre des Hommes (1939), (Chapitre III: L’Avion)

Contents

Notation and conventions 1

Introduction and motivation 3

I Basic settings of a gravitational theory 11

1 A framework for relativistic theories of gravity 13 1.1 Deﬁningtheframework ...... 13 1.2 Immediateconsequences ...... 15

2 Observers and reference frames 17 2.1 Observers ...... 18 2.2 Derivatives ...... 21 2.2.1 Fermiderivative ...... 21 2.2.2 Comoving and corotating derivative ...... 26 2.3 Observercharacterization ...... 29 2.3.1 Acceleration, rotation, shear, and expansion ...... 30 2.3.2 Synchronization ...... 36 2.4 Referenceframes ...... 37

3 Observer-referred kinematics and dynamics 39 3.1 Thetimelikecase: testparticles...... 40 3.1.1 Gamma-factorandrelativevelocity...... 40 3.1.2 Observer derivatives and relative acceleration ...... 41 3.1.2.1 Characterization and uniqueness of the observer derivative...... 44 3.1.3 Projectionoftheequationofmotion ...... 47 3.2 Thelightlikecase:lightrays...... 51 3.2.1 Frequencyandpropagationdirection ...... 51 3.2.2 Projectionof the propagationequation...... 52

4 Spherically symmetric spacetimes 55 4.1 Sphericalsymmetry ...... 55 4.2 Conservedcurrents and conservedcharges ...... 57 4.2.1 TheMisner–Sharpenergy ...... 57 4.2.2 The Kodamavectorfieldand the arealvolume ...... 58 4.2.3 The Kodama current and the Misner–Sharp energy ...... 59 4.3 MoreontheKodamavectorfield ...... 60 4.3.1 Kodama vector field and Killing fields ...... 60 4.3.2 PropertiesoftheKodamaobserverfield ...... 61

vii viii Contents

II Applications to general-relativistic cosmology and space physics 63

5 Spherical symmetry in General Relativity 65 5.1 Misner–Sharp energyin General Relativity ...... 65 5.1.1 Einstein equation in case of spherical symmetry ...... 65 5.1.2 Interpretation of the Misner–Sharp energy ...... 66 5.1.2.1 Misner–Sharp energy and other energy deﬁnitions . 66 5.1.2.2 Ricci and Weyl decomposition of the Misner–Sharp energy...... 67 5.1.3 Spherically symmetric perfect ﬂuids ...... 69 5.2 Spherically-symmetric matchings ...... 72 5.2.1 Matching procedure for spherically symmetric spacetimes . . 76

6 Spherically symmetric inhomogeneities in cosmological spacetimes77 6.1 Findingthemodel ...... 77 6.2 Matchedsolutions ...... 80 6.2.1 Homogeneouscosmologicalmodels ...... 81 6.2.2 TheEisenstaedttheorem ...... 85 6.2.3 The Einstein–Straus–Schücking vacuole revisited ...... 85 6.3 TheMcVittiemodels...... 87 6.3.1 TheoriginalMcVittiemodel ...... 88 6.3.2 GeometryoftheMcVittieansatz ...... 91 6.3.2.1 Relation to conformal Schwarzschild class ...... 91 6.3.2.2 SpatialRicci-isotropy ...... 92 6.3.2.3 Misner–Sharp energy and Kodama observer field . . 94 6.3.2.4 Singularitiesand trappedsurfaces ...... 95 6.3.2.5 Otherglobalaspects...... 98 6.4 Attempts to generalize McVittie’s model ...... 98 6.4.1 Einstein’s equation for the McVittie ansatz ...... 99 6.4.2 Perfectfluid...... 100 6.4.3 Perfectfluidplusheatflow ...... 101 6.4.4 Perfectfluidplusnullfluid ...... 104 6.4.5 Conclusion ...... 104

7 Cosmologicaleﬀectsonlocaldynamics 107 7.1 Electromagnetically-boundedsystems ...... 107 7.1.1 EquationofmotioninFLRWspacetimes ...... 108 7.1.2 Exact condition for non-expanding circular orbits ...... 109 7.2 Gravitationally-boundedsystems ...... 111 7.2.1 Equation of motion in McVittie spacetime ...... 111 7.2.2 Exact condition for non-expanding circular orbits ...... 112 7.3 Next-to-Newtoniananalysis ...... 113 7.3.1 Specifying the initial-value problem ...... 114 7.3.2 Discussion of the reduced eﬀective potential ...... 115

8 Cosmologicaleﬀectsonkinematics 119 8.1 InﬂuenceonDopplertracking ...... 119 8.1.1 Two-way Doppler tracking in a FLRW spacetime ...... 120 8.1.2 Two-way Doppler tracking in a McVittie spacetime ...... 124 8.2 Conclusion ...... 126 Contents ix

III Massive gravitational theories 129

9 Linear massive theories 131 9.1 Geometricbackground ...... 132 9.2 The van Dam–Veltman–Zakharovargument ...... 133 9.3 Massivetensortheoryforgravity ...... 134 9.3.1 Static spherically symmetric vacuum solution ...... 134 9.3.2 Lagrangian formulation and matter couplings ...... 136 9.3.3 TheEquivalencePrinciple ...... 137 9.3.4 Field equations with matter and their Cauchy problem . . . . 138 9.3.5 Couplingtoapointparticle ...... 140 9.3.6 Coupling to the electromagnetic ﬁeld ...... 142 9.3.7 Deﬂection of light rays in the massive spin-2 theory . . . . . 143 9.3.8 Themasslesslimitanditsproblems ...... 144 9.4 Massive scalar-tensor theories for gravity...... 145 9.4.1 Fieldequationsforageneralmassterm ...... 145 9.4.2 Scalar-tensor decomposition and Cauchy problem ...... 147 9.4.3 Thescalarisaghost ...... 150 9.4.4 Commentongravitationalwaves ...... 151 9.4.5 Formalsolutionandmasslesslimit ...... 153 9.4.6 Static spherically symmetric vacuum solution ...... 154 9.5 Conclusions about linear massive theories ...... 155

10 Nonlinear massive theories 157 10.1Freetheory ...... 157 10.2Mattercoupling...... 160 10.3 Static spherically-symmetric conﬁgurations ...... 161 10.3.1 Sphericalsymmetryandstaticity ...... 161 10.3.2 Spatiallyisotropiccoordinates ...... 163 10.3.3 Curvature-coordinates ...... 163 10.3.4 Numerical and analytical investigations ...... 164

Appendices 165

A Diﬀerential geometry 167 A.1 Basicnotation ...... 167 A.2 Tensorﬁeldsalongamap ...... 169 A.3 Derivations ...... 170 A.4 Antisymmetric multilinear forms ...... 172 A.4.1 TheHodgestar...... 174 A.4.2 Hodge star for two-dimensional Lorentzian manifolds ..... 175 A.5 Symmetric multilinear forms ...... 176 A.6 Connectionsandcurvature ...... 176 A.6.1 Decompositionof the curvaturetensor ...... 177 A.6.2 Sectionalcurvature...... 179 A.6.3 Curvature of two-dimensional Lorentzian manifolds ...... 180 A.7 Submanifolds ...... 180 A.8 Conformaltransformations ...... 182 A.9 Warpedproducts ...... 183 A.10Sphericalsymmetricspacetimes ...... 185 x Contents

B Proofs 189 B.1 ProofofTheorem6.2...... 189 B.2 ProofofProposition6.3 ...... 190 B.3 ProofofProposition6.5 ...... 193

C General Relativity 195 C.1 Basicsequations ...... 195 C.2 Variationalformulation ...... 196 C.3 Linearization ...... 197

D Observational data 199 D.1 Cosmologicaldata ...... 199 D.2 Pioneer10and11data...... 200

Bibliography 201

Publications and preprints 209

Index 211 Notation and conventions

We first fix some conventions. We mostly use geometric units, in which G = c = 1, reintroducing G’s and c’s when needed. The differential-geometric notation and conventions, which mostly adhere to the monograph [Str84]1, are collected in Ap- pendix A.1. In the rest of Appendix A we collect the mathematical definitions, statements, and formulae which are needed in the text. For the spacetime metric we adopt the ‘mostly minus’ signature choice. Hence, a vector v is timelike, lightlike, or spacelike w.r.t. the metric g if g(v, v) is positive, zero, or negative, respectively. In the present work we make large use of the index-free notation therefore, to help the understanding, we distinguish between scalar functions, written in italic, and tensors (fields), written in boldface italic. The differential operators which are denoted by capital Latin letters (like, e.g., the Lie and the Fermi derivative) are written in boldface Roman. In the Chapters 2 and 3 the spacetime dimension does not really matter and therefore it is left unspecified and denoted with n. When dealing with spherical symmetry (Chapter 4) and when considering applications (Parts II and III) the spacetime is taken to be four-dimensional. When using indices we adopt the following convention: spacetime indices are Greek and running from 0 to 3 (or, in general, to n 1) whereas space indices are Latin and running from 1 to 3 (or to − n 1). The time index takes thus the value 0. When considering semi-Riemannian − manifolds in general (as we do in Appendix A) the indices are Latin and running from 1 to n. The Einstein summation convention is always used, unless stated otherwise. Notice that when dealing with field theories on Minkowski spacetime (Chapter 9) the indices refer always to a Minkowskian coordinate-basis, hence one for which it holds η(∂/∂xµ, ∂/∂xν ) = diag(1, 1, 1, 1), where η denotes the − − − Minkowski metric. The following abbreviations are used throughout the text: ‘iff’ stands for ‘if and only if’, ‘w.l.o.g.’ for ‘without loss of generality’, ‘w.r.t.’ for ‘with respect to’, and ‘r.h.s.’ and ‘l.h.s.’ for ‘right-’ and ‘left-hand side’, respectively.

1 We refer also to the new edition [Str04] which, however, makes use of the ‘mostly plus’ signature.

1 Acknowledgments

I am grateful to my supervisor Prof. Dr. Domenico Giulini for his patient advice, support, and encouragement, and for helping me discovering the beauty of physics. I wish to express my gratitude also to Prof. Dr. Hartmann Römer and to his successor Prof. Dr. Stefan Dittmaier for the generous acceptance in their groups. This work has been supported by the following institutions: the Deutsche Forschungsgemeischaft (research project ‘Quantentheorie und Gravitation’, reference no. RO 864/7), the Swiss National Science Foundation (fellowships for prospec- tive researchers ‘Uber¨ die Frage der dynamischen Konsistenz massiver Gravitations- felder’, reference no. PBSK2–106579), and the European Space Agency (Ariadna Project of the Advanced Concepts Team ‘On the effect of the global cosmological expansion on the local dynamics in the Solar System’, ID 04/1302, contract no. 18913/05/NL/MV). I am also grateful to the European Space Agency (ESTEC) and the Albert-Einstein-Institute in Golm for their hospitality and support. I would like to thank Prof. Dr. Felix Finster, Prof. Dr. Oscar Lanford III, and PD. Dr. Claus Lämmerzahl for useful discussions and for pointing out relevant references. I owe my deepest gratitude to my parents, Marco and Wanda Carrera, which always encouraged me to continue my way. Without their generous support and warm encouragement the conclusion of this work would not have been possible. In equal measure I am grateful to Felicitas Lingnau for her love, understanding, encouragement, and for keeping me grounded. My gratitude extends also to her family for the understanding and encouragement. Finally, warm thanks go also to all the professors, colleagues, students, and friends which enriched the last years with stimulating and motivating discussions about physics, mathematics, and the other nice things of life. Among them I ac- knowledge in particular, and in alphabetical order, Florian Becher, Svea Beiser, Fabrizio Bernasconi, Yvonne Brühlmann, Michael Carl, Tommaso Cereghetti, Gus- tav Holzegel, Stefan Jansen, Stefano Lecchini, the late Nikolai Neumaier, the late Prof. Dr. Klaus Pohlmeyer, Prof. Dr. Hartmann Römer, Sebastian Schlicht, Imke Schneider, Stefan Waldmann, Stefan Weiss, and the friends of the ‘Kleiner Kreis’. Introduction and motivation

The main part of the present work is motivated by the following simple questions: given that the universe is expanding at large scales, what are the structures which resist the expansion? Are there any ‘local’ experiments or observations which are influenced by it? One might of course expects that, for example, the atoms and molecules from which we are made up do not expand in a measurable way; and the same we believe to be true for our Solar System. But it remains uncertain up to which scale astronomical structures remain insensitive to cosmological expansion. These questions are so fundamental that one should expect to find them discussed in any good introductory book on cosmology. These, however, mostly limit themselves to a pictorial explanation in which the bounded structures (identified roughly with galaxy clusters) are described as raisins in an expanding pancake or pennies attached to an expanding two-dimensional rubber membrane or inflating balloon (see, e.g., Sect. 27.5 in [MTW73]2). Here, the pancake and the membrane represent the expanding space, whereas the raisins and the pennies stand for the mass over- densities which do not expand. To our knowledge, the only introductory book which is more specific on this issue is that of Stephani [Ste80] (see [Ste04] for a recent En- glish edition). There, the above questions are answered by means of the so-called vacuole model due to Einstein & Straus [ES45, ES46] and Schücking [Sch54]. This is an exact solution to Einstein’s equation which is given by the Schwarzschild spacetime (i.e. static, spherically symmetric, and vacuum) inside a ball of some radius

Rv (called the ‘vacuole radius’) and is a dust-filled Friedman–Lemaˆıtre–Robertson– Walker (FLRW) cosmological model (i.e. spatially homogeneous and isotropic) outside this ball (see Section 6.2.3). This is perhaps the simplest model which combines together the two regimes covered by the two most paradigmatic solutions to Ein- stein’s equations: Schwarzschild’s solution, which describes the spacetime around a spherically symmetric mass distribution like a star, and the FLRW models, on which our picture of the Universe at large scales is based. If the vacuole model could be applied, it would imply that there is no effect at all of the cosmological expansion inside the vacuole. This seems, at first sight, very satisfying but, as we will discuss in Chapter 6, cannot be the definitive answer to our questions. Beside their fundamental character, our questions also have a very practical relevance. For example: should people at a space agency be concerned about cosmological expansion when computing trajectories for spacecrafts which are intended to fly somewhere in the outermost regions of the Solar System? Or when measuring velocity or acceleration (relative to us, on the Earth) of spacecrafts by means of some tracking procedure via the exchange of electromagnetic signals?

2 To be precise, at the end of Sect. 27.5 of [MTW73] one is referred, for a more detailed treatment of the topic, to a paper of Noerdlinger & Petrosian [NP71]. The latter authors base their analysis on the McVittie model, which we will extensively discuss in Chapter 6.

3 4 Introduction and motivation

To address the raised issues one has first of all to make precise some very fundamental notions like, e.g., what do we mean by saying that the orbit of a planet is non-expanding and what do we mean by relative velocity and acceleration. At this point it is important to realize that the latter is an issue which urges to be addressed, as is evident from the confusions which are present in the literature originating from a misuse of the concepts of relative velocity and acceleration. One of the most fundamental lessons of General Relativity (GR) is that one has to give up the idea of global inertial system. This concept has to be replaced by local reference systems (to be defined below) which, in general, will be accelerating or rotating with respect to each other. Accordingly, a rethinking is needed regarding the concepts of relative velocity and relative acceleration. This we do in Part I, where in full generality (meaning independently of any choice of model and, to some extent, independently of the choice of the field equation for the gravitational field) we lay down a theoretical framework for kinematics and dynamics, which allows to precisely formulate (and answer) the questions raised above. The next step, which is tackled in Part II, is then to find appropriate models, in the setting of GR, for quasi-isolated structures in a cosmological spacetime. Here, by quasi-isolated structures we mean bounded systems (where the binding can be of gravitational or electromagnetic nature) or geometrical structures like the horizon of a black hole. Finally, we apply to the considered models the developed formalism in order to estimate the effects of the cosmological expansion on such structures.

In the last part of the present work (Part III) we address another issue exploring the possibility of generalizing the field equation of GR, i.e. Einstein’s equation, allowing for a ‘mass’ of the gravitational field. Thereby the motivation is to provide candidate theories for gravity which allow to test the experimental validity of GR in a ‘neighborhood’ of it. This possibility is first studied at linear level, where the field equations are taken to be those of a massive spin-2 field on flat Minkowski spacetime. There, we critically review the so-called van Dam–Veltman–Zakharov (vDVZ) discontinuity, which states that, as the mass tends to zero, the deflection of light rays converge to 3/4 of the value predicted by the massless theory (linearized GR). This discontinuity, together with the present-day observational tests of the light bending, is often believed to prove that the mass of the gravitational field must be exactly zero. This would imply that gravity, at linear level, is a massless spin-2 field and herewith, following a famous argument (see [Des70] and references therein), uniquely leading to Einstein equation if one includes self-interaction. Accordingly, the vDVZ discontinuity would provide an important argument in favor of Einstein’s equation and thus of GR. That is why we critically analyze the status of the vDVZ discontinuity giving an alternative, geometrical approach which shows the subtleties and the fragility of this argument in the linear setting. Next we extend our analysis to the nonlinear setting where it is sometime believed that the vDVZ discontinuity is cured. One should mention that a certain interest has come up in the last years on massive gravitational theories in particular in regard to gravitational waves [Wil98, WY04, FS02, SF02]. This is also a reason for studying the theoretical consistency of such theories, an issue which we address in Chapter 9. 5

Outline and main results

In Part I we introduce a setting which is suitable for the study of kinematics and dynamics within a class of test theories of gravity, that is gravitational theories which are ‘in the neighborhood’ of GR (the meaning of this is specified in Chapter 1). Our approach, exposed in Chapter 2 and which goes back to [JCB92], consist in the local spacetime decomposition (‘split’) in space and time which arise by the choice, at each point of spacetime, of a preferred time direction. This involves the specification of a timelike future-pointing (and, w.l.o.g., normalized) vector field, which we will call an ‘observer field’. The local space plus time split is then just the decomposition of the tangent space in the orthogonal subspaces w.r.t. the considered observer field. This decomposition is then systematically extended to the basic geometric objects (tensor fields) as well as to derivations. For the latter we take a new approach, based on our orthogonal decomposition lemma for derivations (Lemma 2.5), which gives a particularly elegant and unifying way to construct derivatives which have a direct physical interpretation (as, e.g., the Fermi derivative). All the derivatives introduced in this work are based on this construction. In Chapter 3 we apply the observer-referred decomposition of spacetime to the kinematics and dynamics of particles and, respectively, light rays. We give thereby the definitions of ‘relative velocity’ and ‘relative acceleration’ which allow to re- formulate the spacetime equation of motions (as, e.g., the geodesic and the Lorentz equations) in term of these spatial quantities. This re-formulated equation of motion is then to be seen as suitable generalization of Newton’s (second) equation. It is essential to realize that the notions of relative velocity and relative acceleration introduced here are to be understood as local quantities since they give the relative velocity (and acceleration) between a worldline and an observer field at the very same spacetime point. Accordingly, the reformulated equation of motion is to be understood as a local relation between quantities defined at the same spacetime point. An important issue concerns the definition of the relative acceleration: if we want it to be defined as the ‘variation’ of the relative velocity we have to specify which derivative operator we have to choose for taking the variation. In Section 3.1.2 we argue in favor of the derivative defined in (3.8), which we call ‘observer derivative’ and for which we give a uniqueness characterization in Section 3.1.2.1. The rich geometric structure of spherically symmetric spacetimes is presented in Chapter 4. We first recall their description in terms of warped product spaces. Next we review how two conserved currents (the so-called ‘Kodama vector field’ and the ‘Kodama current’, whose original definition goes back to Kodama [Kod80]) arise from the intrinsic geometric structure of spherically symmetric spacetimes. We derive their properties, as well as their related charges, which are given by the so-called areal volume and by the Misner–Sharp (MS) energy, respectively. As we will show in Chapter 5, the latter is a quantity which can be interpreted as a concept of quasi-local mass in the setting of GR via Einstein’s equation. However, we stress here that these objects are defined and useful without the need of any field equation. In particular, we stress the importance, and derive the kinematical properties, of the Kodama vector field and of its normalization, called the ‘Kodama observer field’. The former exhibit many of the properties of a timelike Killing field and can therefore be seen as the generalization of the latter for an arbitrary spherically symmetric spacetime. Accordingly, the Kodama observer field can be seen as the 6 Introduction and motivation natural substitute for a stationary observer field (the latter being defined as the normalization of a future-pointing timelike Killing field) and herewith is recognized to be the natural choice of observer field in an arbitrary spherically symmetric spacetime. This also gives, as by-product, an elegant proof of Birkhoff’s Theorem (see Section 4.3). In Chapter 5 we specialize to GR and review the interpretation of the MS energy as a concept of quasi-local mass (Section 5.1). For this purpose we first review how the Einstein equation simplifies in case of spherical symmetry. Moreover, we give the decomposition of the MS energy in its Ricci and Weyl part and discuss their physical interpretation. The former is locally determined—via Einstein equation— by the matter distribution, whereas the latter is recognized to be a measure for the integrated mass contrasts, furnishing herewith a very useful geometrical tool for detecting inhomogeneities. In Section 5.2 we then show how the tools developed in Chapter 4 can be used to greatly simplify the problem of matching spacetimes, under the assumptions that these are spherically symmetric. We derive there spherically symmetric junction conditions (SSJC) which are equivalent, in case of spherical symmetry, to the general (Darmois) junction conditions. These new junction conditions, beside having a nice physical interpretation, involve only scalar quantities and are easy to verify, which made them to a very efficient tool for constructing new solutions to Einstein’s equation in situations of spherical symmetry. These results have been published in [CG10b, CG10a]. Armed with these tools, in Chapter 6 we tackle the problem of finding an appropriate model for a bound system in a cosmological spacetime. In the case where the binding is of electromagnetic nature as, for example, between a proton and an electron (where the gravitational binding can be neglected), it is quite obvious what we have to do: we solve Maxwell’s equation for a point charge (the proton) in a given cosmological background (taken to be a FLRW spacetime) and then integrate the Lorentz equation for the electron. This direct approach, which goes back to Bonnor [Bon99], is reviewed in Section 7.1. For electromagnetically-bounded systems the situation is so simple since the binding force and the cosmological expansion are governed by separate equations (Maxwell’s and Einstein’s equations, respectively) and the bounded system, in the considered simplified example, does not influence at all the cosmological background. Things are not so simple if one considers gravitationally-bounded systems: in this case, both the binding force as well as the cosmological expansion are governed by the same equation (Einstein’s equation), which is nonlinear. This makes particularly challenging the task of finding solutions modeling the simultaneous presence of two (or more) structures like, in our case, a gravitationally-bounded system in a cosmological spacetime, since there is no superposition principle. At this point one should recall the general situation which is encountered in physics: we are hardly ever in the position to mathemati- cally rigorously model physically realistic scenarios. Normally, we are at best able to provide either exact solutions for approximate models or approximate solutions for realistic models. The art of physics consists then in finding the right balance between these two strategies. However, one should stress that the most important models on which our physical understanding of the World rests are given by exact solutions. Moreover, exact solutions are also of fundamental importance for the validation of approximation schemes and numerical outcomes. That is why, in trying to model an inhomogeneity immersed in a cosmological spacetime, we con- 7 centrate in this work to exact solutions. For simplicity the inhomogeneity is taken to be electrically neutral and non-rotating. The latter simplification motivates the assumption of spherical symmetry, to which we shall restrict here. The known exact solutions for these two structures are, as is well-known, the Schwarzschild and the FLRW solutions, respectively. In Chapter 6 we will explain how one can combine together this two solutions (we recall again that, due to the nonlinearity of Einstein’s equation, this cannot be done via the superposition principle). For this purpose there are two ways one can follow, which can be called the ‘matching’ and the ‘melting’ approach, respectively. In the first, one matches together the Schwarzschild and the FLRW solutions along a suitable matching hypersurface. This leads to the Einstein–Straus–Schücking (ESS) vacuole model, for which we give an alternative, simplified derivation with help of our spherically symmetric junction conditions (see Section 6.2.3). The latter conditions give also, as by-product, an immediate proof of the Eisenstaedt Theorem (see Section 6.2.2). This theorem, due to Eisenstaedt [Eis77], says that a necessary condition for the matching of a spherically symmetric inhomogeneity in a FLRW spacetime is that its ‘average energy density’ (measured as the ratio of the MS energy to the areal volume) must be equal to the energy density of the FLRW spacetime. In the second approach, one tries to ‘melt together’ the Schwarzschild and the FLRW solutions. This means that the sought solution is required to be approximately equal to Schwarzschild’s solution for small radii and to approach a FLRW solution for large radii. In the region in-between, then, one expects to have a kind of mixture of both. A model which meets these requirements is the so-called McVittie solution (cf. (6.39)), which we extensively study in Section 6.3 exploring then its possibles generalizations in Section 6.4. In this analysis we take advantage of the tools developed in the Chapters 4 and 5, which prove themselves useful for computational as well as for interpretational purposes. The advantages and disadvantages of the ESS vacuole and of the McVittie model are discussed in Section 6.1, where we also give a survey of the (few) known exact solutions for the considered physical situation. There we conclude by arguing that the McVittie model is, at present, the most suitable exact solution to Einstein’s equation for estimating of the effects of the cosmological expansion on gravitationally-bounded local systems. The estimation of such effects on dynamics is then carried out in Chapter 7. We start by considering the case of electromagnetically-bounded systems where we consider the example of a proton-electron system (Section 7.1). For the case where the binding is of gravitational nature we consider the motion of a test particle (which stands for a planet or a spacecraft) in a McVittie spacetime (Sec- tion 7.2). In both cases we derive the spatial version (in the sense explained in Chapter 3) of the general-relativistic equation of motion. The key point here is that the term ‘spatial’ is to be referred to the Kodama observer field which, as we already stressed above, is recognized to be the natural choice of observer field in an arbitrary spherically symmetric spacetime. In fact, this choice of observer gives, in an appropriate approximation scheme, the Newtonian equation of motion improved with a cosmological correction term which is also derived in several other ways in the literature [Pac63, Pac64, CFV98, Pri05, AMF07]. This improved Newtonian equation is then studied in Section 7.3: our discussion, which has been published in [CG05, CG10b], complements the perturbative analysis of [CFV98] which misses all orbits which are unstable under cosmological expansion (which do exist). In 8 Introduction and motivation this respect we follow a very similar strategy as, e.g., in the more recent papers by Price [Pri05] (the basic idea of which goes back at least to Pachner [Pac63, Pac64]) and also [AMF07], though we think that our approach has definite advantages. The result of this analysis is that there is a critical radius above which no closed orbits (and hence no bound configurations) exist and that below it the effect of cosmological expansion shows up as an increase of the orbital radius and orbital period, as compared to the situation without cosmological expansion. This is also to be expected by a Newtonian picture of cosmological expansion in which an accelerated (decelerated) expansion corresponds to outwardly-pulling (inwardly-pushing) forces. These effects are of quadratic order in H and can be safely neglected for planets in the Solar Systems or for atoms (ideally modeled by a classical proton-electron system). To be concrete, these effects would give relative deviations for orbital radius and period of at most 10−16 for a planet in the Solar System (i.e. for an orbital radius smaller than 100 AU). Similarly, for an atom whose radius is smaller than, say, 104 Bohr radii, these relative deviations would be smaller than 10−57. In the last chapter of Part II, Chapter 8, we discuss the effects of cosmological expansion on the kinematics of Doppler tracking. Here we derive the exact formula for the two-way Doppler-tracking in FLRW spacetimes (see eq. (8.12) in Section 8.1.1). This formula is then generalized, in a controlled approximation, for the McVittie spacetime in Section 8.1.2. These Doppler formulas give the relation between the observable frequency-shift rate (measured ‘here’, on Earth) and the local kinematical properties of the spacecraft (defined ‘there’, at the spacetime point occupied by the spacecraft) in the considered spacetime. We show that the leading-order cosmological correction to the Newtonian contribution is given by an acceleration term of magnitude Hβ, where β is the ‘radial velocity’ (over c) of the spacecraft (a term which is made precise below). Accordingly, the correction to the Doppler formula is of linear order in H, depending thus only on the expansion of the universe. This is to be opposed to the case of dynamics analyzed in Chapter 7, where the correction to the Newtonian equation of motion due to the cosmological expansion is of quadratic order in H and depends only on the acceleration of that expansion. For typical spacecrafts within the Solar System the contribution of the cosmological correction in the Doppler formula relative to the Newtonian one is shown to be at most of order 10−7. This is negligible but is ‘only’ four order of magnitude smaller than the so-called ‘anomalous acceleration’ of the spacecrafts Pioneer 10 and 11 (usually referred to as ‘Pioneer anomaly’ and henceforth abbreviated by PA), which happens to be surprisingly close in magnitude to the value Hc 7 10−10 m/s2 (see ≈ × Appendix D.2) and which we take as reference for the present-day measurement precision. These results have been published in [CG06, CG10b]. In Part III we address the question of whether it is possible to modify the field equation of gravity with the addition of a ‘mass term’. In Chapter 9 we argue that, in the framework of linear Poincaré-covariant field theories, one cannot meet the requirements of both theoretical consistency and accordance with observations. For details we refer to the introduction of that chapter and to its conclusions in Sec- tion 9.5. That is why in Chapter 10 we extend our analysis to nonlinear theories, where sometimes it is believed that these problems are cured. Here, we first introduce a parametrization for nonlinear generalizations of the massive Pauli–Fierz spin-2 theory which covers all generalizations present in the literature. This, how- 9 ever, shows also the arbitrariness of this generalization process. We then focus on static spherically symmetric situations, where we review the only known solutions (due to Salam & Strathdee [SS77] together with their generalizations by Isham & Storey [IS78]), identifying the geometrical origin of their classification of such solutions. It has long been debated about the existence of solutions in the nonlinear theory which for small mass parameters approach the Schwarzschild solution of GR (implying herewith the vanishing of the vDVZ discontinuity) [Vai72, DDGV02, DKP03] and the debate still goes on [BDZ10, BDZ09]. We approached this problem both with a numerical analysis as well as exploring the possibility of establishing some ‘no-go theorem’ for asymptotically flat solutions. The results of our numerical analysis agree with those of Damour and collaborators [DKP03]: it seems not to be possible to numerically construct a solution with the above properties. However, the establishment of a ‘no-go theorem’ based on analytic methods was not be possible up to now.

Conclusions and outlook

In this work we give a thorough and careful treatment of the problem of estimating the effect of cosmological expansion on dynamics and kinematics, the latter on the example of Doppler tracking. Thereby we lay down an appropriate framework for relativistic kinematic and dynamics within which this problem can be precisely formulated and solved. For the case of dynamics this leads to a new, genuinely relativistic derivation of the correction term due to the cosmological expansion to the Newton equation of motion, which is derived in several other ways in the literature. We show that this correction term give rise to deviations of quadratic order in the Hubble’s constant H which are, within the Solar System, many order of magnitude smaller than the present-day measurement precision and thus safely negligible. For the case of Doppler tracking the kinematical framework allow to set up a Doppler formula which we explicitely derive for the relevant spacetimes. Here, the effect of the cosmological expansion shows up as a correction term of linear order in H which, for typical situations in the Solar System, is also negligible but ‘only’ about four order of magnitude smaller than the measurement precision. This gives a satisfactory answer our initial questions. In the last part of the work we explore the possibility of modifying the field equations for gravity by the addition of a mass term. In the framework of Poincaré- covariant linear field theories we argue that one cannot meet the requirements of both theoretical consistency and accordance with observations. In regard to nonlinear generalizations the question still remains open, as also pointed out by Thibault Damour in the section about experimental tests of gravitational theory of the last edition (January 2010) of The Review of Particle Physics [A+08]: “The addition of a mass-term in Einstein’s field equations leads to a score of theoretical difficulties which have not yet received any consensual solution”. However, we have also to point out that during the proof-reading phase of this work the explicit analysis of [BDZ10] was made available. This seem to prove the existence of an asymptotically flat solution which for small mass parameter shows a recovery of the corresponding solution of GR, giving thus a numerical confirmation of the original claim of Vainshtein. As main direction for future research we feel that an effort should be done in finding more realistic models for inhomogeneities in cosmological spacetimes. 10 Introduction and motivation

Believing in the importance of exact solutions, the first step would be to start with spherically symmetric situations, where the tools exposed in the Chapters 4 and 5 promise to be useful for the construction of new solutions, as well as for their physical interpretation. In a second step, then, one may try to weaken the assumption of spherical symmetry and to go over to axial symmetry in order to account also for the rotation of the modeled astronomical structures. An interesting aspect which we did not address in this work is the effect that local inhomogeneities have on the global dynamics of spacetime (‘back-reaction’). Moreover, such inhomogeneities might also have an impact on the interpretation of our cosmological observations which, at present, relies on the FLRW homogeneous and isotropic models. Recent serious efforts to interpret cosmological data in models with realistic inhomogeneities have been done in [Buc00, Räs06, Wil07, BC03, Buc08]. It has even been speculated that the current measurement of the cosmological constant could be re-interpreted as an effect of inhomogeneities [Cél00, Cél07, Wil07, Wil08]. Also in regard to this issues it would be very useful to have at disposal exact solutions which model realistic inhomogeneous situations. Part I

Basic settings of a gravitational theory

Chapter 1

A framework for relativistic theories of gravity

The purpose of this chapter is to specify a framework which allows to study candidate theories of gravity, as well as the interaction of gravity with matter, in a unified manner. Thereby, we are not supposed to lay down the most general setting pretending to motivate the necessity of every assumption made. Our scope here is to focus attention to candidate theories which are, in a sense to be specified below, in ‘the neighborhood’ of the theory of gravity which, by nearly a century now, passes with excellence all the theoretical and experimental scrutinies: General Relativity (GR). We aim thus to define a framework which is restrictive enough to already imply some interesting theoretical consequences but, on the same time, also not too restrictive to allow only for GR. We refer also to [Wil93, Dic64, TLL73], and references therein, for the general discussion on the theoretical fundament for a theory of gravity.

1.1 Deﬁning the framework

As for GR, motivated by the universality of the free-fall (meaning that the motion of a structureless1 test body in a gravitational ﬁeld is independent of its mass and composition) and the equality of inertial and gravitational mass, we require the following principle to hold (see, e.g., [Str04] for the formulation and [Wil01, Wil93] for its experimental status):

Einstein Equivalence Principle (EEP). In an arbitrary gravitational ﬁeld no local non-gravitational experiment can distinguish a freely falling non-rotating system from a uniformly moving system in the absence of gravitational ﬁelds.

Assuming the special relativistic picture for the description of the non-gravitational interactions, the mathematical implementation of the EEP is obtained generalizing the flat Minkowski spacetime (R4, η) to a Lorentzian manifold ( , g), hence a M differentiable manifold endowed with a metric g (i.e. a non-degenerate smooth M symmetric (0, 2) tensor field) with the same signature as the Minkowski metric η. The pair ( , g), to which we shall refer to as the spacetime, is the minimal set M 1 By ‘structureless’ we mean a body whose spin and higher momenta are negligible.

13 14 Chapter 1. A framework for relativistic theories of gravity of ingredients for a theory of gravity: should be thought of as the ‘set of all M events’ and g is necessary in order to measure lengths and times intervals, as well as angles. Moreover, in order to differentiate tensor fields (i.e. compare them at different nearby spacetime points), one needs an affine connection. Among all possible connections, there is one in particular which is distinguished since it can be constructed from the metric (via the Koszul formula). This is the so-called Levi-Civita connection, which is uniquely characterized by the metricity and torsion-freeness requirements (see Appendix A.6). Although neither metricity nor torsion-freeness seem to be implied by the EEP (see, e.g., [MTGC07] and references therein), according to the strategy ‘first things first’ we chose here to take, like in GR, the Levi-Civita connection, denoted in the following by ∇. Hence, summarizing, we have the following framework:

1. The ‘set of all events’ is modeled by a four-dimensional diﬀerentiable manifold , which is also assumed to be oriented. M

2. There is a set of fields on which can be classified in ‘matter fields’ (de- M scribing proper matter, electromagnetic fields, etc.) and ‘gravitational fields’ (all the rest). The gravitational fields are subdivided in:

(i) a Lorentzian metric g, called ‘physical metric’, which defines the causal structure and enables to measure lengths, times intervals, and angles (ii) the Levi-Civita connection ∇ to g, which defines parallel transportation and allows to differentiate tensor fields (iii) eventually other fields that describe gravity or are needed in order to set up the field equations for g. These can be either of dynamical nature (i.e. specified by some differential equations plus initial conditions) or of static nature (i.e. specified by some algebraic equation).

3. The equations for the fields in point 2. shall be generally covariant. By this we mean that they are covariant under the action of the group of diffeomorphisms of , without adding any extra fields. Moreover, the field equations for the M matter fields shall be derived from those valid in Special Relativity via the mathematical implementation of the EEP hence, via the substitutions ‘η g’ → and ‘∂ ∇’. → We call a theory satisfying the above requirements a relativistic theory of gravity. It is worth noting that GR is in some sense the simplest theory which fits the framework. By this we mean that if for the gravitational fields one does not take any fields other than g, then, under some additional but very natural assumptions, Lovelock’s Theorem [Lov71] leads directly to Einstein’s field equation for g and herewith to GR. The other theories in this framework have either extra fields for the description of gravity other than the metric g or other conditions on the metric. Of the second type is, for example, the Einstein–Fokker theory, where one puts the additional condition of vanishing of the Weyl tensor to g. Examples of the first type are the family of ‘massive’ gravitational theories we study in Part III (the additional field being a background metric), scalar-tensor theories as, e.g., the Brans–Dicke– Jordan theory, and Ni theory (see e.g. [MTW73] and references therein). 1.2. Immediate consequences 15

1.2 Immediate consequences

From point 3. of the above framework it follows, in particular, that the matter equation of motions that are implied by the divergence-freeness of the matter energy- momentum tensor are the same as in GR. The easiest example for this are perfect ﬂuids, which are described by a velocity ﬁeld u (timelike, future-pointing, and normalized), energy density ̺ and pressure p (which have to satisfy some equation of state in the form f(̺,p) = 0). The energy-momentum tensor is then given by

T = ̺ u u + p (u u g) , (1.1) ⊗ ⊗ − where we notice that the (0, 2) tensor after p is the positive-deﬁnite spatial metric on the subspace of the tangent space orthogonal to u (see (2.7)). Decomposing the divergence equation div T = 0 for (1.1) in its orthogonal parts w.r.t. u one shows that the former is equivalent to the following set of equations:

u(̺)= (̺ + p) div u , (1.2a) − (̺ + p)∇uu = Pu ∇p , (1.2b) where Pu is the projector onto the subspace of the tangent space orthogonal to u (see (2.1b)). These are the relativistic Euler equations for a perfect fluid which, together with the equation of state and appropriate initial conditions on a spacelike Cauchy hypersurface, determine uniquely the evolution for u, ̺, and p. That free-falling structureless test bodies move along timelike geodesics of g follows from the EEP via the standard argument of writing the special-relativistic equation of motion, valid in a local inertial frame, and then correcting this to a covariant equation (see, e.g., Sect. 1.4.1 in [Str04]). Alternatively, one can derive this from the divergence-freeness of the energy-momentum tensor of a point particle (see Sect. 22.3 in [Ste04]). This result can be generalized also for any ‘small bodies’, requiring, in addition, that the related energy-momentum tensor satisfies the strong energy condition [GJ75]. The same standard argument used for free-falling structureless test bodies can be applied to show that light rays move along lightlike geodesics for g. This can be also derived in the geometric-optic approximation of Maxwell’s equations, from which it also follows that the wave-vector field (defined as the gradient of the phase) is lightlike and geodesic (see Sect. 1.8 in [Str04]). In the remaining chapters of Part I we derive some fundamental features which already follows from the above framework, thereby only assuming that we are given the minimal set ( , g). M

Chapter 2

Observers and reference frames

In this chapter we introduce the concept of ‘observer (field)’, which refers to a timelike future-pointing normalized vector (field). This give rise to a local decomposition (‘split’) of spacetime in space and time which is extended to the relevant geometric objects, like tensor fields (Section 2.1), as well as to derivations (Section 2.2). The latter enables the construction of derivations, like the Fermi and the comoving derivative introduced in this chapter as well as the observer derivative introduced in the next chapter, which have a nice physical interpretation and application. These constructions are all based on our main Lemma 2.5. In Section 2.3 we then introduce and characterize some particular observers and review their kinematic properties and evolution equations in terms of the physically relevant derivatives. In Section 2.4 we then conclude with a discussion on reference frames. At this point we remark that a spacetime decomposition in space and time is also possible by giving a so-called ‘time function’, that is a function on spacetime whose gradient is timelike and future-pointing. Clearly, if some time function t is given, one has automatically an observer field (by defining u := ∇t −1∇t) which is, per k k definition, hypersurface-orthogonal (and hence synchronizable, see Section 2.3.2). Conversely, if an observer u is given, one can introduce a time function which synchronize u, only if the latter is hypersurface-orthogonal. Hence, the concept of observer field is more general since it allows to consider also rotating ‘reference systems’. We recall that, as already anticipated in Chapter 1, in the following chapters of Part I we are assuming the framework exposed in Section 1.1 in its minimal version which consists in the pair ( , g) denotes a spacetime, that is a four-dimensional1 M differentiable manifold (which we will assume space- and time-orientable) together with a Lorentzian metric g. The Levi-Civita connection, ∇, is to be seen as being a structure which can be build up out of ( , g) via the Koszul formula. Recall M also that we adopt the ‘mostly minus’ signature convention where the restriction of g to spacelike directions is negative definite and that we use geometric units, in which c = 1.

1 Actually, the considerations that follow in the present and in the next chapter are independent of the dimension of spacetime.

17 18 Chapter 2. Observers and reference frames

2.1 Observers

A smooth curve γ : I R is called timelike, lightlike, or spacelike if its ⊂ → M tangent vector ﬁeld is, respectively, timelike, lightlike, or spacelike everywhere along the curve. A lightlike worldline is a smooth lightlike curve which we assume to be future-directed and a timelike worldline is a smooth timelike curve which, w.l.o.g., we assume to be future-directed and parametrized by its arc length.

Definition 2.1 (Observers). By an observer at p we understand a timelike ∈ M future-pointing normalized vector in T . An observer field is a timelike future- pM pointing normalized vector field on some open subset of . M Clearly, the tangential vector at p of a timelike worldline through p is an observer field at p. Moreover, the integral curves of an observer field are timelike worldlines. An observer u at p give rise to a natural decomposition of the tangent ∈M space T in ‘space’ and ‘time’. On T we define the temporal and the spatial pM pM projectors by

Qu := u u (2.1a) ⊗ Pu := id Qu , (2.1b) − which project onto the subspaces parallel and orthogonal to u, respectively. Here u := g(u, ) is the one-form corresponding to u via g and id is the identity map on · the tangential space. It is immediate to check that P are indeed complementary, orthogonal projectors, i.e. are endomorphism of T which satisfy the following pM properties:

Q, P are self-adjoint (w.r.t. g) endomorphism (2.2a) Q Q = Q and P P = P (idempotent) (2.2b) ◦ ◦ Q P = P Q = 0 (independent) (2.2c) ◦ ◦ Q + P = id (complementary) (2.2d)

We call any pair (Q, P ) satisfying (2.2) complementary, orthogonal projectors. The local rest space of u at p is then deﬁned as

u(p) := Pu(T ) . (2.3) R pM

The crucial point here is to realize the fact that vectors in u(p) represent the R directions to the nearby events to p that are Einstein-synchronized (by exchange of light-signals) w.r.t. the observer u at p (see Exercise 5.2.6 in [SW77]). This justifies the name of u(p) as the ‘local rest space’ of u. Analogously, we denote R with u(p) := Qu(T ) the local temporal subspace of u at p. An observer u at T pM p enables thus the orthogonal decomposition T = u(p) u(p) of the tangent pM T ⊕ R space in a temporal and a spatial part w.r.t. u. We call the collection ( u(p), u(p)) T R the local rest reference system of u at p. We note that u defines a distribution of R codimension one on which is (locally) integrable iff u is irrotational (this follows M from Lemma 2.17 below together with Frobenius-Theorem). The above projectors can be naturally extended to tensor fields of arbitrary rank in the following way. Given a projector P , one extends it to an operator Pˆ 2.1. Observers 19 defined on tensor fields requiring, for any function f, vector field X, one-form α, and arbitrary tensor fields S, T :

Pˆf = f , (2.4a)

PXˆ = PX , (2.4b)

Pˆ α = α P , (2.4c) ◦ Pˆ(S T ) = (PSˆ ) (PTˆ ) . (2.4d) ⊗ ⊗

In the ‘index language’ this corresponds to the projection of every free index. We call this extension of a projector, denoted by a hat, projective extension. (This is to be distinguished from the derivative extension defined in Appendix A.3 and denoted by a check.) As an example, for an endomorphism L one has PLˆ := P L P . ◦ ◦ Notice that the extensions Qˆ, Pˆ of a pair of complementary, orthogonal projectors (Q, P ) satisfy the idempotence and the independence properties on arbitrary tensor fields (with the only exception of functions for the latter) but, in general, do not satisfy the complementarity property (see the next paragraph). Be careful too, that Qˆ and Pˆ do not commute with contractions: for a vector field X and a one-form α, one has, because of (2.4a), that Pˆ(α(X)) = α(X), which is in general different from (Pˆ α)(PXˆ ) = α(P PX) = α(PX). A tensor field T is called spatial ◦ w.r.t. the observer u or, in short, u-spatial if PûT = T . Correspondingly, if it holds QûT = T , then T is called temporal w.r.t. u, or simply u-temporal. In the following, sometimes we will drop the hat since it should be clear from the contest what is meant.

As already anticipated, the identity Qu + Pu = id valid on vectors fields does not, in general, extend to arbitrary tensor fields—with the only exception of one- forms. One gets the spatial part and the temporal part of a tensor by applying the extensions Pû and, respectively, Qû to it. It is clear that adding together this two parts one does not get back the original tensor, since all the mixed parts are missing. On forms the situation is simpler due to the fact that Qu is a projection onto a one- dimensional subspace, hence Qûω vanishes for every p-form ω with p 2. Because ≥ of its antisymmetry, it is clear that a form ω is spatial iff iuω = 0. Moreover, on forms one has the following useful orthogonal decomposition identity

ω = Pûω + u iuω , (2.5) ∧ which decomposes a p-form ω in its spatial part Pûω — called magnetic part, in analogy with electrodynamics — and a mixed rest part which contains the purely spatial (p 1)-form iuω — called electric part of ω w.r.t. u. This decomposition is − easily proven defining an operator P˜u via P˜uω := ω u iuω and showing that this − ∧ is indeed the spatial projector Pû: i.e. that this is linear, it satisfies the idempotence property, and that its image is spatial (i.e. that iu(P˜uω) = 0). Linearity is obvious; the other two properties follow easily noticing that iu(u iuω) = iuω (since u is ∧ normalized and iX iX = 0 on forms). ◦ Now, given an observer u at p we split the spacetime metric g into two degen- 20 Chapter 2. Observers and reference frames erate, positive semi-definite metrics, called u-temporal and u-spatial metric,

tu := Qug = u u , (2.6) ⊗ hu:= Pug = u u g , (2.7) − ⊗ − which measure proper-time- and proper-space-intervals with respect to the observer u. Clearly, the restrictions of tu and hu on u and, respectively, u are positive T R definite metrics. In the same way we decompose the n-dimensional spacetime volume form µg induced by g into a volume 1-form u on u and a volume (n 1)-form T − ǫu := iuµg on u. Finally, we define the u-spatial Hodge-star ⋆u on forms on R u by the defining equation (A.18), where one takes , to be the scalar product R h· ·i induced by the u-spatial metric hu and ǫu as volume form.

Example 2.1 (Electric and magnetic field). In a spacetime , the electromagnetic M field-strength is described by a 2-form F . For an observer u at a point p , the ∈M electric and magnetic field Eu and Bu are nothing but the electric and magnetic part of F w.r.t. u, that is one has

F = Bu + u Eu , (2.8) ∧ where Bu := PûF and Eu := iuF . Here, the electric and magnetic fields are described by a 1- and a 2-form, respectively. In a four-dimensional spacetime, the familiar electric and magnetic vector fields are then, respectively, the u-spatial- metric-dual of Eu and minus the spatial-metric-dual of the spatial-Hodge-dual of Bu. Explicitely, in Minkowski spacetime and w.r.t. Minkowskian coordinates one has: E = F and B = B F (and cyclic permutations). i 0i 1 − 23 ≡− 23 The orthogonal decomposition w.r.t. an observer u applied to a skew-symmetric endomorphism E of the tangent space gives

E = B + R , (2.9) where R := PûE is the purely spatial part and B := E PûE the mixed part. − Clearly, both B and R are skew-symmetric. Hence, B is nothing than an infinitesimal boost in a plane containing u, R an infinitesimal rotation in the rest space of u, and (2.9) is nothing than the decomposition, w.r.t. u, of an infinitesimal isometry in a boost and a rotation. The former can be written as

B := a u u a = (a u) , (2.10) (a,u) ⊗ − ⊗ ◦ ∧ where a is some u-spatial vector. Then exp(B ) is the boost of rapidity a , (a,u) k k in positive a-direction, in the plane spanned by a and u. Similarly, for any two u-spatial vectors a and b, we put

R := a b b a = (a b) . (2.11) (a,b) ⊗ − ⊗ ◦ ∧

If a and b are linearly independent exp(R(a,b)) is the rotation in the plane spanned by a and b with the rotation angle h(a, b) in positive mathematical direction w.r.t. the orientation (a, b). We notice from (2.11) that

⊥ ⊥ R(a,b) = R(a b ,b) = R(a,b a ) , (2.12) 2.2. Derivatives 21

⊥ where a b denotes the part of a orthogonal to b in u. For later reference we R report some basic properties of boost. Let u be an observer, a, b two u-spatial vectors, and λ, µ some constants. Then

B(λa+µb,u) = λB(a,u) + µB(b,u) , (2.13a)

B B = hu(a, b)Qu a b , (2.13b) (a,u) ◦ (b,u) − ⊗ [B(b,u), B(a,u)]= R(a,b) . (2.13c)

The ﬁrst two properties follow by direct computation and the latter is a direct consequence of (2.13b). Equation (2.13c) express the well-known fact that the commutator of boosts in diﬀerent directions gives a rotation in the plane spanned by such directions.

2.2 Derivatives

Beside the covariant and the Lie derivative it is useful to introduce three other derivatives, called Fermi, comoving, and corotating derivative. These derivatives measure the rate of change of tensor fields (the former along a worldline and the latter two along an observer field) as measured in the local rest space and the local time subspace of the worldline and the observer field, respectively. The Fermi and the comoving derivative can be seen as generalizations of the covariant and, respectively, the Lie derivative since for geodesic worldlines and geodesic observer fields they reduce to the latter ones. The corotating derivative can be seen as the length- and angles-preserving ‘correction’ of the comoving derivative. A key property common to this three derivatives which we are going to define below, is that spatial (temporal) tensor fields remain spatial (temporal) under the transportation induced by them. In particular, this allows to construct adapted reference frames (see Section 2.4) along arbitrary (and not only geodesic) worldlines and observer fields. Moreover, as we shall see, these derivations have a direct physical interpretation. Whereas the Fermi derivative is well-known and the corotating derivative can also be found in the literature, the ‘comoving’ one seems not to have been considered so far. In this section we give the definitions of this derivatives, collect their properties, and compute their relations with each other and with the covariant and the Lie derivative on vector fields (equations (2.18), (2.29), (2.24), (2.27), (2.30), and (2.31)). Since all these derivatives coincide on functions, these formulae generalize straightforwardly for arbitrary tensor fields by just putting a check in top of the r.h.s.. This is explained in Appendix A.3 (see the discussion below equation (A.3)).

2.2.1 Fermi derivative The Fermi2 derivative is motivated by the physical problem of ﬁnding the equation of motion for the spin vector of a gyroscope (or some massive particle with spin) which moves along an arbitrary timelike worldline (see, e.g., Sect. 1.10 of [Str84] and Sect. 2.2 of [SW77]). The derivation of this equation of motion exposed in [Str84]

2 In the literature the name ‘Fermi–Walker’ is also in use for this derivative. In fact, Fermi defined the derivative in question in the year 1922 [Fer22] on vector fields and it was Walker [Wal32] who ten years later extended the definition on tensor fields of arbitrary rank (see [Hit06]). We adopt here the name ‘Fermi derivative’ just for brevity. 22 Chapter 2. Observers and reference frames makes only use of the equivalence principle and hence remains valid here: in Special Relativity, the spin is a spatial vector ~s in the local rest system of the gyroscope and there, in absence of external forces, it holds ~s˙ = 0. From the spacetime viewpoint ~s becomes a vector field S along γ which is orthogonal to γ˙ , if γ is the timelike worldline along which the gyroscope moves. (For the definition of vector fields along a map ˙ we refer to Appendix A.2.) The equation ~s = 0 becomes Pγ˙ (∇γ˙ S) = 0, which, using the constraint g(γ˙ , S) = 0, is equivalent to Qγ˙ (∇γ˙ (Qγ˙ S))+Pγ˙ (∇γ˙ (Pγ˙ S)) = 0. With the abbreviation (2.15) this reads:

Fγ˙ S = 0 (2.14) along γ. Note that, using the ﬁrst two properties of (2.19), from (2.14) it follows immediately that the constraint g(γ˙ , S) = 0 is preserved and g(S, S) is constant along γ. This motivates the following

Definition 2.2 (Fermi derivative). Given a timelike worldline γ and a vector field X along it, the Fermi derivative of X along γ is defined as

Fγ X := (Qγ ∇γ Qγ + Pγ ∇γ Pγ )X . (2.15) ˙ ˙ ◦ ˙ ◦ ˙ ˙ ◦ ˙ ◦ ˙

One defines Fγ˙ f := γ˙ (f) on functions and then extends this to a derivation of tensor fields along γ via derivative extension, hence requiring the Leibniz rule w.r.t. the tensor product and commutativity with contractions (see Appendix A.3). A tensor field along γ is called Fermi-transported along γ if its Fermi derivative vanishes along γ.

Clearly, a vector field X along γ is Fermi transported along γ iff the two conditions Pγ˙ (∇γ˙ (Pγ˙ X)) = 0 and Qγ˙ (∇γ˙ (Qγ˙ X)) = 0 hold along γ. This means that the spatial and temporal parts of X do not change (in the sense of parallel transportation), as seen from both the local rest space γ (γ(τ)) and the local temporal R ˙ subspace γ (γ(τ)) of γ˙ at γ(τ), respectively. The above physical motivation implies T ˙ that a Fermi transported vector field along a timelike worldline γ is one that does not change w.r.t. a system of gyroscopes carried along γ. At this point we want to make two remarks on the above definition. Remark 2.3. If one is mainly interested in spatial quantities, one could have dropped the first term involving the temporal projector since it would anyway not contribute. However, the so-defined derivative would be a derivation only on spatial tensor fields, the reason being the failing of the Leibniz rule (see (2.17)). From a somewhat philosophical point of view, the spacetime splitting in space and time introduced here, is just in order to relate spacetime objects to quantities we are familiar with in our everyday experience. Doing this, however, we do not want to forget that the true nature of the stage in which we are acting is that of a spacetime (at least this is our best understanding of it at the moment). Accordingly, we think of all the spatial (or temporal) quantities we introduce in this work as spacetime quantities with the additional property of being spatial (or temporal). Moreover, we want also be able to deal with quantities of arbitrary causal character (spatial, temporal, or mixed) and, for computational convenience, we do not want to give up nor the Leibniz rule nor commutativity with contractions for the derivatives we are going to introduce. 2.2. Derivatives 23

Remark 2.4. Another alternative definition of the Fermi derivative could have been as follows. One defines Fˆ γ := Qˆγ ∇γ Qˆ γ +Pˆγ ∇γ Pˆγ on arbitrary tensor fields ˙ ˙ ◦ ˙ ◦ ˙ ˙ ◦ ˙ ◦ ˙ with the only exception of functions, where one puts Fˆ γ˙ f := γ˙ (f). (Here, the hats on the projectors denote their extensions on arbitrary tensor fields as defined in (2.4) and ∇γ˙ the usual (derivative) extension of the covariant derivative.) Doing this, one first have to check whether or not Fˆ γ˙ satisfies the Leibniz rule and commutes with contractions. In fact, one can show that Fˆ γ˙ satisfies the Leibniz rule w.r.t. the tensor product, i.e. Fˆ γ (A B) = Fˆ γ A B + A Fˆ γ B, only if A and B are ˙ ⊗ ˙ ⊗ ⊗ ˙ both spatial (or both temporal). Restricting to these cases, one shows that Fˆ γ˙ also commutes with contractions. By its very definition, Fˆ γ˙ = Fγ˙ on functions (see (2.4a)) and on spatial (or temporal) vector fields. This implies, due to the uniqueness of derivations (Lemma A.2), that the equality Fˆ γ˙ = Fγ˙ extends also on spatial (or temporal) tensor field of arbitrary rank. In the light of what we had said here, we prefer the original Definition 2.2, since this is by definition a derivative on tensor fields of arbitrary rank and arbitrary causal character (spatial, temporal, and mixed), which commutes with contractions.

To be precise, in this definition we tacitly use the fact that Fγ˙ , as defined by (2.15), satisfies the Leibniz rule on vector fields: since we will again encounter the same combination of a derivation with a pair of complementary, orthogonal projectors as in (2.15), it is a good idea to prove the following more general

Lemma 2.5 (Orthogonal decomposition of derivations). Let D be a derivation of tensor fields on (or of tensor fields along a map φ : from some M N →M manifold to ) and the pair (Q, P ) be a smooth assignment of complementary, N M orthogonal projectors3 on (respectively, along the map φ). Define D′ by D′ := D M on functions in C∞( ) (respectively in C∞( )) and M N D′ := Q D Q + P D P (2.16) ◦ ◦ ◦ ◦ on vector fields (respectively on vector fields along φ). Then D′ is R-linear and satisfies the Leibniz rule on functions and on vector fields and thus it can be uniquely extended to a derivation of tensor fields (respectively of tensor fields along φ) via derivative extension: requiring the Leibniz rule w.r.t. tensor product and commutativity with contractions. The so constructed derivation D′ has the following properties: (i) D′ commutes with the extensions of the projections Q and P on arbitrary tensor fields. (ii) If D is metric (i.e. Dg = 0), then also D′ is metric and the difference E := D D′ on vector fields is a skew-adjoint endomorphism. − (iii) If D is metric and, in addition, one of the two projectors projects onto a one-dimensional non-null subspace, then the unit vector field which spans this subspace (unique up to a sign and denoted by, say, u) is constant w.r.t. D′, that is it holds D′u =0. We notice here that the ‘decomposition’ map ′ : D D′ is idempotent, that is 7→ (D′)′ = D′, as one may easily check, and as one expects to be the case.

3 That is a pair satisfying the properties (2.2). 24 Chapter 2. Observers and reference frames

Proof of Lemma 2.5. We assume in the following that D is a derivation of the tensor fields on . The proof for the case that D is a derivation of tensor fields along a M map is carried out exactly in the same manner. First, R-linearity of D′ on functions and on vector fields follows directly from the R-linearity of D and of the projectors Q, P . The Leibniz rule on functions follows from D′f = Df and the Leibniz rule of D. From the Leibniz rule on vector fields for D, the idempotency of Q and P (see (2.2b)), and that D′f = Df, we have

D′(fX)= fD′X + D′(f)(Q + P )X . (2.17)

Hence, the Leibniz rule on vector field is satisfied because of the complementary property Q + P = id. (For this reason, as already stressed in Remark 2.3, the complementary property is crucial: if one would drop one of the two summand, let say the first, on the r.h.s. of the definition (2.16), then D′ would satisfy the Leibniz rule only on those tensor fields T with PTˆ T .) The uniqueness of the derivative ≡ extension of D′ on arbitrary tensor fields is then ensured by Lemma A.2. It remains to show the properties (i)–(iii). Property (i) is clearly valid on vector fields and on functions. (Recall that on functions the projectors acts trivially by definition—see (2.4a).) Moreover, it is also valid on one-forms: Let α be a one-form, X a vector field, and P any of the two projectors Q, P . Then one has: (D′(Pˆ α))(X) = D′((Pˆ α)(X)) (Pˆ α)(D′X) = − D′(α(PX)) α(P D′X) = D′(α(PX)) α(D′(PX)) = ((D′α)(PX)) = − − (Pˆ(D′α))(X), where we used, in the order, the definition of D′ on one-forms, that of Pˆ, the fact that D′ commutes with P on vector fields, again the definition of D′ on one-forms, and, finally, that of Pˆ. The statement on arbitrary tensor fields follows then by induction: what we just showed is the basis case. For the inductive step, let A, B any tensor fields along γ on which D′ commutes with Pˆ. Then D′ commutes with Pˆ also on A B since D′(Pˆ(A B)) = D′(Pˆ A PBˆ )= ⊗ ⊗ ⊗ D′(Pˆ A) PBˆ +Pˆ A D′(PBˆ )= Pˆ(D′A) PBˆ +Pˆ A Pˆ(D′B)= Pˆ(D′(A B)). ⊗ ⊗ ⊗ ⊗ ⊗ The first part of the statement (ii) follows by a straightforward computation: starting with (D′g)(X, Y ) = D′(g(X, Y )) g(D′X, Y ) g(X, D′Y ) one shows − − that the r.h.s. identically vanishes by expressing there D′ in terms of D, using the metricity of D, and that Q, P are complementary, orthogonal projectors. That the difference E := D D′ is an endomorphism on vector field is true for any − two derivations which agrees on functions since (D D′)(fX) = f(D D′)X + − − (D(f) D′(f))X. Moreover, if both derivations are metric and they agree on − functions, we have: g(D′X, Y )+ g(X, D′Y )= g(DX, Y )+ g(X, DY ) and hence g((D D′)X, Y )+ g(X, (D D′)Y )=0. − − To show (iii) let u be as described above in the lemma and, w.l.o.g., let Q be the projector onto the non-null one-dimensional subspace (spanned by u). One has: Qu = u and Pu =0. By (i) it follows that P D′u = 0, meaning that the part of D′u orthogonal to u vanishes. But since u is normalized and, by (ii), D′ is metric, we have g(u, D′u) = 0, which means that the part of D′u parallel to u vanishes too.

Next proposition collects the properties of the Fermi derivative: 2.2. Derivatives 25

Proposition 2.6 (Properties of the Fermi derivative). Let γ be a timelike worldline, then the following holds: (i) On vector ﬁelds along γ one has

∇γ Fγ = B ∇ , (2.18) ˙ − ˙ ( γ˙ γ˙ ,γ˙ )

where B(·,·) denotes the inﬁnitesimal boost deﬁned by (2.10). In particular, if γ is geodesic, then the Fermi derivative reduces to the covariant derivative. (ii) The following transport equations hold along γ:

Fγ˙ γ˙ =0 , (2.19a)

Fγ˙ g =0 , (2.19b)

Fγ˙ tγ˙ =0 , (2.19c)

Fγ˙ hγ˙ =0 . (2.19d)

(iii) Fγ˙ commutes with the extension of the projections Qγ˙ and Pγ˙ on arbitrary tensor ﬁelds along γ.

(iv) Fγ˙ maps spatial tensor fields to spatial tensor fields and temporal tensor fields to temporal tensor fields.

Proof. The statements (ii)–(iv) follow immediately applying Lemma 2.5 to the

(metric) derivative ∇γ˙ . The ﬁrst property, (2.18), follows with a short computation after insertion of the projectors’ deﬁnitions (2.1) in (2.15).

From (2.19) it follows that Fermi-transport preserves scalar products measured with the spacetime metric g, as well as with the projected metrics tγ˙ and hγ˙ . Notice that both sides of (2.18) are endomorphisms of the tangent space, hence function- linear, although the single summands on the l.h.s. are not (see also Lemma A.4 in Appendix A.3). The meaning of this equation is that the diﬀerence of Fermi transport w.r.t. parallel transport is just the inﬁnitesimal boost B(a(γ˙ ),γ˙ ). This is exactly what is necessary in order to keep a Fermi transported vector which is initially orthogonal to γ˙ , always orthogonal to γ˙ , as it must be, for example, for a gyroscope axis. In this sense, a less prosaic name for the Fermi transport would be ‘boosted (parallel) transport’. We conclude this section with the following nice physical application:

Example 2.2 (Motion of the spin of a particle in an electromagnetic field). We are interested in the equation of motion for the spin of a massive particle moving arbitrarily in an external electromagnetic field. Starting point is the classical, New- tonian equation of motion ~s˙ = (ge/2m)~s B~ , which holds in the local rest system × of the particle. Denoting with γ the particle’s worldline (not further specified here), the spin vector is as above described by a γ˙ -spatial vector field S along γ. If F is the 2-form describing the electromagnetic field then Bγ˙ := Pˆγ˙ F is the magnetic

2-form (see Example 2.1 at page 20). Since iSBγ reduces in the local rest system − ˙ to ~s B~ and in absence of electromagnetic ﬁeld the spin equation of motion is (2.14), × the generalization of the Newtonian equation must be

ge Fγ S = iSBγ . (2.20) ˙ − 2m ˙ 26 Chapter 2. Observers and reference frames

In fact, writing the Fermi-derivative in term of the covariant derivative (see (2.18)) and writing Bγ˙ in terms of F and γ˙ one arrives at the familiar form

ge ∇γ S + g(∇γ γ˙ , S)γ˙ = iSF F (S, γ˙ )γ˙ , (2.21) ˙ ˙ − 2m − cf. equation (6) in [BMT59]. Note that, as for the gyroscope, from the spin equation of motion (2.20) it follows immediately that the constraint g(γ˙ , S) = 0 is preserved and g(S, S) is constant along γ.

2.2.2 Comoving and corotating derivative Let now consider the situation, where a smooth observer field v is given on some open region of the spacetime. We may think of it as the velocity vector field of some fluid. As the Fermi derivative was motivated by the problem of finding the transport equation for the spin vector of a gyroscope, the comoving derivative finds its motivation in the transport equation for the spatial separation vector, s say, between neighboring integral curves of v, hence, thinking at the fluid, the spatial separation of neighboring fluid particles. Perhaps, one would intuitively answer that, of course, s is Lie-transported w.r.t. v (that is Lvs = 0). However, how we will now see, this is true only if v is geodesic. We first have to give a mathematical expression for the spatial separation vector. In order to do this, we pick up an integral curve, sayγ ˜, of v which is our reference worldline. To describe the nearby worldlines we define a bijective map γ : I B γ(I B) : (τ, σ) γ(τ, σ), × → × ⊂M 7→ where I is some open ‘time interval’ in R and B an open ball around 0 in Rn−1, such that, for every fixed σ B, γ ( ) := γ(τ, ) is an integral curve of v and γ =γ ˜. ∈ σ · · 0 Hence, σ is just a ‘label’ for the nearby worldlines or, thinking at the fluid, for the nearby fluid particles. At the time τ (which, by the way, is the proper time of γ0), the infinitesimal separation vector between the reference particle 0 and the nearby particle σ (fixed!), that is the separation vector between the points γ(τ, 0) i i and γ(τ, σ), is q := σ γ∗∂/∂σ . The spatial separation vector is then the projection of q onto the rest space of v at γ0(τ), that is s := Pvq. (Note that both s and q are defined along our reference worldline γ0.) Now, since by construction v = γ∗∂/∂τ, i i it follows that q is Lie transported along v: Lvq = [v, q]= σ γ∗[∂/∂τ, ∂/∂σ ]=0. Hence, for s q g(q, v)v we have: Lvs = v(g(q, v))v. For the coefficient of v on ≡ − − the r.h.s. we have v(g(q, v)) = g(∇vq, v)+g(q, ∇vv). Using the torsion-freeness of

∇ and [v, q] = 0 we have ∇vq = ∇qv and thus the ﬁrst term becomes g(∇qv, v), which vanishes since v is normalized. The normalization of v also implies that its acceleration a := ∇vv is orthogonal to v and thus we can replace q with s in the second term of the above expression. All together, we have: Lvs = g(s, a)v. − Since s is by construction always orthogonal to v the last equation is equivalent to Qv(Lv(Qvs)) + Pv(Lv(Pvs)) = 0 or, with the abbreviation (2.23), to

Cvs =0 . (2.22)

Note that any vector ﬁeld s which is v-spatial on some ‘initial’ hypersurface of codimension one transversal4 to v and is transported via (2.22) it remains v-spatial for all times. This is ensured by Proposition 2.8: Cv(g(v, s)) = (Cvg)(v, s)+0+0 = 2θ(v, s) = 0, since θ is spatial. All this motivates the following

4 An orthogonal hypersurface would not exist if v is allowed to rotate (see Lemma 2.17). 2.2. Derivatives 27

Definition 2.7 (Comoving derivative). Given an observer field v and a vector field X, the comoving derivative of X along v is defined as

CvX := (Qv Lv Qv + Pv Lv Pv)X . (2.23) ◦ ◦ ◦ ◦

As for the Fermi derivative, one extends Cv to a derivation of tensor ﬁelds in the usual way: one puts Cvf := v(f) on functions and requires the Leibniz rule w.r.t. tensor product and commutativity with contractions. We call a tensor ﬁeld comovingly transported along v (or comoving with v or simply v-comoving) if its comoving derivative along v vanishes.

We see from (2.23), that a vector ﬁeld X is comoving with v iﬀ both

Pv(Lv(PvX)) = 0 and Qv(Lv(QvX)) = 0, that is iff its variation measured with the Lie derivative in both the local temporal subspace and in the local rest space of v vanish. According to the above motivation and thinking of v as the velocity vector field of some fluid, a v-comoving vector field is one which does not change as seen by the fluid particles. The comoving derivative is defined very much like the Fermi derivative, where the covariant derivative is replaced by the Lie one: so to say, the comoving derivative stands to the Lie derivative as the Fermi derivative stands to the covariant one. Notice that the Fermi derivative, defined in the first place as a derivation along a timelike worldline (γ, say), generalizes obviously to a derivation along an observer field (v, say), defined on some open spacetime region (just substitute γ˙ by v everywhere in Definition 2.2). On the contrary, since the Lie derivative is defined only along vector fields, it clearly does not make sense to define the comoving derivative along a single worldline. (We recall that, in order to compute the Lie derivative

LX T of some tensor T w.r.t. a vector ﬁeld X at a point p one needs to know X (and, of course, T ) in an open neighborhood of p. On the contrary, in order to compute the covariant derivative ∇X T , one needs to know X only at p. This follows from the fact that the map X ∇X T is function linear whereas the map 7→ X LX T is not.) 7→ We now state the basic properties of the comoving derivative:

Proposition 2.8 (Properties of the comoving derivative). Let v be an observer ﬁeld. Then the following holds: (i) On vector ﬁelds one has

Cv Lv = v a , (2.24) − ⊗ where a := ∇vv is the acceleration of v. In particular, if v is geodesic, then the comoving derivative reduces to the Lie derivative. (ii) The following transport equations hold:

Cvv =0 , Cvv =0 , (2.25a)

Cvg =2 θ , (2.25b)

Cvtv =0 , (2.25c)

Cvhv = 2 θ = +2 θ . (2.25d) − hv

Here, θ is the shear-expansion tensor of v (see Section 2.3) and θ and θhv its covariant versions deﬁned w.r.t. the metrics g and hv, respectively. 28 Chapter 2. Observers and reference frames

(iii) The comoving derivative Cv commutes with the extension of the projections Pv and Qv on arbitrary tensor ﬁelds. This is to be compared with Proposition 2.6, which collects the properties of the Fermi derivative. From the transport equations (2.25b) or (2.25d) it is clear that, in contrast to the Fermi transportation, the comoving transportation fails to be an isomorphism due to shear and expansion. In order to compensate this we deﬁne below in this section the ‘corotating transport’, but before we prove the properties of the comoving derivative:

Proof of Proposition 2.8. For the point (i) we ﬁrst notice that on vector ﬁelds it holds:

[Lv, Qv]= v a (2.26a) ⊗ [Lv, Pv]= v a . (2.26b) − ⊗

Here [Lv, Qv] := Lv Qv Qv Lv. Since Pv = id Qv, equation (2.26b) ◦ − ◦ − follows directly from (2.26a). In turn, the latter equation follows from Lv(QvX)=

Lv(g(X, v)v)= v(g(X, v))v = g(∇vX, v)v + g(X, a)v = g(∇X v + LvX, v)v + g(X, a)v = Qv(LvX)+v a(X). In the last step we used that v is normalized and ⊗ in the step before the torsion-freeness of ∇. Now, using (2.26) in the deﬁnition (2.23) together with the projection properties, one gets immediately (2.24). The last point, (iii), follows directly from Lemma 2.5 and (ii) follows from (i) taking into account its extension to one-forms and (0,2)-tensor ﬁelds (equations (A.5) and (A.7)). Explicitly, for v we have g(v, a) = 0 (since it is normalized) and with (2.24) it follows immediately Cvv = Lvv = 0. Moreover, for the one-form v, with (2.24) and (A.5) we have: Cvv = Lvv v (v a)= a a = 0, where we used − ◦ ⊗ − the acceleration formula a = Lvv (see (2.36)). Similarly, for the comoving derivative of the metric one has from (2.24) and (A.7): Cvg = Lvg v a =2 θ (see (2.39)). − ∨ Finally, (2.25c) follows from (2.25a) and (2.25d) from (2.25b) and (2.25c).

It is now very instructive to ﬁnd out the relation between the comoving and the Fermi derivative. First, recall that the relation between the Lie and the covariant derivative is given by the torsion-freeness of ∇:

∇v Lv = Pˆv∇v + a v . (2.27) − ⊗ Here, on the r.h.s. we decomposed the endomorphism ∇v in orthogonal parts with respect to v:

∇v = Pˆv∇v + a v , (2.28) ⊗

The spatial endomorphism Pˆv∇v = Pv ∇v Pv will be interpreted below as the ◦ ◦ endomorphism of the (infinitesimal) spatial motion of v in its local rest space. To see (2.28) just note first that, because of the normalization of v, the image of ∇v is purely spatial. Thus: ∇v = Pv ∇v = Pv ∇v Pv +Pv ∇v Qv = Pˆv∇v+a v. ◦ ◦ ◦ ◦ ◦ ⊗ Now, subtracting (2.18) and (2.24) from (2.27), one gets immediately Proposition 2.9. The difference between the Fermi and the comoving derivative with respect to an observer field v is the endomorphism of the infinitesimal spatial motion of v, hence it holds

Fv Cv = Pˆv∇v (2.29) − 2.3. Observer characterization 29 on vector ﬁelds.

The above relation is an important point of this section: it builds the bridge between Fermi- and comoving transport, and the infinitesimal spatial motion of v. Thinking again at v being the velocity vector field of some fluid, and since we showed at the beginning of this subsection that the spatial separation vector s between two nearby fluid particles is comovingly transported, equation (2.29) implies that the motion of s w.r.t. gyroscopic axes is given by the spatial endomorphism Pˆv∇v. This shows that Pˆv∇v is indeed the endomorphism of the infinitesimal spatial motion of v in the rest space. For completeness, we give the last two of the six relations between the four derivatives Fermi, comoving, covariant, an Lie. The relation between the comoving and the covariant derivative follows immediately by adding (2.18) and (2.29):

∇v Cv = Pˆv∇v + B , (2.30) − (a,v) and, ﬁnally, the relation between the Fermi and the Lie derivative,

Fv Lv = Pˆv∇v + v a , (2.31) − ⊗ as follows by taking the diﬀerence between (2.27) and (2.18). Concluding, we deﬁne the corotating derivative5 by putting

Rv := Fv ω (2.32) − on vector fields, Rvf := v(f) on functions, and extending derivatively. Note that in view of (2.29) and of the decomposition (2.33) we have Fv ω = Cv + θ on vector − fields. Hence, corotating transportation corresponds to Fermi transportation plus rotation or, equivalently, to comoving transportation without shear and expansion. Since Rv differs from the Fermi derivative by a spatial skew-adjoint endomorphism, it follows at once that the properties (ii)–(iv) of Proposition 2.6 apply also to the corotating derivative. In particular, the corotating transportation preserves angles and lengths and hence it can be used to define a corotating proper reference frame (see Section 2.4).

2.3 Observer characterization

In this section we want to introduce some particular observers which are distinguished by their kinematical properties or can be constructed by means of spacetime symmetries. In fact, as we will see, all type of observer ﬁelds we introduce below, can be characterized by means of conditions on the acceleration and on the endomorphism of the spatial motions. We start therefore with the decomposition of the latter endomorphism in rotation, shear, and expansion—as is familiar from ﬂuid dynamic—and give then the evolution equations for these kinematical quantities in terms of both the Fermi and the comoving derivative.

Definition 2.10 (Motion’s decomposition). Let u be an observer field defined on some open region of the spacetime. Then ∇u is an endomorphism of T . As we M saw above, its spatial projection Pû∇u := Pu ∇u Pu gives the infinitesimal ◦ ◦ 5 See also equation (2.16) of [JCB92]. 30 Chapter 2. Observers and reference frames

motion of nearby worldlines. We decompose Pu∇u in its self-adjoint part, the shear-expansion tensor, denoted by θ, and in its skew-adjoint part, the rotation tensor, denoted by ω:

Pˆu∇u = θ + ω . (2.33)

The shear-expansion tensor θ can be further decomposed in its traceless part, the shear tensor, denoted by σ, and its trace, the expansion, denoted by θ. If σ, θ, or ω vanishes then we call the observer ﬁeld (as well as the motion related with it) shearless, non-expanding, or irrotational, respectively.

Note that, in the above definition, it does not make any difference if we take the self- and skew-adjoint part w.r.t. the spacetime metric g or to the spatial metric hu (see Notation and conventions). For the same reason, the decomposition (2.33) is not affected by the signature choice. Sometimes, for computational convenience, it is useful to see the shear-expansion and rotation tensors as (0, 2)-tensors instead of endomorphisms. Hence, according to (A.43), we define, for any vector fields X and Y (not necessarily orthogonal to u):

θ(X, Y ) := g(θ(X), Y ) , (2.34) ω(X, Y ) := g(ω(X), Y ) . (2.35)

Since θ and ω are self- and, respectively, skew-adjoint w.r.t. g, it follows that θ and ω are symmetric and antisymmetric bilinear forms, respectively. In terms of ˆ ∇ 1 ˆ ∇ ˆ ∇ Pu u one has explicitly: θ(X, Y )= 2 g((Pu u)(X), Y )+ g((Pu u)(Y ), X) 1 ˆ ˆ and ω(X, Y )= g((Pu∇u)(X), Y ) g((Pu∇u)(Y ), X) , respectively. 2 − 2.3.1 Acceleration, rotation, shear, and expansion

If an observer ﬁeld u is given on some open region of the spacetime and one is asked to compute its acceleration, rotation, shear, and expansion, it is generally with the language of forms that one achieves this goal in the fastest and more elegant way—without the need of computing any Christoﬀel symbols. For the acceleration one-form the we have:

a := ∇uu = ∇uu = iu∇u = iuLug = iudu = Luu . (2.36)

The most rapid way to compute the acceleration is perhaps via the formula a = iudu. In the ﬁrst step of (2.36) we used the metricity of ∇, whereas in the last one we used Cartan’s formula and that u is normalized. For the rest we just note that, since u is normalized, we have: (∇u)( , u)= g(∇ u, u) = 0. Hence one has: · · iu∇u = iu2 (∇u) = iuLug (see (A.45)) as well as: iu∇u = iu2 (∇u) = iudu S A (see (A.46)). For the shear-expansion and rotation tensors the formulae are:

1 θ = Pu (∇u) = Pu(Lug) , (2.37) S 2 1 ω = Pu (∇u)= Pudu . (2.38) A 2 These are a direct consequence of the deﬁnitions (2.34) and (2.35) together with the relations (A.45) and (A.46). Writing out the spatial projection in the above 2.3. Observer characterization 31 formulae, one gets:

1 θ = (Lug u a) , (2.39) 2 − ∨ ω = 1 (du u a) . (2.40) 2 − ∧ The formula for the rotation follows immediately with the decomposition (2.5) and (2.36). In the expression for the shear-expansion tensor one has, using the symmetry of g, that Pu(Lug)= Lug(Pu , Pu )= Lug Lug( , Qu ) Lug(Qu , )+ · · − · · − · · Lug(Qu , Qu )= Lug u iuLug + (iuiuLug)u u. Then, with (2.36) it follows · · − ∨ ⊗ directly (2.39).

Evolution equations for rotation, shear, expansion, and spatial separation

If one wants to set up evolution equations for rotation, shear, and expansion of an observer field, one has the choice to compute the variation of the above quantities with any of the derivatives we introduced in the last section. Among them, we chose here the Fermi and the comoving derivative since, how we have seen, they have a direct physical interpretation. These evolution equations allow to find out the effect of gravity on motion and, for some particular motions (for example in case of stationary motions and of ideal fluids), to derive conserved quantities. We sketch here the (perhaps shortest) path one may follow, labeling the steps from (i) to (vi): (i) compute the covariant derivative along u of the endomorphism ∇u. Since the Levi-Civita satisfies the Leibniz rule and commutes with contraction one has: (∇u(∇u))(X)= ∇u(∇u(X)) ∇u(∇uX). Then, in the first summand, − which equals ∇u(∇X u), one exchanges the places of ∇u and ∇X and gets a curvature term. Using the torsion-freeness of ∇ one finally gets:

∇u(∇u)= ∇u ∇u + Au + ∇a , (2.41) − ◦ where

Au := R(u, )u (2.42) · is the tidal operator associated with the observer u. This is a u-spatial, self-adjoint endomorphism whose trace is given by trAu = Ric(u, u). (ii) With (2.18), − together with (A.6), express on the l.h.s. of (2.41) ∇u in terms of Fu and find a formula for Fu(∇u). (iii) On the r.h.s. (but not on the l.h.s.) of the equation derived in the last point decompose ∇u as in (2.28). (iv) Use the fact that Fu commutes with the projections Pu, Qu, and with their extensions on arbitrary tensor fields, to compute Fu(Pû∇u) from the point (iii). One gets:

Fu(Pû∇u)= (Pû∇u) (Pû∇u)+ Au + Pû∇a a a (2.43) − ◦ − ⊗ (v) Use the fact that the Fermi derivative commutes with the operations of taking the self- and the skew-adjoint part, and with taking the trace, to compute Fuω, Fuθ, and Fuθ. For the rotation and shear-expansion tensors one gets, respectively:

Fuω = θ, ω + g(Pˆu∇a) (2.44) −{ } A Fuθ = θ θ ω ω + Au a a + g(Pˆu∇a) (2.45) − ◦ − ◦ − ⊗ S 32 Chapter 2. Observers and reference frames and for the expansion scalar one gets the famous Raychaudhuri equation:

2 2 Fuθ = θ + ω Ric(u, u)+ div a . (2.46) −k k k k −

Here, g(L) and g(L) denote the skew- and, respectively, the self-adjoint part of A S the endomorphism L w.r.t. the metric g. In passing, we notice that every term in (2.44) is skew-adjoint, as well as every term in (2.45) is self-adjoint. (vi) Finally, use (2.29), together with (A.6), to compute the comoving version of the above equations. One gets:

Cuω = 2 θ ω + g(Pû∇a) (2.47) − ◦ A Cuθ = θ θ ω ω [ω, θ]+ Au a a + g(Pû∇a) (2.48) − ◦ − ◦ − − ⊗ S Notice, that the first term on the r.h.s. of (2.47) is neither skew- nor self-adjoint: this should not surprise, since the comoving derivative does not, in general, commute with the operations g, g. These, in fact, depend explicitely on the metric and A S for the comoving derivative it holds Cug =2θ, which is in general not zero. We conclude with the evolution equation for spatial separation vectors, which directly follows from (2.43). Recall that a spatial separation vector, say s, of the observer field u is an u-spatial vector field which propagate via (2.22). In terms of the Fermi derivative, by (2.29), this propagation equation reads

Fus = (Pˆu∇u)s . (2.49)

Now, the relative acceleration of the two nearby integral curves of u with spatial sep- 2 ˆ ∇ ˆ ∇ aration vector s is given by Fus = Fu(Fus) = (Fu(Pu u))s + (Pu u)(Fus)=

(Fu(Pû∇u))s + (Pû∇u) (Pû∇u)s, where we used twice the propagation equa- ◦ tion (2.49). With (2.43) it follows immediately

2 F s = (Au + Pˆu∇a a a)s (2.50) u − ⊗ which, for the special case that u is geodesic, reduces to the so-called geodesic deviation equation ∇2 us = Au(s) . (2.51)

This justiﬁes the name ‘tidal operator’ for the spatial endomorphism Au deﬁned in (2.42).

Characterizations of observers We are now in the position to define some particular observers which are distinguished by their kinematical properties or can be constructed by means of spacetime symmetries. Notice that sometimes, instead of dealing with observer fields, one prefers to deal with the flow generated by them, often called ‘motion’. Since the relation between vector field and its flow is one-to-one, we are free to switch between these two pictures. Accordingly, the terminology introduced in the following for observer fields apply also for motions.

Deﬁnition 2.11 (Geodesic, uniformly accelerated, stationary, and static observers). Let u be an observer ﬁeld and a := ∇uu its acceleration. We say that 2.3. Observer characterization 33

u is geodesic if a = 0 and uniformly-accelerated if Pu(∇ua) = 0. If the spacetime is stationary, hence if there exists a (non-vanishing) timelike Killing field K, we call u stationary if u = K/ K . If, in addition, the Killing field is hypersurface- k k orthogonal (that is the spacetime is static), we call such an observer field static. We extend all this terminology to worldlines putting u = γ˙ and understanding the above conditions as conditions along γ.

Perhaps the definition of uniformly accelerated observer needs some words of comment. The condition Pu(∇ua) = 0 just says that the variation of the observer’s acceleration, as measured in its rest space, vanishes. This is indeed the only meaningful way to define uniformly accelerated observers. An immediate consequence of uniformly acceleration is, in particular, that the norm of the acceleration a remains constant along the worldline, since we have ∇u(g(a, a)) = k k 2 g(∇ua, a)=2 g(Pu∇ua, a) (last identity holds since Pua = a). With (2.15) we have Fua = Pu∇ua. This gives an alternative characterization of uniformly accelerated observers: Proposition 2.12 (Uniformly accelerated observers). An observer field (or a timelike worldline) is uniformly accelerated iff its acceleration vector is Fermi- transported.

From the physical point of view this is very satisfying, since it means that an observer field (or a timelike worldline) is uniformly accelerated iff its acceleration vector does not change w.r.t. a system of gyroscopes carried along with it. Two important examples of uniformly accelerated observers are the so called ‘hyperbolic motion’ in Minkowski spacetime (given by the boost-orbits) and a static observer in any static spacetime (see Proposition 2.19 below). To show the latter, we need to recall some basic facts about Killing fields. First of all, recall that a vector field K is a Killing field in some open subset of if it is an infinitesimal U M isometry, hence if LK g = 0 in . This condition is equivalent with the requirement U that ∇K is skew-adjoint w.r.t. g:

g(∇X K, Y )+ g(∇Y K, X) = 0 (2.52) for any vector ﬁelds X, Y . The latter is often referred to as the Killing equation and it is an immediate consequence of the identity (A.45) of Appendix A.6. Notice that the l.h.s. of (2.52) is just 2 (∇K)(X, Y ), the symmetric part of ∇K (see S Appendix A.6). The next lemma collects some useful properties related to Killing ﬁelds:

Lemma 2.13. Let K be a Killing ﬁeld deﬁned on some open subset of . Then U M on it holds: (iU) K( K 2)=0 , k k (ii) ∇KK = (1/2) ∇(g(K, K)) , and − (iii) div K =0 .

Proof. For the ﬁrst property notice that K(g(K, K)) = 2g(∇KK, K), and the latter vanishes because of the Killing equation. For (ii), using the Killing equation, we have g(∇K K, Y ) = g(∇Y K, K) = (1/2)Y (g(K, K)) = − − (1/2) d(g(K, K))(Y ). The last property is obvious, since the divergence of K, − being the trace of a skew-adjoint endomorphism (∇K), vanishes identically. 34 Chapter 2. Observers and reference frames

Now we can derive the basic properties of stationary and static observers. An immediate consequence of the previous lemma is Proposition 2.14 (Stationary observers, I). The acceleration of a stationary observer is given by a = ∇ ln K , (2.53) − k k where K is the timelike Killing field related to u, hence K = K u. In particular, k k the acceleration one-form of a stationary observer field is exact. This acceleration formula is very useful since it allows one to compute quickly the acceleration without need to compute any Christoffel-symbol. More generally, we have the following uniquely characterization of stationary observers: Proposition 2.15 (Stationary observers, II). An observer is stationary iff it shear- expansion tensor vanishes and its acceleration is exact. Proof. An observer is stationary iff is collinear to a timelike Killing field (the latter w.l.o.g. future-pointing) hence iff it exists a positive smooth function f with Lfug = 0. In view of (2.39) and (A.37) this is equivalent to

2 θ + (d ln f + a) u = 0 (2.54) ∨ which, in turn, is equivalent to θ = 0 and a = d ln f. The last equivalence is − because the two summands in (2.54) are orthogonal.

In particular, the above proposition says that a stationary observer is shearless and non-expanding, hence, the only spatial motions a stationary observer is allowed to do, are rotations. Now, a static observer is a special case of a stationary one and can be characterized as follows: Proposition 2.16 (Static observers, I). A stationary observer is static iff irrotational. Proof. This follows directly from the following two facts: First, as we will prove below (Lemma 2.17), an observer field is hypersurface orthogonal iff irrotational. Secondly, since (fdu) d(fu) = f 2u du for any function f, hypersurface or- ∧ ∧ thogonality is a property which is not altered by a rescaling of the vector field. Summarizing: A stationary observer is static iff the related Killing field K is hypersurface orthogonal. This is the case iff u = K −1K is hypersurface orthogonal, k k which, in turns, is the case iff u is irrotational.

Hence, recalling Proposition 2.15, a static observer is shearless, non-expanding, and non-rotating. It remains to prove Lemma 2.17. An observer ﬁeld is hypersurface orthogonal iﬀ irrotational. Proof. We have to show that u du = 0 is equivalent to ω = 0, where ω is ∧ the rotation of u. From the orthogonal decomposition (2.5) we have that du =

Pudu + u iudu =2 ω + u iudu (see (2.38)), and hence ∧ ∧ u du =2 u ω , (2.55) ∧ ∧ from which the statement of the lemma follows immediately. (One direction is obvious, the other follows applying iu to the above equation.) 2.3. Observer characterization 35

As a concrete example to Proposition 2.16, one may consider an observer in a Schwarzschild spacetime ((6.33) with Λ = 0) which follows an integral curve of the Killing ﬁeld K := ∂/∂t+Ω∂/∂ϕ (Ω is a constant) in the ‘equatorial’ plane θ = π/2. Its acceleration is easily computed with (2.53) and its rotation with (2.38)—all without need to compute any Christoﬀel-symbol. We now show that the acceleration and the rotation of a stationary observers are comoving.

Proposition 2.18 (Stationary observers, III). The acceleration vector a and the rotation tensor ω of a stationary observer ﬁeld u are u-comoving (in the sense of Deﬁnition 2.7), that is:

Cua =0 , (2.56a)

Cuω =0 . (2.56b)

Proof. The second equation is a direct consequence of the propagation equation (2.47), together with the fact that a stationary observer is shearless and non- expanding (θ = 0) and its acceleration one-form is exact (see Proposition 2.15). For the acceleration (one-form) of an arbitrary observer one has, just using Cartan’s formula, that

Lua = iuda . (2.57) Since for a stationary observer a is exact, we have that a is Lie transported along u. From (2.24), together with (A.5), one gets immediately Cua = 0, hence that a is comoving. That this is also true for the acceleration vector a is clear because

Cug = 0 for a stationary observer (see (2.25) and Proposition 2.15), and hence Cu commutes with the operations and .

First notice that at this point one can appreciate the usefulness of the comoving derivative: expressing (2.56b) in terms of the Lie derivative one would get, by (2.24) and (A.6), Luω + (u a) ω = 0, which looks indeed not as nice as (2.56b). ⊗ ◦ The Fermi versions of the above evolution equations (2.56) are immediately derived with (2.29), together with (A.6), and recalling that a stationary observer performs just a rotation (Pˆu∇u = ω). Hence, one gets, for any stationary observer:

Fua = ω(a) , (2.58a)

Fuω =0 . (2.58b)

Hence the acceleration vector, being comoving as computed above, rotates with ω w.r.t. a system of gyroscopic axes. This is clear, since the only motion performed is a rotation. In particular, the acceleration’s modulus remains constant along a worldline in u: u( a )=0 , (2.59) k k since u(g(a, a)) = 2g(∇ua, a)=2g(Fua, a)=2g(ω(a), a) = 0, using that ω is skew-adjoint. We can now give a further property of static observers. Since from Proposi- tion 2.16 we know that a static observer is irrotational, (2.58a) implies that in this case Fua = 0 or, in other words: 36 Chapter 2. Observers and reference frames

Proposition 2.19 (Static observers, II). A static observer is uniformly accelerated. We conclude this section giving, for completeness, the deﬁnition of rigid observers due to Born [Bor09]:

Definition 2.20 (Rigid observers). An observer field is called rigid if Luhu = 0. The above condition means that the spatial shape of some body transported with the flow of a rigid observer field does not change. Since

Luhu = Lu(Pug)= Pu(Lug) , (2.60) − − the rigidity condition is equivalent with the spatial part of the Killing equation, hence the former is weaker as the stationarity condition. In view of (2.37), equation (2.60) implies Proposition 2.21 (Rigid observers). A observer is rigid iﬀ it shear and expansion vanish (θ = 0).

This characterizes uniquely a rigid observer by means of a condition on Pˆu∇u and implies that the only motions a rigid observer can perform is to rotate and, eventually, be accelerating. For more on rigid observers we refer to the very lucid exposition [Giu06].

2.3.2 Synchronization

Definition 2.22 (Synchronization). An observer field u on some open subset U ⊆ is said to be locally synchronizable if it is hypersurface orthogonal, hence if M u du = 0, locally proper-time synchronizable if du = 0, synchronizable if there ∧ are smooth functions f and t such that f > 0 and u = fdt, and proper-time synchronizable if there exists a smooth function t such that u = dt. The previous equations are understood to be valid on . In the last two cases a function t as U above is called a time function. It is clear that, since d d = 0, proper-time synchronizable implies locally proper- ◦ time synchronizable. Similarly, using (fdt) d(fdt) 0, synchronizable implies ∧ ≡ locally synchronizable. Moreover, Poincaré-Lemma and Frobenius-Theorem ensure that the respective conversions hold in a sufficiently small open subset of . If, in U addition, is simply connected, then the conversions hold globally on . U U First of all, the above terminology is justified since if an observer u is synchronizable, there exists a time function (whose level-sets are orthogonal to u) that allows to assign a ‘global’ time to each event along the worldline of every single observer of u. Moreover, if the observer field is proper-time synchronizable, then it follows that this time, restricted along any worldline of the observer field u, coincides with its proper time (up to a trivial translation). The latter holds since, if u is proper-time synchronizable, then from γ˙ := γ ∂/∂τ = u = (∇t) it follows that dτ = γ∗dt. ∗ |γ |γ Here τ is the proper time of γ. At this point we stress that in the theoretical framework of Section 1.1 this terminology has also a deeper physical justification, since proper-time synchronizable means then that nearby worldlines of u are synchronizable to each other via the Einstein prescription (‘Einstein synchronization’) by the exchange of light (or radar) signals (cf. the discussion below (2.3)). A necessary and sufficient condition for locally synchronization is given by the following proposition, which is just Lemma 2.17 restated in other words: 2.4. Reference frames 37

Proposition 2.23. An observer ﬁeld is locally synchronizable iﬀ it is irrotational.

Analogously, for locally proper-time synchronization one has the following result, which will be important in cosmology, where it is often implicitly used:

Proposition 2.24. An observer ﬁeld is locally proper-time synchronizable iﬀ it is geodesic and irrotational.

Proof. We need to show that the acceleration a := ∇uu and the rotation ω of u both vanish iﬀ du does. This is indeed a direct consequence of the equation du = 2 ω + u a (see (2.40)), since the two summands on the r.h.s. are linearly ∧ independent (and even orthogonal).

2.4 Reference frames

An observer, sometimes, may want to deﬁne a ‘reference frame’ in its rest space. By this we mean the following:

Deﬁnition 2.25 (Reference frames). Given an observer u at p , a proper ∈ M reference frame at p is an n-bein e µ = 0,...,n 1 for which it holds that { µ| − } e := u (w.l.o.g.) and the e ’s (i = 1,...,n 1) are an orthonormal basis in 0 i − the rest space of u (i.e. g(u, ei) = 0 and hu(ei, ej ) = δij , together: g(eµ, eν ) = diag(1, 1,..., 1)). Given an observer worldline γ,a proper reference frame along − − γ is a smooth assignment of a proper reference frame to each point of γ.

Now, given an arbitrary observer worldline γ and a proper reference frame e { µ} at some point of γ we can construct a proper reference frame along γ by Fermi- transporting the eµ’s along γ. The properties of the Fermi transport (2.19) then ensure that the transported frame is indeed a proper reference frame along γ. Such a reference frame is then called non-rotating with respect to γ. The name is motivated by the fact that in any gravitational theory satisfying the requirements in Chapter 1, the spin vector of a gyroscope moving along a worldline γ is Fermi-transported. Hence, the spatial part of such a bein is physically realized if the observer carries a system of n 1 gyroscopes (with spin vectors orthogonal to each other) along with − it. If, instead of a single worldline, an observer field u is given in some open region of spacetime, one can construct a proper reference frame along any integral curve γ of u not only via Fermi transportation but also via corotating transportation (see (2.32)). As noted there, this transport preserves angles and lengths and thus a frame which is corotatingly transported is indeed a proper reference frame along γ, provided it is so at the beginning. Such a reference frame is called corotating with u along γ. If the observer field u is stationary, then a corotating proper reference frame along a worldline γ in u is called stationary (sometimes this is also called Copernican [Str84]). In this case, since the shear-expansion of a stationary observer vanishes, the corotating transport coincides with the comoving one. Finally, if u is static, a corotating proper reference frame along a worldline γ in u is called static. In this case, since a static observer is non-rotating, the corotating transport coincides with the Fermi one, and hence the static proper reference frame coincides with the Fermi proper reference frame. In some sense, a stationary proper reference frame can be considered to be ‘at rest’ w.r.t. the spacetime, since the properties of 38 Chapter 2. Observers and reference frames the latter remains unchanged along the worldlines of the stationary observer field. A static proper reference frame can then be considered ‘at rest’ and ‘non rotating’. Chapter 3

Observer-referred kinematics and dynamics

In this chapter we apply the observer-referred decomposition of spacetime in space and time introduced in Chapter 2.1 to the kinematics and dynamics of particles (Section 3.1) and, respectively, light rays (Section 3.2). In the former case we review the decomposition of the spacetime velocity in terms of the ‘gamma-factor’ and the (spatial) relative velocity (cf. (3.7)) as is familiar from Special Relativity. The corresponding relation for the lightlike case is given by the decomposition of the wave-vector in terms of frequency and (spatial) propagation direction (cf. (3.42)). In order to refer the variation of the spatial objects, like the relative velocity for a particle and the propagation direction for a light ray, to clocks and rods at rest w.r.t. the considered observer field, we define the so-called observer derivative, whose construction bases on the decomposition lemma for derivations (Lemma 2.5 in Section 2.2.1). In Section 3.1.2.1 we give a uniqueness characterization of the observer derivative (Proposition 3.5) which is then recognized as being the suitable derivation for the purpose described above. This is used for the definition of relative acceleration in terms of variation of the relative velocity and is applied to dynamics in order to set up the spatial version of the equation of motion which, for the case of particles, is to be seen as generalized version of Newton’s equation. The first systematic work on this topic was made by the Rome’s group on Robert Jantzen, see [JCB92, BCJ95]. Our approach here share the same starting point with Jantzen and collaborators, but differs in the construction of the observer-referred derivatives. In our approach these are constructed in a unifying way (which is based on the orthogonal decomposition lemma for derivations, see Lemma 2.5) as derivations on arbitrary tensor fields (of arbitrary rank and causal character) which commute with contractions, whereas in [JCB92] the derivatives are derivations (in the sense that they satisfy Leibniz rule) only on spatial tensor fields. On the advantages of our approach we already commented in the Remarks 2.3 and 2.4. Moreover, our approach allows to establish the uniqueness of the so-called ‘observer derivative’ which plays a fundamental role in the definition of the relative acceleration. We note also that here, the concepts of relative velocity and relative acceleration, as well as the spatial version of the equation of motion, are to be understood as local quantities—and relations between such—at the same spacetime point. Of course, one can also properly define concepts of ‘relative velocity’ and ‘relative acceleration’

39 40 Chapter 3. Observer-referred kinematics and dynamics between, say, worldlines which are at different spacetime points, however, in this case there is no unique natural way to define such notions: rather, there are several ways to define ‘relative velocity’ and ‘relative acceleration’, each of which is to the same grade ‘natural’ as the others. For a lucid discussion on this point, see the nice work of Bolos [Bol07].

3.1 The timelike case: test particles

3.1.1 Gamma-factor and relative velocity The decomposition of the tangential bundle in space and time discussed in Sec- tion 2.1 enable us to define the concepts of relative velocity and acceleration between two observers. These definitions are the natural generalization of the familiar space+time-split of Special Relativity. Definition 3.1 (Relative velocity). Let u and v be two observers at p. The relative velocity of v w.r.t. u is given by

Puv βu(v)p := , (3.1) Quv p k k and its modulus1 by

hu(v, v) βu(v) := βu(v) = hu(βu(v), βu(v)) = . (3.2) p k kp p tu(v, v) p p p Given a unit vector e in the rest space of u (i.e. Pue = e and e = 1), the relative k k velocity of v w.r.t. u at p in direction e is given by

e g(e, v) β (v) := hu βu(v), e = g βu(v), e = − . (3.3) u p p − p g(u, v) p

A little manipulation with the projectors shows that 1 Quv = g(u, v)= . (3.4) 2 k k 1 βu(v) − This is just the ‘gamma-factor’, familiar fromp Special Relativity, between the observers u and v. The inverse of the gamma-factor in front of the u-spatial vector

Puv in (3.1) is introduced in order to refer the velocity in terms of the proper time of u rather than the proper time of v. In Minkowski spacetime, (3.1) reduces to the familiar spatial velocity β of Special Relativity (see (3.20) in Example 3.1 below). Since for any two observers u and v it holds g(u, v) 1, expression (3.4) implies e ≥ that 0 βu(v) < 1 and 1 <β (v) < 1. Moreover we can put ≤ − u g(u, v) = cosh χ(u, v) , (3.5) which deﬁnes χ(u, v) (assuming this to be positive), the rapidity of v w.r.t. u. With (3.4) then one has the following familiar relation between rapidity and modulus of the relative velocity:

βu(v) = tanh χ(u, v) . (3.6)

1 Please note that the vector β is written in boldface, whereas its modulus β is written in normal font. 3.1. The timelike case: test particles 41

Note that since the gamma-factor is symmetric in u and v, one have that both the rapidity and the modulus of the relative velocity of v w.r.t. u equals that of u w.r.t. v. Note, however, that the former velocity is measured with clocks and rods at rest w.r.t. u and is represented by a vector in u(p), whereas the latter R is measured with clocks and rods at rest w.r.t. v and is represented by a vector in v(p). R If an observer field u is defined on some open region of the spacetime which contains the trace of the worldline γ, the velocity γ˙ along γ is uniquely determined by its gamma-factor Quγ˙ = g(u, γ˙ ) and its relative velocity βu(γ˙ ) w.r.t. the k k observer field u: γ˙ = Quγ˙ (u + βu(γ˙ )) . (3.7) k k This is just the generalization of the special-relativistic expression of the four- velocity in term of the gamma-factor and the three-velocity (see also Example 3.1 below).

3.1.2 Observer derivatives and relative acceleration In order to define the relative acceleration of a worldline γ, say, w.r.t. an observer field u, say, we need a derivation which measures variations along γ in terms of clocks and rods at rest w.r.t. u. For this purpose, and with in mind what we have learned about derivatives on the Chapter 2, we are led to introduce the following Definition 3.2 (Observer derivative). Let γ be a worldline (timelike or lightlike), u an observer field along γ, and X a vector field along γ. We define the u-observer derivative of X along γ as

∇u 1 ∇ ∇ γ˙ X := (Qu γ˙ Qu + Pu γ˙ Pu) X . (3.8) Quγ˙ ◦ ◦ ◦ ◦ k k u −1 u We then define ∇ f := Quγ˙ γ˙ (f) on functions and extend ∇ to a derivation γ˙ k k γ˙ of tensor fields along γ in the standard way (as we did for the Fermi and the comoving derivative): requiring the Leibniz rule w.r.t. tensor product and commutativity with contractions. A tensor field along γ is called u-observer-transported along γ, if its u-observer derivative vanishes along γ. Definition (3.8) resembles that of a rescaled (with the inverse of the gamma- factor Quγ˙ ) Fermi-derivative along γ (c.f. (2.15)). Notice, however, that the pro- k k jections are here taken w.r.t. the observer field u instead of γ˙ . Rather, the observer derivative is to be seen as the orthogonal projection (in the sense of Lemma 2.5) −1 of the derivation Quγ˙ ∇γ. Clearly, a vector field X is u-observer transported k k ˙ along γ iff Pu(∇γ˙ (PuX)) = 0 and Qu(∇γ˙ (QuX)) = 0 along γ, that is iff the spatial and temporal parts of X do not change—with respect to the covariant derivative—as seen separately from the local rest space u and the local tempo- R ral subspace u along γ, respectively. The inverse of the gamma-factor in (3.8) is T introduced in order to refer the variation to the proper time of u rather than to the proper time of γ (in case this is timelike), or to the affine parameter (in case γ is lightlike). In the next proposition we collect the properties of the observer derivative along a timelike worldline. Since this is an object which will recur in the following, we define the kinematical operator Ku of the observer field u by

KuX := ∇uu + (Pû∇u)X , (3.9) 42 Chapter 3. Observer-referred kinematics and dynamics for any vector field X. This contains the kinematical quantities (acceleration, rotation, shear, and expansion) of the observer field u. Clearly, it is an affine operator on the tangent space and its range is u-spatial.

Proposition 3.3 (Properties of the observer derivative, timelike case). Let γ be a timelike worldline and u an observer field defined on a tubular neighborhood of γ. Then the following properties hold: (i) On vector fields along γ one has

1 ∇ ∇u γ˙ γ˙ = B(Kuβu,u) , (3.10) Quγ˙ − k k

where βu := βu(γ˙ ), Ku is the kinematical operator (3.9) of u, and B(·,·) denotes the boost deﬁned by (2.10). In particular, if the acceleration, the rotation, the shear, and the expansion of u all vanish, the observer derivative reduces to the covariant derivative along γ rescaled with the inverse of the gamma-factor. (ii) The following transport equations hold along γ: ∇u γ˙ u = 0 (3.11a) ∇u γ˙ g = 0 (3.11b) ∇u γ˙ tu = 0 (3.11c) ∇u γ˙ hu = 0 (3.11d)

∇u (iii) γ˙ commutes with the extension of the projections Pu and Qu on arbitrary tensor fields along γ. ∇u (iv) γ˙ maps u-spatial tensor fields to u-spatial tensor fields and u-temporal tensor fields to u-temporal tensor fields.

Proof. The points (ii)–(iv) follow immediately applying Lemma 2.5 to the (met- −1 ric) derivation Quγ˙ ∇γ. It remains to show property (i). Evaluating the k k ˙ l.h.s. of (3.10) on a vector ﬁeld X (deﬁned along γ), inserting the identity

Pu + Qu = id before and after the covariant derivative in the ﬁrst term, and −1 expanding everything, one gets Quγ˙ (Pu(∇γ (QuX)) + Qu(∇γ (PuX))) = k k ˙ ˙ −1 Quγ˙ (Pu(∇γ Qu)X + Qu(∇γ Pu)X). With decomposition (3.7), one has k k ˙ ˙

∇γu = Quγ˙ Kuβu , (3.12) ˙ k k and hence: ∇γ Qu = Quγ˙ (Kuβu u + u Kuβu) . (3.13) ˙ k k ⊗ ⊗

Now, with ∇γ Pu = ∇γ(id Qu) = ∇γQu and using the fact that Kuβu is ˙ ˙ − − ˙ u-spatial one gets immediately the r.h.s. of (3.10).

The above proposition is to be compared with Proposition 2.6 for the Fermi derivative. In particular, it follows that one can build a proper reference frame, which one may call observer reference frame along γ, deﬁning at some point of γ an orthonormal n-bein e with e = u and then transporting it along γ by means { µ} 0 of the u-observer transportation. The properties (3.11a) and (3.11b) then ensure that the n-bein is orthonormal and e0 = u everywhere along γ. 3.1. The timelike case: test particles 43

We now are going to define the relative acceleration as the variation of the relative velocity, where the variation is taken with respect to the observer derivative. Why are we choosing here the observer derivative and not another derivative? What makes the observer derivative the ‘right one’ for this task? This is a crucial point which clearly needs a justification. In Example 3.1 below we show that defining the relative acceleration in this way we indeed recover, in the case of a flat spacetime and a covariantly constant observer field, the usual expression for the spatial acceleration familiar from Special Relativity. Moreover, in the paragraph below that example, we give a natural characterization of the observer derivative, which uniquely determines it. All this provides a justification for the following Definition 3.4 (Relative acceleration). The relative acceleration of a worldline γ w.r.t. an observer field u along γ is defined by the vector field ∇u αu(γ) := γ˙ βu(γ˙ ) (3.14) along γ.

Since the relative velocity βu(γ˙ ) is u-spatial along γ per definition, the relative acceleration αu(γ) is also u-spatial. If one chose the observer field u to be γ˙ itself, then αγ (γ) 0 (3.15) ˙ ≡ for any worldline γ. In words, this means that the relative acceleration of a worldline w.r.t. itself vanishes identically, which is indeed a very satisfying property. For later use we give the derivatives of the relative velocity’s modulus and projection and of the gamma-factor: ∇u 2 γ˙ βu(γ˙ ) =2 hu βu(γ˙ ), αu(γ) , (3.16) ∇u e ∇u γ˙ βu(γ˙ ) = hu α u(γ), e + hu βu(γ˙ ), γ˙ e , (3.17) u 3 ∇ Quγ˙ = Quγ˙ hu(α u(γ),βu(γ˙ )) . (3.18) γ˙ k k k k In (3.17), e is some u-spatial unit vector-field along γ. For the derivation of the ∇u above formulae we just used the metricity of γ˙ , the definition of the relative 2 acceleration (3.14), and βu(γ˙ ) = hu(βu(γ˙ ), βu(γ˙ )). In the following example we show how the notions introduced in this section reduce, in the case where the spacetime is flat and the observer field is covariantly constant (i.e. geodesic, non-rotating and with vanishing shear and expansion), to those familiar from the 3+1-splitting of Special Relativity. Example 3.1 (Covariantly-constant observer in Minkowski spacetime). Let = M Rn and g = η. As usual, one introduces Minkowskian coordinates, i.e. a global chart xµ = (t, xi) with respect to which η(∂ , ∂ ) = diag(1, 1,..., 1). If a µ ν − − timelike worldline γ is expressed in coordinates by zµ(τ) = (z0(τ),zi(τ)), then its velocity is γ˙ =z ˙µ∂ γ, where the dot denotes a derivation w.r.t. the proper time µ ◦ τ and ∂ γ denotes the vector field ∂ along the map γ (see Appendix A.2). Now µ ◦ µ we refer the quantities introduced in this section with respect to the covariantly constant observer field u = ∂/∂t. For this, we parametrize the worldline w.r.t. the 0 global time t, more precisely w.r.t. tγ defined by tγ(τ) := t(γ(τ)) = z (τ). The gamma-factor between u and γ˙ is then

g(u, γ˙ ) =z ˙0(τ)= t˙ (τ) (dt /dτ)(τ) (3.19) γ ≡ γ 44 Chapter 3. Observer-referred kinematics and dynamics and the relative velocity of γ˙ w.r.t. u is

z˙i dzi β γ ∂ ∂ u( ˙ )= 0 i γ = i γ . (3.20) z˙ ◦ dtγ ◦

u −1 Since u is covariantly constant, by (3.10) it follows that ∇ reduces to Quγ˙ ∇γ . γ˙ k k ˙ Recall that a vector ﬁeld X along γ can be locally expressed as X = Xµ∂ γ, where µ ◦ µ µ µ ν the X are functions of τ. Moreover, we have ∇γX = X˙ ∂ γ +X z˙ (∇∂ ∂ ) γ ˙ µ ◦ ν µ ◦ µ (see equation (A.2)), which reduces here to ∇γ X = X˙ ∂ γ. Hence, the u-observer ˙ µ◦ derivative of X is simply given by

µ ∇u dX γ˙ X = ∂µ γ (3.21) dtγ ◦ and, with (3.20), the relative acceleration of γ w.r.t. u is given by

d2zi α ∂ u(γ)= 2 i γ . (3.22) dtγ ◦

This shows that gamma-factor, relative velocity, and relative acceleration reduce indeed to the corresponding quantities familiar from Special Relativity.

3.1.2.1 Characterization and uniqueness of the observer derivative At this point we can summarize the important properties of the observer derivative. We consider in the following γ to be a timelike worldline, however what follows also ∇u applies in the case of a lightlike worldline. We have: (a) γ˙ is a derivation on u −1 tensor fields along γ which commutes with contractions, (b) ∇ = Quγ˙ γ˙ on γ˙ k k ∇u ∇u functions along γ, (c) γ˙ u = 0, (d) γ˙ g = 0, and (e) if the spacetime is flat and the observer field u is geodesic, non-rotating and with vanishing shear and ∇u expansion, γ˙ reduces to (3.21). These are all properties one would expect to hold ∇u for an observer derivative which is used to define the relative acceleration as γ˙ β. (a) is in some sense for computational convenience, (b) ensures that the variation is referred to the proper time of the observer field u rather than to the proper time of γ, (c) and (d) ensure that one can define a proper reference frame along γ (see ∇u Section 2.4), and (e) ensures that γ˙ β reduces, in the case of a flat spacetime and a covariantly constant observer field, to the spatial acceleration of Special Relativity. ∇u The nice news is that these properties already characterize γ˙ uniquely, as we now show.

u Proposition 3.5 (Uniqueness of the observer derivative). Let Oγ˙ be a derivation on tensor fields along a timelike worldline γ, which commutes with contractions and for which it holds: u −1 (i) O = Quγ˙ γ˙ on functions along γ, γ˙ k k u (ii) Oγ˙ u =0, u (iii) Oγ˙ g =0, (iv) if the spacetime is flat and the observer field u is geodesic, non-rotating and u with vanishing shear and expansion, Oγ˙ reduces to (3.21). 3.1. The timelike case: test particles 45

u ∇u Then Oγ˙ = γ˙ on tensor ﬁelds along γ.

u ∇u Proof. The equality between Oγ˙ and γ˙ holds, by (i), already on functions. Because of the uniqueness result of Lemma A.2 it suffices to show this equal- u ity on vector fields. Since Oγ˙ is a derivative and because of (i) we have that −1 u Quγ˙ ∇γ O = E on vector fields along γ, where E is an endomorphism of k k ˙ − γ˙ the tangential space of along γ. Now (iii), together with the metricity of ∇, im- M −1 u plies that E is skew-adjoint: 0 = Quγ˙ ∇γg O g = Eˇ g = g(E , ) g( , E ) k k ˙ − γ˙ − · · − · · (see (A.7)). Therefore, E can be uniquely decomposed in its orthogonal components w.r.t. u as E = B + R, where B is an infinitesimal boost in a plane containing u and R is an infinitesimal rotation in the rest space of u (see (2.9)). The former can be written as B(Z,u) (see (2.10)), for some u-spatial vector field Z along γ. −1 Now, (ii) implies that Z = E(u) = Quγ˙ ∇γ u = Kuβu, where we used the k k ˙ decomposition (3.7) and the definition of the kinematical operator (3.9). Hence, summarizing, (i)–(iii) imply

1 ∇ u γ˙ Oγ˙ = B(Kuβu,u) + R (3.23) Quγ˙ − k k on vector ﬁelds, where R remains undetermined. This, together with (3.10), means u ∇u that Oγ˙ is determined, and is equal to γ˙ , up to a rotation in the local rest frame of u. The last requirement, (iv), now implies that this rotation must vanish, which, u ∇u in turn, implies the equality between Oγ˙ and γ˙ .

From what we just saw, a derivation which commutes with contraction differ- ∇u ent from the observer derivative γ˙ and which satisfies the properties (i)–(iii) of Proposition 3.5 must differ from the latter by a rotation in the local rest space. An example for this is what one would call the Fermi observer-derivative, which can be defined along a timelike worldline. One puts

u 1 Fγ˙ X := (Qu Fγ˙ Qu + Pu Fγ˙ Pu) X (3.24) Quγ˙ ◦ ◦ ◦ ◦ k k u −1 on vector fields, F f := Quγ˙ γ˙ (f) on functions, and then extends derivatively γ˙ k k requiring commutativity with contractions. This derivation clearly satisfies the points (i)–(iii) of Proposition 3.5: (i) by its very definition, (ii) and (iii) as con- −1 sequence of Lemma 2.5 applied to the (metric) derivation Quγ˙ Fγ . Moreover, k k ˙ using (2.18) to express the Fermi derivative in (3.24) in terms of the covariant one, we get u u ∇ ⊥ γ˙ Fγ˙ = R ∇ β (3.25) − ((Pu γ˙ γ˙ ) ,β) on vector fields along γ. Here β := βu(γ˙ ) and R(·,·) is the rotation defined in (2.11). u Defining the acceleration as Fγ˙ β one would then have in (3.22) an unwanted additional term given by the rotation on the r.h.s. of (3.25). In the remaining part of this section we comment on an alternative definition of the observer derivative we used in a former work [CG06] and which is also in use elsewhere in the literature (see e.g. Section V in [JCB92]). We show below that, on u-spatial tensor fields, this alternative derivative coincides with the observer 46 Chapter 3. Observer-referred kinematics and dynamics derivative defined above. Hence, when dealing with u-spatial tensor fields (as we did in [CG06]), the two derivatives are interchangeable. In particular, all considerations made in [CG06] remain valid if restated in terms of the observer derivative defined above. The latter derivative has indeed the advantage of being a derivation which commutes with contractions on arbitrary tensor fields (of arbitrary rank and causal character) and this by its very definition (see Definition 3.2). In our view, this is convenient for computations. Moreover it allows, together with the properties (3.11), to construct an observer-transported proper reference frame w.r.t. any observer fields u and along arbitrary worldlines: for this one defines at some point of γ an n-bein e with e := u and transport this along γ requiring ∇ue = 0. { µ} 0 γ˙ µ

Alternative definition for the observer derivative In [CG06] we used as ‘observer derivative’ the operator defined on arbitrary tensor fields by ∇ˆ u 1 ˆ ∇ ˆ γ˙ := Pu γ˙ Pu . (3.26) Quγ˙ ◦ ◦ k k

Recall that Pû stands for the extension of the spatial projector on the respective tensor space (i.e. every free index is to be projected, see (2.4)). Notice that on u −1 functions (3.26) reduces to ∇ˆ f = Quγ˙ γ˙ (f). γ˙ k k In Remark 2.3 we already discussed the implications of projecting the derivative only onto spatial directions and in Remark 2.4 the implications of defining the projected ‘derivative operator’ on arbitrary tensor fields via the brute-force projection on every slot (index), instead of taking the usual derivative extension. Despite this, definition (3.26) is not as bad as one may now think, since, fortunately, on ∇ˆ u u-spatial tensor fields γ˙ is indeed a derivation which commutes with contractions ∇u and, moreover, equals the observer derivative γ˙ . This we will show below. An ∇u ∇ˆ u advantage of γ˙ w.r.t. γ˙ is certainly that the former is by definition a derivation on arbitrary tensor fields (of any rank and causal character) which commutes with ∇ˆ u contractions. For γ˙ these features must be proven, which we do in the following

∇ˆ u ∇ˆ u Lemma 3.6 (Properties of γ˙ ). For the operator γ˙ it holds: (i) it is a derivation2 on u-spatial tensor fields along γ (ii) it commutes with contractions on u-spatial tensor fields along γ ∇ˆ u (iii) it satisfies the metricity property γ˙ hu =0.

∇ˆ u R Proof. We first show the derivation properties. That γ˙ is -linear and rank- preserving is clear—and this holds not only on spatial tensor fields but also on arbitrary tensor fields along γ. The Leibniz rule, however, holds just for spatial tensor u u fields. This is because, as one can easily see, ∇ˆ (A B) = (∇ˆ A) PuB +PuA γ˙ ⊗ γ˙ ⊗ ⊗ ∇ˆ u ∇ˆ u ( γ˙ B) for any tensor fields A, B along γ. To see that γ˙ commutes with contractions on spatial tensors let consider the contraction C(α X) := α(X) between a ⊗ one-form α and a vector field X along γ, which do not need to be spatial. Using what we said above, we have, from one hand, that C(∇ˆ u(α X)) = C(∇ˆ uα γ˙ ⊗ γ˙ ⊗ 2 In the sense of Definition A.1. 3.1. The timelike case: test particles 47

u u u u u PuX + Puα ∇ˆ X) = (∇ˆ α)(PuX) + (Puα)(∇ˆ X) = (∇ˆ α)(X)+ α(∇ˆ X), ⊗ γ˙ γ˙ γ˙ γ˙ γ˙ where in the last step we used the definition of the projector on one-forms ∇ˆ u and that the image of γ˙ is spatial. On the other hand, one computes that u u −1 −1 ∇ˆ (C(α X)) = ∇ˆ (α(X)) = Quγ˙ γ˙ (α(X)) = Quγ˙ ((∇γ α)(X) + γ˙ ⊗ γ˙ k k k k ˙ u u −1 α(∇γ X)) = (∇ˆ α)(X)+ α(∇ˆ X)+ Quγ˙ ((∇γ α)(QuX) + (Quα)(∇γ X)). ˙ γ˙ γ˙ k k ˙ ˙ This shows that C(∇ˆ u(α X)) = ∇ˆ u(C(α X)) iff both α and X are u- γ˙ ⊗ γ˙ ⊗ spatial. This argumentation generalizes straightforwardly for tensors of higher ∇ˆ u rank. Finally, we conclude the proof by showing the metricity property: γ˙ hu = −1 −1 Quγ˙ Pû(∇γ (u u g)) = Quγ˙ Pû (∇γ u) u = 0, which is valid on k k ˙ ⊗ − k k ˙ ∨ arbitrary pairs of vector fields along γ (not necessarily u-spatial). ∇ˆ u ∇u Now, from the definitions (3.26) and (3.8), it is clear that γ˙ = γ˙ on functions and on u-spatial vector fields along γ. But since both are derivations and commute with contractions on u-spatial tensor fields along γ, the uniqueness of derivations ∇ˆ u ∇u (Lemma A.2) implies that the equality γ˙ = γ˙ holds also on u-spatial tensor fields of arbitrary rank.

3.1.3 Projection of the equation of motion

Consider a particle whose (timelike) worldline γ is determined by the equation of motion ∇γ γ˙ = Pγ F γ =: f , (3.27) ˙ ˙ ◦ plus some initial conditions. Here, F is a vector field on (representing an external M force per unit mass acting on the particle) and f is introduced just as a short- hand notation. Notice that on the r.h.s. the γ˙ -spatial projector is needed in order to preserve the normalization condition γ˙ = 1 under the evolution described k k by (3.27). Let us suppose an observer field u is given on some open tubular neighborhood of the trace of the worldline γ. With the notions just introduced we can now rewrite the equation of motion (3.27) in terms of spatial-defined quantities with respect to the observer field u. Decomposing carefully everything in parts parallel and orthogonal to u by means of the projectors (2.1) and expressing all in terms of the spatial observer-referred kinematical quantities αu(γ) and βu(γ˙ ) (the latter from now on abbreviated with βu) and of quantities proper to u, one gets the following Proposition 3.7 (Spatial equation of motion, timelike case). The spacetime equation of motion (3.27) is equivalent to the following spatial equation of motion:

αu(γ)= Sβu (fu Kuβu) − (3.28) = Sβ fu au ωu(βu) θu(βu) . u − − − −2 Here fu := Quγ˙ Puf is the with the gamma-factor rescaled u-spatial-projected k k force, Ku the kinematical operator of u (see (3.9)), au, ωu, θu the acceleration, rotation, and shear-expansion of u, and ♭ Sβ := id βu β u Ru − ⊗ u (3.29) ♭ −2 ♭ = id βu β + Quγ˙ βu β Ru − ⊗ u k k ⊗ u b b b b 48 Chapter 3. Observer-referred kinematics and dynamics

♭ is an endomorphism of the rest space u along γ. Thereby, β := hu(βu, ) is the R u · −1 spatial one-form related to βu via the spatial metric hu, βu := βu βu is the k k unit vector in direction of βu, and idRu is the identity map in u. b R

We notice from (3.29) that the endomorphism Sβu acts trivially on the two- dimensional subspace of u orthogonal to βu and on the one-dimensional subspace R parallel to βu acts as rescaling by the inverse squared of the gamma-factor. Formula (3.28) generalizes the spatial equation of motion familiar from Special Relativity. The latter is indeed recovered if the spacetime is flat and the observer field u is geodesic and its rotation, shear, and expansion all vanish (see Example 3.1). We note also that (3.28) resemble very much the Newtonian equation of motion in a non-inertial reference frame (in the Newtonian sense), provided one keep only linear terms in β and the shear and expansion of u vanish. The terms au and ωu(β) correspond to the terms due to the linear acceleration and, respectively, the rotation of the non-inertial reference frame (in Newtonian sense), the latter term including both Coriolis- and centrifugal acceleration. This provides also a physical justification for the definitions of the spatial quantities introduced so far.

Proof of Proposition 3.7. The proof follows the same steps (of course with the appropriate generalizations) one has to perform in the 3+1-split of the equation of motion in Special Relativity. The ﬁrst step is to express velocity and acceleration in terms of spatial quantities. For the former, this was already done in (3.7). For the latter one has, using (3.7) and (3.18):

2 −1 4 ∇γγ˙ = Quγ˙ Quγ˙ ∇γ β + ∇uu + ∇βu + Quγ˙ h(α, β)(u+β) (3.30) ˙ k k k k ˙ k k where here, and henceforth, we write β for βu(γ˙ ), α for αu(γ), and h for hu. The u-temporal and u-spatial part of the acceleration are then

2 4 g(u, ∇γ γ˙ )= Quγ˙ h ∇uu + (Pˆu∇u)β, β + Quγ˙ h(α, β) (3.31a) ˙ k k k k and

2 4 Pu∇γγ˙ = Quγ˙ α + ∇uu + (Pû∇u)β + Quγ˙ h(α, β)β , (3.31b) ˙ k k k k respectively. Equation (3.31b) is straightforward. For (3.31a) we have rewritten, in the projection of the first term of (3.30), g(u, ∇γβ) = g(∇γ u, β) = ˙ − ˙ Quγ˙ g ∇uu + (Pû∇u)β, β =+ Quγ˙ h ∇uu + (Pû∇u)β, β . Here, in the −k k k k third step, we used again the decomposition (3.7). Now, in order to write everything in term of spatial quantities, we need the relation g(u, f)= h(Puf, β) , (3.32) which follows using the decomposition (3.7) in the identity g(γ˙ , f) = 0. With (3.32) and decomposition (3.31), the temporal and spatial projection of the equation of motion (3.27) become, after a little rearrangement,

4 2 Quγ˙ h(α, β)= h Puf Quγ˙ (∇uu + (Pˆu∇u)β), β (3.33) k k −k k 3.1. The timelike case: test particles 49 and

2 2 Quγ˙ α = Sβ Puf Quγ˙ (∇uu + (Pˆu∇u)β) , (3.34) k k −k k respectively. In order to get the ﬁnal form (3.34) we used (3.33). The proof is concluded noticing that indeed (3.33) is already implied by (3.34). This is immediately seen projecting (3.34) along β and looking at the second expression for the operator

Sβ in (3.29).

In the following, motivated by the similarity of the spatial equation of motion (3.28) with the Newtonian equation in a non-inertial reference frame, we im- port from Newtonian mechanics the concept of ‘inertial reference frame’ and give an equivalent characterization for it. Recall that in Newtonian mechanics, an inertial reference frame is a reference frame in which the acceleration of free-falling particles vanishes. With this motivation in mind we introduce Definition 3.8 (Inertial observers). Denote with the set of all timelike geodesic Gp worldlines through p . We call an observer field u inertial at p, if the relative ∈M acceleration αu(γ) at p vanishes for every γ . p ∈ Gp An example of inertial observer field is ∂/∂t in Minkowski spacetime (for the notation see Example 3.1). More generally, one has the following characterization for inertial observers: Proposition 3.9 (Inertial observers). Let p be a spacetime point and u an observer field defined on an open neighborhood of p. Then u is inertial at p iff it is geodesic

(∇uu = 0), irrotational, shearless, and non-expanding (Pˆu∇u = 0) at p. This, in turn, is the case iﬀ u is covariantly constant (∇u = 0) at p.

2 Proof. This is a direct consequence of (3.28). Since det(Sβ )=1 βu > 0, it u −k k follows that αu(γ) = 0 iﬀ ∇uu + (Pˆu∇u)(βu) = 0. Since this should hold for all

γ , the last equation should then hold for all βu in the open unit ball (in the ∈ Gp rest space (p) at p). A little linear algebra then shows that αu(γ) = 0 iﬀ both Ru the vector ∇uu and the endomorphism Pˆu∇u vanish. The last statement of the proposition follows then immediately from the decomposition (2.28).

The above proposition tell us that it is possible to find an inertial observer at a given point p iff there is an observer field which is covariantly constant at p. The latter, in turn, is the case iff there exists a timelike vector field Z which is covariantly constant at p (the normalization were dropped). This equivalence is obvious in one direction. For the other just note that ∇( Z −1Z)= Z −1∇Z + Z d( Z −1). k k k k ⊗ k k Since ∇Z = 0 implies also ∇ Z = 0, then u := Z −1Z defines a covariantly k k ±k k constant, timelike unit vector field. (The sign of u is to be chosen such that u is future-pointing.) Hence, summarizing, the existence of an inertial observer field is equivalent with the existence of a timelike covariantly constant vector field. As we will show in the next paragraph, at a given point p it is always possible to find such a vector field and thus it is always possible to find an inertial observer field at p: this is just a reformulation of the fact that one can always find coordinates with respect to which the Christoffel-symbols vanish at the given point. On the contrary, in order 50 Chapter 3. Observer-referred kinematics and dynamics to have an inertial observer field in some open neighborhood, one must impose the equation ∇Z = 0 on that neighborhood. This of course results in constraints for the curvature: it must hold R(X, Y )Z = 0 for every X, Y , which can be paraphrased saying that the sectional curvature of every two-plane containing Z must vanish. This, of course, cannot be achieved in every spacetime. To conclude this section we show that at a given point p, it is always possible to find a timelike vector field Z with ∇Z =0 at p. A quick way to show this is to use a local chart xµ (with the usual convention that x0 denotes the timelike coordinate) λ on a neighborhood of p w.r.t. which the Christoffel-symbols Γµν vanishes at p. Then, for example, Z := ∂0 is a timelike vector field (defined on the domain of the chart) µ ∇ µ∇ µ λ which, for any vector X = X ∂µ, satisfies X Z = X ∂µ ∂0 = X Γµ0∂λ = 0 at p. The easiest example, as we have already anticipated, is that of Special Relativity with the observer field u = ∂/∂t (where t is the Minkowskian time coordinate, see Example 3.1). Then (3.28) reduce to the familiar equation of motion

α = Sβfu (3.35)

i 2 i 2 of the 3+1-split, where β = (dz /dtγ )∂i and α = (d z /dtγ )∂i. Taking as reference, instead of the covariantly constant ∂/∂t, an observer-ﬁeld with acceleration, rotation, shear, or expansion, one then sees corresponding terms arising on the r.h.s. of (3.28). For example, in case of a rotating observer-ﬁeld, one gets a term ω(β), which is nothing but the Coriolis-centrifugal-term. An easy and instructive example is the following:

Example 3.2 (Static observer in Schwarzschild spacetime). Recall that the Schwarzschild spacetime is described by the relations (6.33) (putting Λ = 0 there). In the static region of this spacetime, the Killing field ∂/∂t is a geometrically distinguished vector field which, via u := ∂/∂t −1∂/∂t, defines a static observer k k field. Because of the staticity, the spatial endomorphism Pû∇u vanishes identically (Propositions 2.15 and 2.16) and its acceleration is ∇uu = ∇ ln ∂/∂t = − k k ′ (√V ) eR (see (2.53)). Here, the prime denotes a differentiation w.r.t. the argument of V (the areal radius R) and e := ∂/∂R −1∂/∂R is the unit vector field in radial R k k direction. Notice, that since V ′(R) = (m/R2)/ V (R) is positive, the acceleration of the observer field u points in outward radialp direction: clearly, this is needed in order that u remains stationary. Now, the spatial equation of motion (3.28) for a geodesic worldline γ becomes

m −1/2 αu(γ)= V (R) Sβ e . (3.36) −R2 u R This looks very much as the Newton equation but notice that this equation is exact. In the slow-motion (small β) and weak-ﬁeld approximation (small m/R) of (3.36) one recovers the familiar Newtonian equation m αu(γ)= e . (3.37) −R2 R More examples are found in Chapter 7, where this formalism is applied to the dynamics of local systems in some cosmological spacetime. 3.2. The lightlike case: light rays 51

3.2 The lightlike case: light rays

3.2.1 Frequency and propagation direction

As we discussed in Section 1.2, light3 can be described in the geometric-optics limit by a geodesic lightlike vector ﬁeld k, the so-called wave-vector ﬁeld. An integral curve of k, that is a curve λ satisfying

λ˙ = k λ , (3.38) ◦ describes the worldline of a light pulse (photon), which is consequently a lightlike geodesic: ∇ ˙ λ˙ λ =0 . (3.39) Given an observer field u defined on some open region of the spacetime containing λ, the wave-vector field along λ, that is λ˙ , is uniquely determined by the two following quantities defined along λ: the frequency

ωu(λ˙ )= g(u, λ˙ ) (3.40) measured by the observer u and the spatial propagation direction of the light ray

P λ˙ kˆ := u (3.41) Puλ˙ k k measured by u in its rest spaces u along λ. Since λ˙ is lightlike we have Puλ˙ = R k k Quλ˙ = g(u, λ˙ )= ωu(λ˙ ) and hence the decomposition k k

λ˙ = ωu(λ˙ ) u + kˆ (3.42) along λ. This is the lightlike analogon to the decomposition (3.7). In this sense, the frequency and the spatial propagation direction are the lightlike counterparts of the gamma-factor and the relative velocity, respectively. In order to set up the evolution equations for frequency and spatial propagation direction, we need a derivative on tensor fields along λ. The uniqueness characterization of the observer derivative along a timelike worldline given in Proposition 3.5 extends trivially also for a lightlike worldline (in the proof one just has to use the decomposition (3.42) instead of (3.7)). Hence, also for the lightlike case the observer derivative is singled out. By the way, we notice that the Fermi observer derivative is not defined along a lightlike worldline (since the Fermi derivative itself is not defined in that case). Note that the factor Quλ˙ in the definition of the observer k k derivative (Definition 3.2), which is nothing than the frequency measured by u, implies that ∇u measures indeed the variation of tensor fields along λ in terms of λ˙ the proper time of u, rather than the affine parameter of λ. The analogon of Proposition 3.3 for the lightlike case is now

Proposition 3.10 (Properties of the observer derivative, lightlike case). Let λ be a lightlike worldline and u an observer ﬁeld deﬁned on a tubular neighborhood of λ. Then the following properties hold:

3 Here, and in the following, ‘light’ stands for electromagnetic waves in general. 52 Chapter 3. Observer-referred kinematics and dynamics

(i) On vector ﬁelds along λ one has

1 ∇ ∇u = B , (3.43) λ˙ λ˙ (Kukˆ,u) Quλ˙ − k k

where kˆ is the spatial propagation direction (3.41), Ku the kinematical opera-

tor (3.9) of u, and B(·,·) denotes the boost deﬁned by (2.10). In particular, if the acceleration, the rotation, the shear, and the expansion of u all vanish, the observer derivative reduces to the covariant derivative along λ rescaled with the inverse of the frequency measured by u. (ii) The following transport equations hold along λ: ∇u λ˙ u = 0 (3.44a) ∇u λ˙ g = 0 (3.44b) ∇u λ˙ tu = 0 (3.44c) ∇u λ˙ hu = 0 (3.44d)

(iii) ∇u commutes with the extension of the projections P and Q on arbitrary λ˙ u u tensor fields along λ. ∇u u u u (iv) λ˙ maps -spatial tensor fields to -spatial tensor fields and -temporal tensor fields to u-temporal tensor fields. Proof. The properties (ii)–(iv) follow, exactly as in the timelike case, applying −1 Lemma 2.5 to the (metric) derivation Quλ˙ ∇ ˙ . (Note that this does not in- k k λ volve at all any assumption on the causal character of the worldline.) It remains to show property (i). The proof of this property in Proposition 3.3 involves the causal character of the worldline only implicitly when one makes use of the decomposition (3.7). However, the very same steps can be done using instead the decomposition (3.42), which means that one has only to replace in the final result gamma-factor and relative velocity by, respectively, frequency and spatial propagation direction.

3.2.2 Projection of the propagation equation Similarly to what we did in the timelike case, we want to express the propagation equation (3.39) in terms of spatial quantities referred to a reference observer-ﬁeld u. This is given by the following Proposition 3.11 (Spatial equations of motion, lightlike case). The spacetime propagation equation (3.39) for the wave-vector ﬁeld of a light ray is equivalent to the following system of evolution equations for the frequency (3.40) and the spatial propagation direction (3.41):

u ∇ ln ωu = hu(Kukˆ, kˆ) (3.45a) λ˙ − u ∇ kˆ = S Kukˆ . (3.45b) λ˙ − kˆ Here Ku is the kinematical operator of u (see (3.9)) and

S := id kˆ kˆ♭ (3.46) kˆ Ru − ⊗ 3.2. The lightlike case: light rays 53 is the projector onto the two-dimensional subspace orthogonal to kˆ in the rest ♭ space u along λ and kˆ := hu(kˆ, ). R · ˆ Notice that, because of the projector Skˆ, the variation of k in (3.45b) is orthogonal to kˆ, and hence kˆ is preserved under the evolution described by (3.45b)—as it k k should be. Moreover, notice that S is indeed the limit for β 1 of the operator kˆ k k → Sβ deﬁned by (3.29) in Proposition 3.7.

Proof. Using the decompositions (3.42) for λ˙ and (2.28) for the observer ﬁeld u we get ∇ ˙ ∇ ˆ 2 ∇ ˆ ∇ ˆ 1 ∇ ˆ λ˙ λ = ( λ˙ ω)(u + k)+ ω uu + (Pu u)k + ω λ˙ k . (3.47) Here, and in the following, we write ω for ωu(λ˙ ). The u-temporal part of (3.47) is then g(u, ∇ ˙ λ˙ )= ∇ ˙ ω + ω g(u, ∇ ˙ kˆ)= ∇ ˙ ω ω g(∇ ˙ u, kˆ). Using again the λ λ λ λ − λ ∇ ∇ ˙ decomposition (2.28) for u we can write the temporal part of λ˙ λ as

∇ ˙ ∇ 2 ∇ ˆ ∇ ˆ ˆ g(u, λ˙ λ)= λ˙ ω + ω hu( uu + (Pu u)k, k) . (3.48a)

In view of the deﬁnition of the observer derivative the u-spatial part of (3.47) follows at once:

∇ ˙ ∇ ˆ 2 ∇ ˆ ∇ ˆ ∇u ˆ Pu λ˙ λ = ( λ˙ ω)k + ω uu + (Pu u)k + λ˙ k . (3.48b) Note that ∇ ω = ω2∇u ln ω. Using this in (3.48) the equivalency of (3.39) λ˙ λ˙ with (3.45) is easily seen.

As example we consider the propagation of a light ray in a FLRW spacetime (see Section 6.2.1, to which we refer for the notation and the deﬁnitions):

Example 3.3 (Propagation of light in a FLRW spacetime). As reference ﬁeld we take the cosmological vector ﬁeld u = ∂/∂t, whose kinematical properties are summarized in the formula: a˙ ∇u = P . (3.49) a u

u The propagation equation for the frequency is then ∇ ln ωu = hu((˙a/a)Pukˆ, kˆ) = λ˙ − |λ (˙a/a) λ = (ln a)· λ = ∇u ln(a λ) which becomes ∇u ln(a λω) = 0 or, − ◦ − ◦ − λ˙ ◦ λ˙ ◦ equivalently,

(a λ) ωu = const. , (3.50) ◦ which is the well-known red-shift formula in FLRW spacetimes. The propagation equation for kˆ takes the simple form

∇u ˆ λ˙ k =0 , (3.51) which immediately follows from (3.45b), (3.46), and (3.49). This equation can be interpreted saying that the observer ﬁeld u sees the spatial projection of the light path as ‘straight’.

Chapter 4

Spherically symmetric spacetimes

In this chapter we first recall the definition of spherical symmetric spacetime (which we assume throughout being four-dimensional1) and its representation in terms of a warped product manifold. In Section 4.2 we present some objects which arise from the geometric structure of a spherically symmetric spacetime: Misner–Sharp (MS) energy, Kodama vector field, and Kodama current. In particular, we will see that in spherically symmetric spacetimes there is an observer field which is naturally distinguished: the Kodama observer field. How we will show below, this observer field is to be seen as the natural generalization of a stationary observer field (i.e. an observer field collinear with a timelike Killing field) in arbitrary spherically symmetric spacetimes.

4.1 Spherical symmetry

We recall that the isometry group, Isom( , g), of a spacetime ( , g) is the sub- M M group of the diffeomorphism group of , Diff( ), which leaves the metric g in- M M variant: Isom( , g) := φ Diff( ) φ∗g = g . M { ∈ M | } Definition 4.1 (Spherical symmetry). A four-dimensional Lorentzian manifold ( , g) is said to be spherically symmetric if its isometry group, Isom( , g), con- M M tains a subgroup G with the following two properties: (i) G is isomorphic to SO(3) and (ii) each orbit of G is spacelike and two-dimensional (up to some closed proper subset of fixed points). A tensor field T on a spherically symmetric spacetime is said to be spherically symmetric if it is invariant under G, hence if φ∗T = T for all φ G. ∈ From this definition it follows (excluding the case where the orbits of G are diffeomorphic to the two-dimensional real-projective space, see Sect. 3.10 of [Str04]) that a four-dimensional spherically symmetric Lorentzian manifold ( , g) can, at M least locally, be expressed as a warped product = S2 between a two- M B ×R dimensional Lorentzian manifold ( , gB), the ‘base’, and the standard unit two- B 1 In fact, most of the material presented in this chapter can be easily generalized to higher- dimensional spacetimes, provided one substitute in Definition 4.1 SO(3) by SO(n−1), where n is the spacetime dimension.

55 56 Chapter 4. Spherically symmetric spacetimes

2 sphere (S , gS2 ), the ‘ﬁber’ (see Appendix A.9). This means that, at least locally, the manifold is a product = S2 (4.1) M B× and the metric is given by

∗ 2 ∗ g = π (gB ) (R π) σ (g 2 ) . (4.2) − ◦ S Here π and σ are the projections of S2 onto and S2, respectively, and π∗, σ∗ B× B their pull-backs. The warping function R is nothing but the areal radius, since, for a point p , the area of the ﬁber S := p S2, that is the SO(3)-orbit through ∈ B p × p, is just 2 area(Sp)=4πR(p) , (4.3) which is the deﬁning equation for the areal radius. No need to say that R has, in general, nothing to do with the ‘proper distance to the center’, which does not need to exist. For later reference, we note that when we need to coordinatize the sphere we use the coordinates (θ, ϕ) (0, π) [0, 2π) such that the metric takes the form ∈ × 2 2 2 gS2 = dθ + sin θ dϕ . (4.4)

In this contest, a vector field X on at some point (p, q) S2 has then M ∈B× a unique decomposition X = XkB + XkS2 in a component tangent to the ‘leaves’ q = σ−1(q) and a component tangent to the ‘fibers’ p S2 = π−1(p). Arbitrary B× × tensor fields on and on S2 can be lifted to tensor fields on in the standard way (see AppendixB A.9). For brevity, where this does not implMy ambiguities, we often omit lifts or projections and not explicitly distinguish between original and lifted quantities. For example, when referring to a vector ‘tangent to ’ we refer to B a vector in the tangent space of or to the lift thereof in the tangent space of . B M If X is a spherically symmetric vector field on , then the component tangent M to the fibers must vanish: XkS2 = 0 and hence it can be written as the lift of a vector field on . In short, this is because a spherically symmetric tangent vector B field on the two-sphere must vanish identically. To see this just pick up a point q on S2 and consider the stabilizer subgroup (or little group) G G of q, that is q ⊂ the subgroup of G which fixes q. This is isomorphic to SO(2) and, in particular, contains the a rotation of angle π which sends X 2 into X 2 . The invariance kS − kS then implies XkS2 = 0, as stated. Similarly, a spherically symmetric one-form on must necessarily be tangent to and thus it can be written as the lift of some M B one-form on . In particular, if u is a spherically symmetric observer field in a B ∗ four-dimensional spherically symmetric spacetime, this implies that u = π (αB ) for some one-form αB on . From this it immediately follows that u du = ∗ ∗ ∗ B ∧ π (αB ) d(π αB ) = π (αB dαB) = 0, since a three-form on a two-dimensional ∧ ∧ space (the tangent space of ) vanishes identically. Hence we have: B Proposition 4.2 (Spherically symmetric observer, I). In a four-dimensional spherically symmetric spacetime a spherically symmetric observer field is necessarily hypersurface orthogonal.

Since hypersurface orthogonality is equivalent with irrotationality (see Lemma 2.17) we have also: 4.2. Conserved currents and conserved charges 57

Corollary 4.3 (Spherically symmetric observers, II). In a four-dimensional spherically-symmetric spacetime a spherically-symmetric observer ﬁeld is necessarily irrotational. This can also be proved directly by noticing that the rotation two-form (see (2.38)) of a spherically symmetric observer ﬁeld is the projection of a two-form onto a one-dimensional space (the orthogonal complement of u in ) and hence identically zero. B

4.2 Conserved currents and conserved charges

4.2.1 The Misner–Sharp energy In a spherically symmetric spacetime, beside the areal radius R, there is another distinguished function. This is the so-called Misner–Sharp (MS) energy [MS64, HM66], denoted here by E, and defined in purely geometrical manner [CM70a, CM70b] as follows: given a point p of the spacetime, compute the sectional curvature of the plane TSp tangent to the SO(3)-orbit Sp through p at p and multiply this with minus2 the half of the third power of the areal radius. In formulae: E(p) := 1 R3(p)sec(TS ) . (4.5) − 2 p Note that, because of spherical symmetry, the function E is constant on each SO(3)- orbit and hence it can be seen as (the lift of) a function on . From (A.100) we B immediately read off R E = 1+ dR, dR , (4.6) 2 h i which provides a convenient expression for the computation of the MS energy. As we shall see in Section 5.1, in the contest of GR the MS energy has the interpretation of a quasi-local mass which turns out to be useful for both computational and interpretational purposes. At the moment, the MS energy is to be seen just as a function which can be geometrically defined in any spherically symmetric spacetime. Its mathematical properties, which we are now going to show, have nothing to do with any assumptions about the gravitational field equations: they solely rely on the geometric structure of spherically symmetric spacetimes. In Section 5.1 we will show how its physical interpretation is intimately related to Einstein’s field equation. The name ‘Misner–Sharp energy’ seems now to be established in the literature, however one should say that this mass concept goes back at least to [Lem33]3, which gives a coordinate expression for it. Its geometric definition (4.5) was first given in [HM66] and its interpretation as the charge of a conserved current was first derived in [Kod80]. Later, an alternative definition was given in [Zan90]: there it is showed that the MS energy can be defined in terms of the norm of the Killing fields generating the isometry group SO(3), leading directly to (4.6). Further relevant studies of the MS energy are [CM70a, CM70b] and, more recently, [Hay96, Hay98, Bur91]. We show below that E is the charge to a conserved current and, moreover, we show that its differential is related to the Ricci curvature of the spacetime (Lemma 4.4). 2 The minus sign here is just a relict of our signature choice. 3 For an English translation see [Lem97]. 58 Chapter 4. Spherically symmetric spacetimes

4.2.2 The Kodama vector field and the areal volume In a spherically symmetric spacetime one defines the Kodama vector field [Kod80] as the (unique up to a sign) spherically symmetric vector field orthogonal to, and of the same norm of, the gradient of R; hence we put (dropping the lift in the r.h.s.)

k := ⋆dR . (4.7)

Here, and throughout this and the next chapter, ⋆ denotes the Hodge star operator in the base manifold ( , gB) w.r.t. the volume form µB induced by gB ; thereby the B orientation is chosen such that an ordered frame t, s , where t is timelike future- 4 { } pointing and s spacelike outwardly-pointing , is positively oriented: µB(t, s) > 0. The sign choice in (4.7) is such that k is timelike and future-pointing if the gradient of R is spacelike, and the latter is the case if R > 2E (see (4.6)). (We notice also that (4.7) determines k in the region R = 2E.) The orthogonality between k and 6 ∇R, that is k(R)=0 , (4.8) follows immediately from (4.7) using the fact that ⋆ is a skew-adjoint endomorphism on one-forms on a two-dimensional Lorentzian manifold, as discussed in Ap- pendix A.4.1. The lesson to learn at this point is that in a spherically symmetric spacetime, in the region where ∇R is spacelike (i.e. where R > 2E), there is a naturally distinguished observer field given by the normalization of the Kodama vector field. We denote it by u := k −1k (4.9) K k k and refer to it as to the Kodama observer field. The integral curves of uK are then worldlines which are orthogonal to the SO(3)-orbits and which ‘stay’ at constant areal radius. This a key property of the Kodama observer field which makes it the natural substitute for a Killing observer field in an arbitrary spherically symmetric spacetime. Further properties of uK, as well as the features the Kodama vector field has in common with a Killing field, are derived below in Section 4.3. Returning to the Kodama vector field: an immediate but important property of it is that it is conserved, div k =0 . (4.10)

Indeed, using (A.105), (4.8), and (A.26), one has: div k = divB k = δk = ⋆ − d ⋆⋆dR 0. The associated charge is then computed as follows. Let Σ be some ≡ spatial three-dimensional hypersurface (possibly with boundary) which, because of spherical symmetry, decomposes as Σ = σ S2, where σ is some spatial curve × (segment) in . Recall that the charge associated to a conserved current X is given B by QX (Σ) := iX µ, where µ is the volume form on . Because of spherical Σ M 2 symmetry, the latter decomposes as µ = µB R µ 2 , where µB and µ 2 are the R ∧ S S volume forms on and on the unit two-sphere, respectively. After integration of B 2 the spherical part and since ikµB = ⋆k = dR, the charge to k is simply σ 4πR dR and hence R 4π Q (Σ) = d R3 , (4.11) k 3 Zσ 4 A vector ﬁeld X on a spherically symmetric spacetime is called outwardly-pointing if it points in a direction of increasing areal radius, that is if dR(X) > 0. 4.2. Conserved currents and conserved charges 59 which says that the charge to k is the so-called ‘areal volume’. This is the ‘volume’ computed via the ﬂat-space volume formula from the areal radius. No need to say that this has, in general, nothing to do with the ‘proper volume contained inside the SO(3)-orbits’, which does not need to exist.

4.2.3 The Kodama current and the Misner–Sharp energy The Kodama current is deﬁned as the vector ﬁeld j on tangent to with M B j := 1 Ein(k, ) , (4.12) 8π · where Ein is the spacetime’s Einstein tensor and the numerical factor is introduced for later convenience. To show that this is a conserved current we need the following fundamental Lemma 4.4. In a spherically symmetric spacetime the following relation holds between the MS energy and the Kodama current:

dE =4πR2 ⋆ j . (4.13)

Proof. One has just to compute the differential of (4.6). For this we first recall the identity d dR, dR = 2 BHess(R)(∇R, ), which follows from d( dR, dR )(X) = h i · B h i X(g(∇R, ∇R)) = 2 gB(∇X ∇R, ∇R) = 2 Hess(R)(∇R, X) for any vector field X tangent to . Then we have dE = (E/R)dR + (R/2)d dR, dR = B h i 2 B Ri∇R (E/R )gB + Hess(R) . The two-tensor in square brackets is nothing but

(R/2) ⋆ Ein B ⋆ (see (A.103)), where ⋆ Ein B ⋆ is the double dual (in ) of the − k k B -part of the spacetime’s Einstein tensor. Hence we have: B 1 2 dE = R i∇ ⋆ Ein B ⋆ . (4.14) − 2 R k 5 The proof is concluded noticing that i∇ ⋆ Ein B ⋆ = dR ⋆ Ein B ⋆ = (⋆dR) − R k − · k · Ein B ⋆ = k Ein B ⋆ = ⋆(Ein(k, ))=8π⋆ j. Where we used that ⋆ is skew-adjoint k · k · on one-forms, the deﬁnition of the Kodama vector ﬁeld (4.7), and of the Kodama current (4.12).

Equation (4.13) is a key relation for the study of spherically symmetric spacetimes. It will also be of crucial importance for the interpretation of the MS energy in the contest of GR. Now, from (4.13) it follows immediately that

j(E)=0 , (4.15) which means that the vector ﬁeld j is tangent to the hypersurfaces in (curves M in ) of constant MS energy. Moreover, (4.13) implies that j is conserved, B div j =0 , (4.16) where the divergence is here taken on the spacetime ( , g). To see this, just M compute the divergence of j with (A.105) and use the Hodge-dual version of (4.13): −2 2 −2 2 2 −1 div j = R divB(R j)= R δ(R j)=(4πR ) δ ⋆ dE 0. ≡ 5 Recall that the dot denotes the contraction, with the help of the metric, of the last slot of the tensor on the left of the dot with the ﬁrst slot of the tensor on the right. 60 Chapter 4. Spherically symmetric spacetimes

We can now compute the charge to the conserved current j. As above in Sec- tion 4.2.2, let Σ = σ S2 be some spatial spherically-symmetric three-dimensional × 2 hypersurface. After integration of the spherical part and since 4πR ij µB = 4πR2 ⋆ j = dE, we get

Qj (Σ) = dE (4.17) Zσ which means that the charge to j is given by the MS energy. This also justiﬁes the interpretation of the MS energy as a quantity associated to the ‘interior’ of the considered sphere of symmetry (SO(3)-orbit): due to (4.16), the charge does not depend how one choose the spatial slice to deﬁne the interior.

4.3 More on the Kodama vector ﬁeld

We look now in more detail at the Kodama vector ﬁeld and at the related observer ﬁeld. This turns out to be very instructive.

4.3.1 Kodama vector ﬁeld and Killing ﬁelds

The Kodama vector field has some features which make it resemble very much a spherically symmetric Killing field. First, like a Killing field, it has vanishing divergence (see (4.10)). Second, like a spherically symmetric Killing field, it is orthogonal to the gradient of the areal radius6 (see (4.8)). Moreover, since is B two-dimensional, the (projection onto of the) Kodama vector field spans the B orthogonal complement to ∇R in the tangential space of . That is, we have the B following

Proposition 4.5. Let ( , g) be a spherically symmetric spacetime. Then, in the M region where ∇R = 0, a spherically symmetric Killing field (that is a Killing field 6 for ( , g) tangent to ) is necessarily collinear with the Kodama vector field. M B Since the Kodama vector field k is orthogonal to ∇R, to find out under which conditions it is a Killing field, it suffices to consider the Lie derivative LkgB of the metric on (Lemma A.9). The latter, in the region where R =2E, can be written as B 6 2E −1 gB = 1 (k k dR dR) . (4.18) − R ⊗ − ⊗ This is because k and dR are per definition orthogonal and their squared norm is given by (1 2E/R) (see (4.6)). For the computation of the Lie derivative of (4.18) − along k one first computes, at best with Cartan’s formula (see (A.14)):

LkdR =0 , (4.19)

1 Lkk = Ri∇ Ein B trgB (Ein B) gB , (4.20) R k − 2 k In the second equation one also uses (4.7), (A.102), and (4.14). Then it follows,

6 That a spherically symmetric Killing field (that is a Killing field for (M, g) which is tangent to B) is orthogonal to the gradient of the areal radius is a direct consequence of Lemma A.9. 4.3. More on the Kodama vector field 61 using again (4.14):

R 2E −1 LkgB = 1 Ein(k, ∇R)(k k + dR dR) 2 − R ∨ ∨ (4.21) Ein(k, k)+ Ein(∇R, ∇R) k dR . − ∨ We note that the coefficient of the last term on the r.h.s. of (4.21) is not proportional to the trace of EinkB: this will become important below. From (4.21) and (4.8) it follows, in particular, that if the spacetime is Ricci-flat7 the Kodama vector field is a Killing field for g. Since in the region R> 2E the Kodama vector field is timelike and, because of spherical symmetry, it is also hypersurface orthogonal, we have the following

Theorem 4.6 (Birkhoff Theorem, I). A Ricci-flat spherically symmetric spacetime, in the region R> 2E, is necessarily static and the timelike Killing field is given by the Kodama vector field. This is certainly a very nice alternative derivation of Birkhoff’s theorem, where as by-product the timelike Killing field is explicitely provided. Looking at the r.h.s. of (4.21) we notice that the above theorem remains valid if we substitute the condition ‘Ricci-flat’ by ‘Ricci-isotropic’. With Ricci-isotropic we denote a semi-Riemannian manifold whose Ricci tensor is proportional8 to the metric: Ric g. (To be precise, a little linear algebra shows that, assuming ∝ R =2E, the r.h.s. of (4.21) vanishes iff Ein B gB, that is, iff Ric B gB.) This 6 k ∝ k ∝ gives a slight generalization of the above theorem:

Theorem 4.7 (Birkhoff Theorem, II). A Ricci-isotropic spherically symmetric spacetime, in the region R> 2E, is necessarily static and the timelike Killing field is given by the Kodama vector field. In the contest of GR, this is nothing but the generalization of Birkhoff’s Theorem including the cosmological term in Einstein’s equation.

4.3.2 Properties of the Kodama observer ﬁeld

We turn now to the Kodama observer field u = k −1k and compute its accel- K k k eration, expansion, and shear. We recall that this observer field is defined in the region where the condition R> 2E holds. This we shall assume in the following. First we recall that, because of (4.8), we have obviously

dR(uK)=0 , (4.22) which can be interpreted as saying that the ‘radial velocity’ (referred to the areal radius) of uK vanishes. For the acceleration we have:

−1/2 ∇ 2E E R u uK = 1 + Ein(e , e ) e , (4.23) K − R R2 2 R R R 7 One calls ‘Ricci-flat’ a semi-Riemannian manifold with vanishing Ricci tensor. Note that the Einstein tensor vanishes iff the Ricci tensor does. 8 Note that in dimensions n 6= 2, which is the case here, the proportionality factor must be a constant, as one can show using the divergence-freeness of the Einstein tensor. In this case the manifold is said to be ‘Einstein’. 62 Chapter 4. Spherically symmetric spacetimes where e := ∇R −1∇R (4.24) R −k k is the outwardly-pointing9 normalized vector field collinear with the gradient of the areal radius. Herewith, in the region R> 2E,

eK := u , eK := e (4.25) { 0 K 1 R} defines a positively oriented orthonormal basis of T , to which we will refer to B as the Kodama basis. Expression (4.23) can be computed noticing that, since uK ∇ is normalized, its acceleration must be orthogonal to it, which implies uK uK = ∇ −1/2 B gB( u uK, e )e = (1 2E/R) Hess(R)(uK, uK)e , where in the last − K R R − − R step we use (4.22). We then use (A.103) to express the Hessian of the areal radius in terms of the MS energy and the Einstein tensor. The first term on the r.h.s. of (4.23), in leading order in E/R, looks very much like the Newtonian expression for the acceleration (outwardly pointing) of a rocket which stays at constant distance R from a central object of mass E. In the contest of GR, where the Einstein tensor is related to the matter energy-momentum T tensor via Einstein’s equation, the factor in the big parentheses becomes E/R˜ 2, where E˜ := E +4πR3p and p := T (eR, eR) is the radial pressure. It is interesting to note that E˜ appears in the Tolman–Oppenheimer–Volkoff (TOV) equation10 where it plays the role of the ‘active gravitational mass’ contained inside the sphere of areal radius R. −1 The expansion of uK is given by θu = div uK = k( k ). Using (4.14) we get K k k

R 2E −1/2 θu = 1 Ein(uK, e ) . (4.26) K 2 − R R In particular, if the spacetime is Ricci-isotropic, the expansion vanishes, which is of course in accord to the fact that in this case k is Killing. Finally, recall that the shear of a spherically symmetric observer ﬁeld is determined by a scalar, which can be computed via (5.32). Because of (4.22) we have then that the shear scalar is just minus one-third of the expansion (4.26).

9 Note that, because of our metric signature choice, ∇R is inwardly pointing in the region R> 2E. 10 See e.g. p. 377 in [Str04]. There, the function M is nothing than the MS energy, denoted here by E. Part II

Applications to general-relativistic cosmology and space physics

Chapter 5

Spherical symmetry in General Relativity

In the present chapter we apply the formalism for spherically symmetric spacetimes developed in Chapter 4 to GR. This allows, in particular, to simplify the Einstein equation and, using the latter, to give a physical interpretation to the MS energy. This is done in Section 5.1, where we also introduce a decomposition of the MS energy in its Weyl and Ricci part, which are interpreted in Section 5.1.2.2. In Sec- tion 5.2 we apply the tools of Chapter 4 to the problem of spacetimes matching, under the assumption that these are spherically symmetric. We derive there a set of junction conditions which, under the assumption of spherical symmetry, is equivalent to the Darmois junction conditions. These new junction conditions involve only scalar quantities, are easy to verify, and have a nice physical interpretation. This result has been published in [CG10b].

5.1 Misner–Sharp energy in General Relativity

In this section we ﬁrst give the expression for the Einstein equation in case of spherical symmetry. We then relate the MS energy to the other energy (mass) concepts common to GR. We decompose the MS energy in its Ricci- and Weyl- parts and discuss their physical interpretation. For the latter we also consider the case in which matter is given by a perfect ﬂuid.

5.1.1 Einstein equation in case of spherical symmetry A general spherically symmetric matter energy-momentum tensor has the form

2 T = TB + p R gS2 , (5.1) where p is the spherical part of the pressure. Hence, using the decomposition (A.101) of the Einstein’s tensor found in Appendix A.10 the Einstein equation (C.4) takes the form

2 E B 2 B + ∆ R gB Hess(R)=8πTB (5.2a) R R2 − R BScal 1 B∆ R =8πp . (5.2b) 2 − R

65 66 Chapter 5. Spherical symmetry in General Relativity

Using the trace of the ﬁrst equation,

1 B 2E ∆ R + =4π tr TB , (5.3) R R2 gB to eliminate B∆ R, one can write (5.2) in the equivalent form

1 E B gB + Hess(R) = 4π⋆ TB⋆ (5.4a) R R2 − BScal 2E + =4π tr TB +2p) . (5.4b) 2 R2 gB Here, ⋆ denotes the Hodge dual for ( , gB) introduced in Section 4.2.2. Notice that B on the r.h.s. of the ﬁrst equation we used the identity (A.31). Using Einstein’s equation the Kodama current (4.12) becomes

j = T (k, ) . (5.5) · Herewith the expression (4.13) for the diﬀerential of the MS energy is nothing but the double Hodge-dual of the -part of Einstein’s equation fed with ∇R. B Finally, the integration condition div T = 0 for the energy-momentum tensor (5.1) reads as 2 2 divB(R TB)+ p d(R )=0 . (5.6)

5.1.2 Interpretation of the Misner–Sharp energy

In the following we ﬁrst relate the MS energy to the other relevant energy (mass) concepts of GR.

5.1.2.1 Misner–Sharp energy and other energy definitions An important feature of the MS energy is that, in an asymptotically flat spacetime, it converges to the Bondi-mass at null infinity and to the ADM mass at spatial infinity (see Proposition 3 in [Hay96]). Moreover, as we will now show, the MS energy can be seen as the specialization to the spherical symmetric case of the Hawking quasi-local mass [Haw68]. The latter is a quantity associated to a spatial two-sphere, S, in an arbitrary spacetime. It is defined by

area(S) 1 M (S) := 1+ θ+θ−µ . (5.7) H 16π 2π S r ZS ± ∇ ± Here, θ := trS2 ( l )/2 are, respectively, the expansions of the outgoing and ingoing future-pointing null vector ﬁelds l± normal to S, the latter being partially normalized such that g(l+, l−) = 1 (there remains the freedom to rescale l± → α±1l±, where α is a positive real-valued function). In the special case of spherical symmetry we take S to be an orbit of the rotation group. Then we clearly have area(S)=4πR2. It is also obvious that the metric of the base , evaluated on S, B can simply be written in the form

+ − − + gB = l l + l l . (5.8) ⊗ ⊗ 5.1. Misner–Sharp energy in General Relativity 67

± −1 ± ± Now, for V tangent to S, (A.85b) gives ∇V l = R l (R)V so that θ = R−1l±(R). Hence we have:

2 θ+θ− =2R−2 dR(l+)dR(l−)= dR, dR /R2 , (5.9) h i where we used (5.8), or rather its contravariant version, in the last step. Equa- tion (4.6) now establishes the equality between the MS energy at a point p and the

Hawking quasi-local mass of Sp, where Sp denotes the SO(3)-orbit through p:

E(p)= MH(Sp) . (5.10)

This equality provides another way to show that the MS energy converges to the Bondi-mass at null inﬁnity and to the ADM mass at spatial inﬁnity (see Section 6.1.2 in [Sza04] and references therein).

5.1.2.2 Ricci and Weyl decomposition of the Misner–Sharp energy

As is the case for the Hawking quasi-local mass, the MS energy (4.5) can be naturally decomposed into a Ricci and a Weyl part:

E = ER + EW , (5.11) where E (p) := 1 R3(p)sec (TS ) , (5.12a) R − 2 R p E (p) := 1 R3(p)sec (TS ) . (5.12b) W − 2 W p

Here secR(TSp) and secW(TSp) denote, respectively, the Ricci and the Weyl parts of the sectional curvature of the plane tangential to the SO(3)-orbit Sp through the point p. These are obtained inserting the decomposition of the Riemann tensor (A.48) in the deﬁnition of the sectional curvature (A.55). The Ricci part of the MS energy is determined by the local matter distribution via Einstein’s equation: Using expressions (5.1) and (4.2) for an arbitrary spherically symmetric energy- momentum tensor and, respectively, metric in (C.6) one gets

4π 3 E = R (trBTB + p) . (5.13) R 3

For the Weyl part of the MS energy we have, in view of (A.99a), that

E = 1 R3w , (5.14) W − 2 where w is given by (A.99b) or (A.99c). Hence, in particular, the Weyl tensor vanishes iﬀ E does. Since the square of the Weyl tensor is Weyl, Weyl W h i ≡ αβγδ 2 WαβγδW = 12w , with (5.14) we obtain the nice expression

E2 Weyl, Weyl = 48 W . (5.15) h i R6 From this we see that, in a spherically symmetric spacetime, the non-vanishing of E (that is the non-vanishing of the Weyl tensor) for R 0 implies a curvature W → singularity at R = 0. 68 Chapter 5. Spherical symmetry in General Relativity

To gain a better physical understanding of the Weyl part of the MS energy we ∇2 take a look at the equation of geodesic deviation us = Au(s) (see (2.51)), where u is a geodesic observer ﬁeld, s the u-spatial separation vector between two nearby integral curves of u, and Au the tidal operator deﬁned by (2.42). The latter can now be decomposed as Au = Bu + Cu in its Ricci- and Weyl-part, denoted here by Bu and Cu, respectively. For an arbitrary spherically symmetric spacetime the latter is given by (see (A.99) and (5.14))

EW B S2 Cu = 2P P . (5.16) R3 u − u B S2 2 where P and P are, respectively, the and S parts of the u-spatial projector Pu u u B (see (2.1b)). The former can be seen as the projection in ‘radial direction’, whereas the latter is the projection onto the planes tangential to the SO(3)-orbits. Equa- tion (5.16) is exactly the same expression one gets in Newtonian gravity—provided one identiﬁes EW with the mass of the central object. The spatial endomorphism Cu just describes the familiar volume-preserving tidal deformation which produces an expansion in radial direction and a contraction in the plane tangential to the SO(3)-orbits. 1 Concerning the Ricci part Bu, in the case where u is the velocity ﬁeld of dust , making use of Einstein’s equation (see (C.6)) we have:

4π Bu = ̺ Pu . (5.17) − 3

This just says that the local effect of matter (here given by dust) is an isotropic contraction. At this point, some words of explanation are due in regard to the physical meaning of the split (5.11). Since Einstein’s equation is a local relation between the Ricci tensor and the energy-momentum tensor for the matter, which determines one in terms of the other, it is obvious that the Ricci part of the MS energy is also locally determined by the matter’s energy-momentum tensor. Likewise, it is clear that this cannot hold for the Weyl part, since, by definition, the Weyl curvature forms that part of the Riemann tensor which is complementary in information to the Ricci part. In general, in four spacetime dimensions, each part accounts for ten independent components, which together make up for the 20 independent components of the Riemann tensor. In case of spherical symmetry, the Weyl part has only a single independent component, which can then be identified with the Weyl part of the MS energy (see (5.14)). Hence, the latter depends on other features (parameters) of the solution than those fixed by the local distribution of the matter’s energy and momentum. A simple example is given by the vacuum exterior Schwarzschild solution, where the Ricci part of the MS energy clearly vanishes and the mass of the black hole is given by the Weyl part alone. Now, the last statement remains true if we match the vacuum exterior to a perfect-fluid interior Schwarzschild solution, whenever the MS energy is measured on a sphere outside the star’s surface. In this case the Weyl part is a non-local measure for the integrated mass contrast. We derive below in Section 5.1.3 general formulae for the radial and temporal variation of the MS energy and its Weyl and

1 Recall that dust particles moves along geodesics by the Euler equation. 5.1. Misner–Sharp energy in General Relativity 69

Ricci parts. This formulae show that the radial variation of the Weyl part vanishes at those points where the mass density is (locally) spatially constant (see (5.30b)), which justiﬁes its interpretation as an integrated measure for spatial mass-density contrasts. For example, if we match the exterior Schwarzschild solution to an interior solution with incompressible ﬂuid (i.e. constant mass density) EW will be zero if evaluated inside the star. Hence, in the sense of an integrated mass-density contrast, it receives all its contribution from the radial discontinuity of the mass density at the star’s surface.

5.1.3 Spherically symmetric perfect fluids We specialize now to a perfect fluid, which is described by a four-velocity vector field u, density ̺, and pressure p. Recall that the energy-momentum tensor of a perfect fluid is given by (1.1). In case of spherical symmetry u is tangent to the basis manifold and thus the matter energy-momentum tensor decomposes as B 2 T = ̺ u u + p (u u gB)+ p R g 2 , (5.18) ⊗ ⊗ − S from which one can read off the part tangent to : B

TB = ̺ u u + p (u u gB ) . (5.19) ⊗ ⊗ − Usually, the description is to be completed with the specification of an equation of state relating density and pressure. We will not assume any equation of state yet, since in some cases (e.g. McVittie spacetime) this happens to be determined by Einstein’s equation. Inserting (5.19) in (5.13) we get for the Ricci part of the MS energy the simple expression 4π E = R3̺ . (5.20) R 3 Also the expression (4.13) for the differential of the MS energy simplifies in case of a perfect fluid. Using (5.19), the Kodama current (5.5) becomes

j = (̺ + p)g(k, u) u p k . (5.21) − It is useful to introduce an adapted orthonormal basis u, e tangent to the basis { } manifold, where u is the velocity vector field of the fluid and e is chosen to be outwardly-pointing (meaning that it points in direction of increasing areal radius: dR(e) > 0). Note that, because of our choice of orientation (see the discussion 2 below (4.7)), the volume form on ( , gB) is µB = u e. From this we have B − ∧ ⋆u = e and ⋆e = u. Using this and the definition of the Kodama vector − − field (4.7) we have g(k, u) = k, u = ⋆dR, u = dR,⋆u = dR, e = dR(e) h i h i −h i h i and hence, the Hodge star of the Kodama current becomes

⋆ j = p dR(u)u ̺ dR(e)e , (5.22) − − which, inserted in (4.13), gives the following expression for the diﬀerential of the MS energy for a perfect ﬂuid:

dE = 4πR2 p dR(u)u + ̺ dR(e)e . (5.23) − 2 This implies in fact that {u, e} is positive oriented, since µB(u, e) = 1 > 0. 70 Chapter 5. Spherical symmetry in General Relativity

From this one reads oﬀ the variation of the MS energy along u and, respectively, along e:

dE(u)= 4πR2 p dR(u) , (5.24a) − dE(e) =+4πR2 ̺ dR(e) . (5.24b)

These expressions have a good physical interpretation: since the matter moves along u, (5.24a) expresses the fact that the energy can only increase (decrease) if the motion along u does (releases) work against (with) the action of the pressure. Equation (5.24b) expresses the almost obvious increase (decrease) of gravitational mass with increase (decrease) of volume in the rest system of the matter. We said ‘almost’ because 4πR2dR(e) is not quite the increment of proper volume. The diﬀerence accounts for the fact that kinetic and gravitational binding energy are themselves gravitationally active. To see that this is indeed what (5.24b) implies, let p be some point in spacetime and Sp the two-sphere of spherical symmetry through p. Assume Sp to have a regular interior, that is, that Sp bounds a 3-ball Bp in the hypersurface Σ orthogonal to u. Except for the origin of Bp, we can write B = σ S2, where σ is a spacelike curve in orthogonal to u, going from the center p × B of symmetry to π(p). Using the expression E = (R/2)(1 + (dR(u))2 (dR(e))2) − for the MS energy to eliminate dR(e) in (5.24b), integrating the latter over σ, and re-expressing the result as a volume integral, one gets:

2E 1/2 E(p)= ̺ 1 + (dR(u))2 µ . (5.25) − R Σ ZBp One sees that the MS energy contains the contribution from the proper mass contained in the ball Bp,

M(p)= ̺ µΣ , (5.26) ZBp as well as contributions from the ‘kinetic’ and ‘potential’ energy [MS64, Hay96]. In a Newtonian approximation, that is for small ‘velocity’ dR(u) and weak ﬁeld (small E/R) one can expand the square root in (5.25) and gets, in leading order:

̺M E(p) ̺ + 1 ̺(dR(u))2 µ . (5.27) ≈ 2 − R Σ ZBp In this approximation the MS energy is therefore just the sum of the proper mass and the Newtonian kinetic and potential energies contained in the ball Bp. This provides a sound justification for the interpretation of the MS energy as the active gravitational energy. At this point we can compute also the differentials of the two parts (5.12) of the MS energy separately. The differential of the Ricci part follows directly from (5.20):

2 1 dER =4πR ̺ dR + 3 R d̺ (5.28) and the diﬀerential of the Weyl part is just the diﬀerence of this with (5.23):

dE = 4πR2 (̺ + p)dR(u) u + 1 R d̺ . (5.29) W − 3 5.1. Misner–Sharp energy in General Relativity 71

Its components in the directions u and e are then

dE (u)= 4πR2(̺ + p)dR(u) 4π R3d̺(u) , (5.30a) W − − 3 dE (e) = 4π R3d̺(e) . (5.30b) W − 3 It is now instructive to express the variation along u of the Ricci and Weyl parts of the MS energy in terms of the kinematical properties of the ﬂuid velocity u. Recall that, because of spherical symmetry, the rotation tensor vanishes identically and the shear tensor has only one independent component. The kinematical quantities reduces thus to two scalars: the expansion

θ = div u (5.31) and the shear scalar dR(u) 1 σ := θ . (5.32) R − 3 The shear tensor is then given3 by the u-spatial trace-free endomorphism

σ = σ P S2 2 P B , (5.33) u − u B S2 2 where Pu and Pu are, respectively, the and S parts of the u-spatial projec- B B tor (2.1b) (compare with the discussion below (5.16)). Note that Pu is nothing but the projector onto the one-dimensional space parallel to e. We recall that the divergence-freeness of the energy-momentum tensor (5.18) is equivalent to

(̺ + p) θ = d̺(u) (5.34a) − (̺ + p) b = dp(e) , (5.34b) − where b := g(∇uu, e) is the acceleration (scalar) of u in positive radial direction − (the minus sign in the latter formula is because the metric is negative deﬁnite in spatial directions). Now, using (5.34a) and (5.32) we get: 4π dE (u) = R3(3̺ σ pθ) (5.35a) R 3 − 4π dE (u)= R3(̺ + p)3σ (5.35b) W − 3

With the equations just derived we can now say when the MS energy, and its Ricci and Weyl parts, are temporally or spatially constant. Here, by temporally (spatially) constant we mean that the variation in direction of u (e) vanishes. We collect the results in the following

Theorem 5.1. Consider a spherically symmetric ﬂuid with ̺ + p =0 and restrict 6 to the region where dR is spacelike. Then for the MS energy E and its Ricci and

Weyl parts ER and EW the following statements hold true: (i) E is temporally constant iﬀ p =0 or dR(u)=0; (ii) E is spatially constant iﬀ ̺ =0;

(iii) EW is temporally constant iﬀ σ =0;

3 Expressions (5.32) and (5.33) are a by-product of the computations carried out in Appendix B.1. 72 Chapter 5. Spherical symmetry in General Relativity

(iv) EW is spatially constant iﬀ ̺ is spatially constant; 3 (v) ER is temporally (spatially) constant iﬀ R ̺ is temporally (spatially) constant.

The proof is a straightforward application of the formulae just derived above. Note that the assumption ̺ + p = 0 is needed only for (iii). The assumption that dR is 6 spacelike is needed only for (ii): If dR is spacelike, then for any spacelike spherically symmetric vector e (hence tangent to the basis manifold ) it holds dR(e) = 0, B 6 since in a two-dimensional Lorentzian manifold any two spacelike vectors are linearly dependent.

5.2 Spherically-symmetric matchings

Because of the nonlinearity of Einstein’s equations it is a very difficult task to find exact solutions which model realistic situations. In fact, exact solutions are only known for highly idealized situations, e.g. in presence of some symmetry of the spacetime which reduces the degrees of freedom and hence simplifies the field equations. One way to enlarge the class of exact solutions at our disposal is to glue together pieces of already known solutions across suitably chosen hypersurfaces (of codimension one), along which the matter distribution is in general allowed to become singular if one wants to take into account surface layers. This approach was pioneered by Lanczos [Lan24] and put into geometric form by Darmois [Dar27] (for the case without surface layers) and Israel [Isr66] (for the general case including surface layers); see also Sec. 21.13 of [MTW73] and Sect. 3.8 of [SKM+03]. In this section, under the assumption of spherical symmetry and that the matching hypersurface is non-null, we present a new alternative set of junction conditions which are equivalent to the old ones. The new conditions only involve scalar quantities, are easy to verify, and have a nice physical interpretation. This results have been published in [CG10b]. Here we shall restrict to piecewise continuous matter distributions. This means that we allow for jumps, as e.g. happens at the surface of stars or planets, but exclude singular (δ-distribution-like) surface layers. Einstein’s equation is said to be satisfied for piecewise twice continuously differentiable fields, if this, at the location of the matching hypersurface, is replaced by its one-dimensional ε-interval integral in normal direction to the hypersurface. The conditions that two twice continuously differentiable solutions (in the ordinary sense) can be matched into a piecewise twice continuously differentiable solution (in the sense just explained above) are then simply given by the so-called

Darmois junction conditions (DJC). For a non-null matching hypersurface Γ ,

(i) the induced metric gΓ and (ii) the extrinsic curvature KΓ shall be continuous through Γ .

Before we go on, some explanatory words are needed about the notion of ‘continuity through Γ ’. This is indeed just an abbreviation, which we will now explain. Gluing together two pieces of spacetimes means the following: initially one has two spacetimes, say ( +, g+) and ( −, g−), with oriented boundaries Γ + and Γ −, M M respectively. Given a diﬀeomorphism φ : Γ + Γ − between the boundaries, the → glued spacetime is the quotient of the disjoint union of + and − under the M M identiﬁcation of each point of p Γ + with φ(p) Γ −. The matching hypersurface ∈ ∈ 5.2. Spherically-symmetric matchings 73

Γ is now the common image of Γ + and Γ − after identification in the quotient spacetime. Now, a tensor field T is said to be continuous through Γ if T + equals T − |Γ |Γ under the push-forward action of the diffeomorphism φ, hence if φ (T + )= T − . ∗ |Γ |Γ The physical content of the DJC is understood as follows: they imply, via Ein- stein’s equation, that

T (n, ) is continuous through Γ (5.36) · where n is, here and in the following, a continuous choice of unit normal of Γ . The relation (5.36) follows directly from the expressions (A.71) for the Einstein tensor given in Appendix A.7. In the case where Γ is timelike, and hence n is spacelike, (5.36) just states the continuity of the normal components for the energy-momentum flux-densities, whereas their tangential components together with the energy density may jump across Γ . In the absence of surface layers this continuity condition is just a physically obvious consequence of local energy-momentum conservation, whereas jumps in, say, the energy-density and tangential stress must clearly be allowed for. For completeness, we note that for a spacelike matching hypersurface, n is timelike and can be chosen to be future-pointing. Doing so, n can be interpreted as an observer field defined on Γ ; then (5.36) states the continuity of the densities of energy and momentum as measured by n, whereas the corresponding currents may jump. We now restrict or attention to the case where the spacetimes to be glued are spherically symmetric, as well as their matching hypersurfaces (or boundaries), meaning that the latter are left invariant, as set, under the action of SO(3). Re- calling that the structure of a spherically symmetric spacetime is that of a warped product = S2 between a two-dimensional Lorentzian manifold and the M B×R B two-sphere by means of the warping function (the areal radius) R, the matching hypersurfaces have the structure Γ = γ S2, where γ := π(Γ ) is the matching ×R curve in . The DJC should then reduce to appropriate conditions along the curve B γ. Indeed, Theorem 5.2 below shows that, in the spherically symmetric case, the DJC are equivalent to the following Spherically symmetric junction conditions (SSJC). Let Γ be a smooth, non- null, spherically symmetric matching hypersurface between two spherically symmetric spacetimes and n a continuous choice of unit normal vector field on Γ . Denote with γ the projection of Γ onto . Moreover, let v the (unique up to a sign) B spherically symmetric, unit vector field on Γ orthogonal to n. The following four functions (i) the arc-length of γ,

(ii) the extrinsic curvature of γ in : g(n, ∇vv), B (iii) the areal radius R, (iv) the MS energy E, shall be continuous through the matching curve γ. First, notice that, since n and v are spherically symmetric and hence tangent to , in view of (A.85) and (A.68) we have: B ∇ B∇ B g(n, vv)= gB(nB, vB vB)= ε(nB) Kγ (vB , vB) . (5.37)

Hence, the quantity in (ii) is indeed (up to a possible sign) the extrinsic curvature of the curve γ in . Second, note that this quantity, being quadratic in v, B 74 Chapter 5. Spherical symmetry in General Relativity does not depend on the sign choice of v. Third, since the matching hypersurface has the structure γ S2, the words ‘continuous through the curve γ’ can be in- × terchanged with the words ‘continuous through the hypersurface Γ ’, depending on ones preference to think four- or two-dimensional. A great advantage of the SSJC is that they are very easy to verify: one simply has to impose continuity on four scalars along the matching curve in the two- dimensional base manifold . Dealing with scalars, since their value is independent B on the particular coordinate choice, one does not need to worry about introducing new coordinates in both spacetimes to be glued, in order to get the different metrics in a form which is comparable. This is indeed an ingrate task: in general, these coordinates are only needed in order to check if the junction conditions are satisfied, and for nothing more. In presence of spherical symmetry all this can be circumvented by using our new junction conditions. Furthermore, the SSJC have a very nice physical interpretation. The continuity condition for the MS energy is interpreted as continuity condition for the active gravitational energy. Note that this, in the special case of FLRW spacetimes, is essentially the statement of the Eisenstaedt Theorem (see Theorem 6.1 below). The continuity condition of both, the areal radius as well as of the MS energy, can be read as equilibrium condition for the gravitational pull acting from opposite directions onto (fictitious) test masses at the location of the matching surface. Concerning the continuity of the extrinsic curvature of the matching curve, we note the following: in the case where the matching hypersurface Γ is timelike, γ = π(Γ ) is a timelike curve in and v is tangent to it. Choosing, w.l.o.g., v to be future-pointing, one can think B of it as the ‘matching observer-field’ (the trace of its worldline being equal to γ). Hence, in the timelike case, the extrinsic curvature is nothing but the acceleration of the matching observer. In the spatial case, on the contrary, n is timelike, γ is spacelike (and v tangent to it). One can choose n to be future-pointing and then think of it as an observer field defined along the spatial (1-dimensional) slice γ. In view of (A.69) one then sees that the extrinsic curvature of γ is exactly4 the shear-expansion of n in the ‘radial direction’ v. We now need to prove the following

Theorem 5.2 (Equivalence of the junction conditions). Let Γ be a smooth, non- null, spherically symmetric matching hypersurface between two spherically symmetric spacetimes and n a continuous choice of unit normal vector ﬁeld on Γ . Assume, moreover, that the areal radii of the two spacetimes are C1 functions in an open neighborhood of the matching hypersurfaces. Then the DJC are equivalent to the SSJC.

Proof. The proof essentially consists in writing down induced metric and extrinsic curvature for a (non-null) spherically symmetric hypersurface in a spherically symmetric spacetime. This is most easily done by introducing an adapted orthonormal frame. We first consider the case where Γ is timelike, hence γ = π(Γ ) is a timelike curve in . The following construction shall be carried out in both spacetimes. One B defines v as in the SSJC, hence as the (unique up to a sign) spherically symmetric, unit vector field on Γ orthogonal to n. That is v, seen as a vector field on , is B 4 Recall that, because of our signature choice, the restriction of the metric to spacelike directions is negative definite. 5.2. Spherically-symmetric matchings 75 tangent to γ. Since n is spacelike, v is timelike. The ambient metric can be then written as 2 g = v v n n R g 2 (5.38) ⊗ − ⊗ − S so that the induced metric (compare Appendix A.7 and (A.65) on Γ is then

2 g = v v R g 2 , (5.39) Γ ⊗ − S where we used that n is spacelike and hence ε(n) = 1. For the extrinsic curva- − ture (A.68), using (A.85) and the fact that v is spherically symmetric and hence tangent to , one has the decomposition B ∇ K = g(n, vv) v v R dR(n) gS2 . (5.40) Γ − ⊗ − Notice that there are no mixed terms—as it should be according to spherical symmetry. Now, from the above expressions (5.39) and (5.40) it follows that the DJC, hence the continuity of gΓ and KΓ , are equivalent to the continuity of the following four functions: (a) the arc-length of γ, (b) R, (c) dR(n), and

(d) g(n, ∇vv). The statement of the theorem will now follow from the following expression of the MS energy (4.6): R E = 1 + (dR(v))2 (dR(n))2 . (5.41) 2 − For this, simply note that if R is continuous through Γ (recall the definition of this concept below the definition of DJC in Section 5.2) the same holds for its derivatives tangent to Γ . In particular, dR(v) is continuous through Γ and hence we may substitute dR(n) by the MS energy in the above list (a)–(d). This completes the proof for timelike Γ . In the case of spacelike Γ the unit normal n is timelike and v is chosen as the unique (up to a sign) spherically symmetric unit vector field on Γ orthonormal to n. Then v is a spacelike ‘radial’ unit vector field orthogonal to the SO(3)-orbits. The proof now proceeds analogously to the timelike case. We merely list the expressions for the ambient metric

2 g = n n v v R g 2 , ⊗ − ⊗ − S the induced metric 2 g = v v + R g 2 , (5.42) Γ ⊗ S the extrinsic curvature ∇ K = g(n, vv) v v + R dR(n) gS2 , (5.43) Γ ⊗ and the MS energy

R E = 1 + (dR(n))2 (dR(v))2 , (5.44) 2 − and conclude exactly as in the timelike case. 76 Chapter 5. Spherical symmetry in General Relativity

5.2.1 Matching procedure for spherically symmetric spacetimes Now let us suppose we are faced with the following situation: we are given two spherically symmetric solutions ( , g ) (i = 1, 2) of Einstein’s equation and we Mi i want to know if they can be matched together at all, and if so, how to characterize the curve γ (or, more precisely, the two curves γ in each spacetime ) along i Mi which this is possible. Answers to these questions will be provided by the SSJC, which should be thought of as conditions on the matching curve(s). Note that a timelike or spacelike curve γ in the two-dimensional base manifold ( , gB) can B be described (if ∇R = 0, which we shall assume in the following) simply by a 6 function Rγ(τ) := R(γ(τ)), where R is the areal radius and τ the curve’s arc-length which, in the timelike case, corresponds to the matching observer’s proper time. The conditions (i) and (iii) of the SSJC are then equivalent to the condition that the functional dependences Ri(γi(τi)), abbreviated in the following by Ri(τi), must be the same (up to a trivial constant translation in τ) in both spacetimes to be matched, that is (possibly after a constant time translation):

R1(τ)= R2(τ) . (5.45)

Assuming that the functional dependence of the MS energy as function of the areal radius is known, condition (iv) implies now that the functional dependences of the

MS energies as function of the areal radius are the same: E1(R) = E2(R). If the latter equation can be solved for R, one gets the expression for the matching curve. It then remains to check, whether or not the last condition, (ii), is satisﬁed. In the next chapter we will see an example for this matching procedure. Chapter 6

Spherically symmetric inhomogeneities in cosmological spacetimes

In this chapter we discuss in detail two exact solutions to the Einstein equation which model a single quasi-isolated inhomogeneity ‘immersed’ in a cosmological spacetime. For simplicity the inhomogeneity is taken to be electrically neutral and non-rotating. The latter simplification motivates the assumption of spherical symmetry, to which we shall restrict in the following. We start in Section 6.1 giving a survey of the (few) known exact solutions which could model the above situation. Among these, the Einstein–Straus–Schücking (ESS) vacuole model and the McVittie model arise as the best candidates for our purpose. These models can be seen as the prototypes of the two strategies for combining together a Schwarzschild and a FLRW spacetime: via matching or ‘melting’, respectively. The ESS model goes the ‘matching way’, matching exactly the two solutions along an appropriate matching hypersurface. As opposed to this, the McVittie model can be regarded as the melting of the Schwarzschild and the FLRW spacetime since it metric is, for small radii, approximately equal to the Schwarzschild solution and for large radii it approaches the FLRW metric, whereas in the region in-between it is a kind of mixture of both. In Section 6.2.3 we give a simplified derivation of the ESS model applying our spherically symmetric junction conditions SSJC of Section 5.2. This is also used to give a simple derivation of the Eisenstaedt Theorem (Section 6.2.2). In Section 6.1 we further argue that the ESS vacuole is not suitable for estimating the effect of cosmological expansion on structures below the scales of galaxy clusters. Hence one is left with the McVittie model, whose original form and generalizations we extensively discuss in the Sections 6.3 and 6.4, respectively.

6.1 Finding the model

In the Introduction and motivation we already stressed the importance of exact solutions and how the approaches to ﬁnd exact solutions for modeling an inhomogeneity immersed in a cosmological spacetime can be subdivided into two strategies, called of matching and melting, respectively. The ﬁrst approach to the matching

77 78 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes idea was initiated by Einstein and Straus [ES45, ES46] in 1945 and later worked out in more analytical detail by Schücking [Sch54]. Here the matched solution, called Einstein–Straus–Schücking (ESS) vacuole model, is such that for radii smaller than a certain matching radius, Rv (henceforth called the vacuole or Schücking radius), it is exactly given by the Schwarzschild solution1 and for radii above this radius it is exactly given by a FLRW universe for dust matter without cosmological constant

(this can be generalized, see below). The radius Rv is a function of the gravitational mass m of the Schwarzschild solution and of the cosmological mass-density ̺, through the latter of which it also depends on the cosmological time t. We show below (cf. (6.35)) that the matching radius is determined by the strikingly simple relation 4π R3 ̺ = m . 3 v

This formula holds for spatially flat as well as for a spatially curved FLRW spacetimes, provided we take the ‘radius’ to be the ‘areal radius’. The ESS model can be generalized in several ways. Instead of cutting out one ball, one can cut several non-overlapping ones and fill in the interiors with Schwarzschild geometries of appropriate masses. For obvious reasons these are sometimes referred to as ‘Swiss-Cheese models’. These, in turn, can be generalized to the cases of non-vanishing cosmological constant [BBC88] or non-vanishing pressure [BS87]. Finally, the ESS model can be generalized to spherically symmetric but inhomogeneous Lemaˆıtre–Tolman–Bondi (LTB) cosmological backgrounds [Bon00]. Since for the ESS model the geometry within R R is exactly Schwarzschild ≤ v (for vanishing cosmological constant) or Schwarzschild–de Sitter spacetime (for non- vanishing cosmological constant), it is clear that any dynamical system situated in this background geometry only detects that part of the cosmic expansion that is due to a non-vanishing cosmological constant. In particular, for vanishing cosmological constant, the cosmic expansion that goes on outside the expanding vacuole R = Rv is not felt from within. This tells us that global expansion due to ordinary (local- izable) matter can, in principle, be completely inhibited by local inhomogeneities. There are, however, some severe problems concerning the ESS vacuole approach. First of all, it cannot provide a realistic model for the environment of small structures in our Universe, ‘small’ meaning below the scales of galaxy clusters or super- clusters. To see this, we specialize to a spatially flat universe whose background matter density ̺ is given by the critical density

3H2 ̺ := 0 , (6.1) crit 8πG where G is Newton’s constant. Then above relation for the matching radius gives

1/3 1/3 m R = R R2 400 ly , (6.2) v S H ≈ m ⊙

1 For modeling a black hole one takes just the exterior Schwarzschild solution and for a star the exterior together with the interior Schwarzschild solution. 6.1. Finding the model 79 where, reintroducing G’s and c’s, 2Gm m RS := 2 3 km , (6.3) c ≈ m⊙

c 23 RH := 4Gpc 1.3 10 km , (6.4) H0 ≈ ≈ × are the Schwarzschild radius for the mass m and the Hubble radius, respectively. m =2 1030 kg is the solar mass. ⊙ × For a single solar mass this gives a vacuole radius of almost 400 lightyears, which is almost two orders of magnitude larger than the average distance of stars in our Galaxy. Therefore, the Swiss-Cheese model cannot apply at the scale of stars in galaxies. This changes as one goes to larger scales. For example, the Virgo cluster is estimated to have a mass of approximately 1015 solar masses [FSSB01]2, which makes its vacuole radius 105 times larger than that for a single solar mass, so that it is approximately given by 10 Mpc. This is just a little smaller than the average distance of groups and clusters of galaxies within the Virgo supercluster. Hence the ESS approach might well give viable models above cluster scales. Similar conclusion can be drawn for the vacuole construction in LTB spacetimes [Bon00]: there it is argued that the vacuole might be as big as the Local Group. The ESS solution (as well as its generalization for LTB spacetimes given by Bon- nor) may also be criticized on theoretical grounds. An obvious one is its dynamical instability: slight perturbations of the matching radius to larger radii will let it increase without bound, slight perturbations to smaller radii will let it collapse. This can be proven formally (e.g. Chap. 3 in [Kra97] and [Bon00]) but it is also rather obvious, since Rv is defined by the equal and opposite gravitational pull of the central mass on one side and the cosmological masses on the other. Both pulls increase as one moves towards their side, so that the equilibrium position must correspond to a local maximum of the gravitational potential. Another criticism of the ESS solution concerns the severe restrictions under which it may be generalized to non spherically-symmetric situations; see e.g. [SV97, MTV02, MTV03, MTV05]. This shows that the ESS approach does not give us useful informations regarding the dynamical impact of cosmic expansion on structures well below the scales of galaxy clusters. Therefore we have to look for other solutions. These should contain an inhomogeneity in form of a spatial region with an overdense matter distribution, as compared to that of the approximately homogeneous matter distribution far away from it. Moreover, this inhomogeneity should also be quasi-isolated in order to be distinguishable from a mere local density fluctuation with smooth transition. Typical exact solutions that models the latter are the LTB solutions, in which matter is represented by pressureless dust that freely falls into the local overdense inhomogeneity. In some sense, these form the other extreme to the ESS solutions in that they make the transition as smooth and mild as one wishes. Here we shall be interested in models that somewhat lie in-between these extremes. An attempt to combine an interior Schwarzschild solution (representing a star) and a flat FLRW universe was made by Gautreau [Gau84]. Here the matter model consists of two components, a perfect fluid with pressure and equation of state p = p(̺) outside the star, and the superposition of this with the star’s dust-matter inside the star. However, Gautreau also made the assumption that the matter out-

2 Their considerations are based on a LTB model for the cluster. 80 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes side the star moves on radially infalling geodesics, which is only consistent if the pressure outside is spatially constant. Thus one is reduced to exact FLRW outside the star [vdBW84] or the LTB model. (Further remarks may be found in [Kra97], see p. 113 and 165 there.) Other solutions, modeling a black hole in a cosmological spacetime, have been given in the literature. However, these solutions model objects which are either rotating [Vai77, Vai84, RNV03], charged [GZ04], or both [PT82]. Surveys on the subject of cosmological black holes are [Vis00] and [McC06]. Fur- ther interesting solutions are given in [RMV01] and in [SD05, FJ07]. The solutions proposed in the latter two works can be seen as generalizations of McVittie’s model [McV33], which we extensively discuss in Section 6.3. A crucial feature of these solutions is, however, that the strength of the inhomogeneity3 varies in time, whereas for the original McVittie model it remains constant. These generalizations are analyzed in detail in Section 6.4 These solutions, which are analyzed in detail in Section 6.4, are of interest in their own right and also help the understanding of the peculiarity of the original McVittie solution. However, our goal here is to focus on the effects due to cosmological expansion and not on the effects due to a changing strength of the central inhomogeneity. The solution proposed in the former work [RMV01] is the melting of a Schwarzschild spacetime in an Einstein’s static universe. This is a purely static solution whose properties and geodesics where studied in [RV02]. For our purposes, however, this spacetime is not interesting since it is asymptotically an Einstein universe, and hence not in agreement with the present picture of our Universe at large scales. For these reasons in this work we pay special attention to the McVittie model. This contains a distinguished central object in the sense that the mass within a sphere centered at the inhomogeneity splits into a piece that comes from the continuously distributed cosmological fluid (with pressure) and a constant piece that does not depend on the radius of the enclosing sphere; see our Eq. (6.44). More- over, the latter piece is also constant in time, meaning that the strength of the central inhomogeneity remains constant. By the way, McVittie’s solutions contain the Schwarzschild–de Sitter one as a special case, which was recently used in the literature to estimate the effects of cosmological expansion on local systems [KKL06, HL08]. In the Chapters 7 and 8 we will make use of the McVittie model to estimate the effect of the cosmological expansion on local dynamics and on kinematics, the latter on the example of the Doppler tracking of a spacecraft.

6.2 Matched solutions

As anticipated above, the simplest way to find a model for an inhomogeneity in a cosmological spacetime is trying to match known inhomogeneous solutions to a FLRW spacetime. We show below how, in a spherically symmetric setting, this matching procedure is extremely simplified if we apply our SSJC of Section 5.2. Doing so we can give a simple derivation of the Eisenstaedt Theorem (Theorem 6.1) and of the Einstein–Straus–Schücking vacuole (Section 6.2.3). But first of all, we review the FLRW models.

3 As discussed in Section 5.1.2.2 the Weyl part of the MS energy, EW gives a measure for the integrate mass contrast. Since for the McVittie model, as well as for its generalizations considered in this work, EW is spatially constant, we can identify the latter with the strength of the central inhomogeneity of these models. 6.2. Matched solutions 81

6.2.1 Homogeneous cosmological models The simplest models which attempt to describe the universe seen at large scales are the so-called Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) cosmological models. These are based on the following two assumptions: 1. Matter assumption: the cosmological matter is described by a perfect fluid with velocity field u (called cosmological observer-field), energy density ̺, and pressure p, satisfying an equation of state of the form f(̺,p)=0

2. Copernican principle: the spacetime is isotropic at every point w.r.t. the cosmological observer-field u. Interestingly, the Copernican principle greatly restricts the spacetime’s geometry: in particular, it implies that the ‘space’ (to be defined below) is homogeneous. Recall that a spacetime ( , g) is said to be isotropic w.r.t. u at p if the M ∈ M group of (differential) isometries which fix p and leave up invariant contains SO(3) as a subgroup, that is if φ (p) φ Isom u( , g) SO(3) u. Here we put { ∗ | ∈ p, M } ⊇ p, Isom u( , g) := φ Isom( , g) φ(p) = p , φ u = u and with SO(3) u we p, M { ∈ M | ∗ p p} p, denote the group of all linear transformation of T which leave u invariant and pM p induce (via the projection (2.1b)) special orthogonal transformations in the local rest space ( u(p), hu) of u at p. R With the above assumptions one shows (see [Str74]) that u is geodesic and proper time synchronizable. The latter means that there exists a time function t, called cosmological time, with u = dt (see Section 2.3.2). Moreover, it follows that the slices Σt of constant cosmological time are spacelike hypersurfaces of constant curvature, and the restrictions of the metric on the Σt’s are all equal up to a multiplicative factor (which depend on the hypersurface, i.e. on t). This means that the spacetime has the structure of a warped product manifold,

= I Σ , (6.5) M ×a between a one-dimensional Riemannian space (I, dt2), where I is an open interval in R, and a three-dimensional Riemannian space of constant curvature (Σ, gΣ). Accordingly, the metric has the form

g = dt2 a2g . (6.6) − Σ Note that the warping function a is nothing but the scale factor. Introducing 2 −1 2 2 spherical coordinates (r, θ, ϕ) on Σ one has gΣ = (1 kr ) dr + r gS2 , where the 4 − constant k, which can take the values 0, 1, +1, is the curvature of (Σ, g ) and g 2 − Σ S is the metric on the unit two-sphere (see (4.4)). Summarizing, we have the following expressions for the metric, matter energy-momentum tensor, and velocity-ﬁeld:

2 FLRW 2 2 dr 2 g = dt a (t) + r g 2 (6.7a) − 1 kr2 S − FLRW T = ̺ u u + p hu (6.7b) M ⊗ u = ∂/∂t (6.7c)

4 Note that if the curvature of (Σ, gΣ) is positive (negative), it can be always made equal to 1 (−1) by means of a constant conformal transformation (see (A.81)). The (positive) constant conformal factor can then be absorbed in the scale factor a. 82 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

In (6.7b) hu denotes the spatial metric (2.7) and the energy density ̺ and pressure p are functions of t only. The worldlines of cosmological matter, being integral curves of u, intersect the spatial slices Σt at constant coordinates (r, θ, ϕ), hence the latter are called comoving coordinates and, in particular, r is called comoving radius. The Einstein equation are now easily set up using the material of Appendix A.9 and using the trick with the substitution f = ia for the warping function in order to comply with our signature choice (similarly to what made in the spherically symmetric case in Appendix A.10). Since the basis manifold I is one-dimensional and the fibers Σ have constant curvature, the expressions (A.89) and (A.90) for the Ricci tensor and for the scalar curvature greatly simplifies. For example, one has I Hess a =a ¨ dt2, I∆ a =a ¨ (here, and in the following, the dot denotes differentia- Σ tion with respect to t or, which is the same, along u), and Ric = 2k gΣ (for the latter see (A.58)). The Einstein tensor is then immediately written down:

a˙ 2 k a¨ a˙ 2 k Ein =3 + dt2 2 + + a2g , (6.8) a a2 − a a a2 Σ ! ! and the Einstein equation (C.1) is then given by

a˙ 2 k 3 +3 Λ=8π̺ , (6.9a) a a2 − a¨ a˙ 2 k 2 +Λ=8πp . (6.9b) − a − a − a2 Moreover, the u-part of the divergence-freeness of the energy-momentum tensor is simply a˙ ̺˙ + 3(̺ + p) =0 . (6.9c) a This follows from the Euler equation (1.2a) noticing that

div u = 3˙a/a , (6.10) as one can immediately see from its deﬁning equation Luµg = (div u)µg with 3 µg = a dt dr dθ sin θdϕ. We notice that the spatial part of the divergence ∧ ∧ ∧ equation div T = 0 says that u is geodesic (see (1.2b) and recall that ∇p is parallel to u), as it must be as consequence of the Copernican principle. Now, as one can show, any two of the three equations (6.9) are equivalent to the full set (6.9). One often conveniently opts for the ﬁrst-order equations pair

a˙ 2 k Λ 8π + = ̺ , (6.11a) a a2 − 3 3 a˙ ̺˙ + 3(̺ + p) =0 , (6.11b) a to which we refer as to the Friedmann equations. These, together with an equation of state f(̺,p) = 0, determine the three time dependent functions (a,̺,p)— provided that initial conditions (a0,̺0,p0) are given. 6.2. Matched solutions 83

Example 6.1 (Dust FLRW). In the special, but relevant, case of a dust equation of state, i.e. p = 0, the Friedmann equation (6.11b) can be integrated giving

̺a3 = const. (6.12)

Herewith, the Friedmann equation (6.11a) becomes an ODE for a alone:

a˙ 2 k C Λ + =0 , (6.13) a a2 − a3 − 3 3 where the constant C := 8π̺0a0/3 depends on the initial conditions a0 := a(t0) and ̺0 := ̺(t0) at some ‘initial’ time t0. Returning to the general case (with arbitrary equation of state, arbitrary k, and arbitrary Λ) we note that, beside being homogeneous, the FLRW models are clearly spherically symmetric (and this w.r.t. each point). In the coordinates of (6.7a) the SO(3)-orbits are given by the surfaces of constant comoving radius r and have area 4π(ar)2. The areal radius is thus

R(t, r)= a(t)r . (6.14)

In passing we notice that the expression for the variation of the areal radius along the integral lines of u (that is the velocity of cosmological matter measured in terms of its proper time and the areal radius) is

u(R)= HR , (6.15) where H :=a/a ˙ (6.16) is the Hubble parameter. Expression (6.15) is nothing but Hubble’s law. Moreover, we also notice that the expansion of u is given by

θu =3H , (6.17) as it follows from (6.10) and Deﬁnition 2.10. The other geometric objects presented in Section 4.2 are as follows. The MS energy can be computed from (4.6): with

dR =ar ˙ dt + adr = HRdt + adr (6.18) and dR, dR = gµν R R = (˙ar)2 + kr2 1 one has h i ,µ ,ν − 1 a˙ 2 k EFLRW = (ar)3 + (6.19) 2 a a2 ! and, using Friedmann equation (6.11a) and (C.5a), one gets the intuitively appealing formula 4π EFLRW = R3(̺ + ̺ ) . (6.20) 3 Λ In fact, this is equal to the Ricci part of the MS energy (see (5.20)), which implies that the Weyl part identically vanishes:

FLRW EW =0 . (6.21) 84 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

This is of course in accord to our interpretation of the Weyl part of the MS energy as integrated mass-density contrast, as discussed in Section 5.1.2. The expression for the Kodama vector ﬁeld follows easily from the expression for the diﬀerential of the areal radius (6.18). For this it is convenient to introduce the (positively oriented5) orthonormal frame e , e in the two-dimensional basis { 0 1} manifold coordinatized by t and r, B ∂ 1 ∂ e := u = and e := , (6.22) 0 ∂t 1 a ∂r

µ µ together with the related dual basis (θ (eν )= δν ) of one-forms

θ0 = dt and θ1 = adr . (6.23)

The diﬀerential of the areal radius (6.18) is then dR = HR θ0 + θ1 and herewith the Kodama vector ﬁeld as one-form is

k = ⋆dR = θ0 + HR θ1 (6.24)

(see Appendix A.4.2) and thus

k = e HR e . (6.25) 0 − 1

We notice that the Kodama vector ﬁeld is timelike (and dR is spacelike) iﬀ R < RH , where RH := 1/H denotes the Hubble radius. In this region we have thus the following relation between the Kodama basis (4.25) and the orthonormal coordinate- basis (6.22) on the two-dimensional base manifold : B

K 1 e0 = uK = (e0 HR e1) , (6.26a) 1 (HR)2 − −

K p 1 e = eR = ( HR e0 + e1) . (6.26b) 1 1 (HR)2 − − p In passing we notice that from (6.26a) one sees that the relative velocity (see (3.1)) of the Kodama observer uK w.r.t. the cosmological observer u has the intuitively appealing form

βu(uK)= HR e . (6.27) − 1 This is inwardly pointing and equal, in modulus, to Hubble’s velocity (6.15). The formulae for the acceleration and expansion of the Kodama observer ﬁeld follow now from (4.23) and (4.26), respectively. For simplicity we specialize here to the case k = 0, which is also in accord to the present observations [K+09]. Notice that (6.19) reads EFLRW = (1/2)R3H2 and thus 1 2E/R =1 (HR)2. Acceleration − − and expansion are thus

∇ 2 −3/2 a¨ 4 3 u uK = 1 (HR) R + H R e , (6.28) K − −a R and, respectively, 2 −3/2 2 θu = 1 (HR) R HH˙ . (6.29) K − 5 Since e0 is future-pointing and e1 is outwardly-pointing (dR(e1) = 1 > 0). 6.2. Matched solutions 85

For small values of HR = R/RH these expressions simplifies as follows. We first notice that the expressionsa/a ¨ and H˙ are to be seen as quadratic in H. This is because, recalling the definition of the dimensionless deceleration parameter

a/a¨ q := , (6.30) − (˙a/a)2 one has thata/a ¨ = qH2 and H˙ = (1+q)H2. We recall also that the actual value − − (cf. [K+09]) of the deceleration parameter is roughly q 1/2 (meaning, by the ≈ − way, that the universe is in accelerating expansion). This implies that the leading contributions to the acceleration and expansion are of quadratic and, respectively, third order in R/RH . Hence, neglecting terms of third and higher-order in R/RH we have: ∇ a¨ u uK R e (6.31) K ≈−a R and θu 0 . (6.32) K ≈ That is, to this order of approximation, the Kodama observer ﬁeld is non-expanding. These expressions will be useful in Chapter 7 when writing down the spatial equation of motion w.r.t. the Kodama observer ﬁeld.

6.2.2 The Eisenstaedt theorem Perhaps the simplest attempt to model a compact body (star) in an expanding universe is trying to inglobate it in a FLRW spacetime and to assume, for simplicity, that the body is spherically symmetric. A direct consequence of the SSJC of Section 5.2 is the following intuitive appealing theorem due to Eisenstaedt [Eis77]:

Theorem 6.1 (Eisenstaedt, 1977). Excise the full world-tube of a comoving Wr0 ball of comoving radius r0 from a FLRW spacetime and insert instead a spherical symmetric inhomogeneity (hence a piece of a spherically symmetric spacetime together with a related matter model, satisfying Einstein’s equations). Then a necessary condition for the resulting spacetime to satisfy Einstein’s equation (in the sense explained in Section 5.2) is that the MS energy of the inserted inhomogeneity equals that of the excised ball.

This says that the mean energy density (measured with the MS energy) of spherically-symmetric inhomogeneities must be the same as the one of the FLRW spacetime. That the Eisenstaedt Theorem is a consequence of the Darmois junction conditions was already pointed out in [Har06].

6.2.3 The Einstein–Straus–Sch¨ucking vacuole revisited

As another application of the above described matching procedure we revisit the Einstein–Straus–Sch¨ucking solution [ES45, ES46, Sch54], which originally consists on a Schwarzschild spacetime (called ‘vacuole’) matched to a dust FLRW universe with zero cosmological constant. Later, this model was generalized also to the case of a non-vanishing cosmological constant [BBC88]. We treat here the general case of an arbitrary cosmological constant and show that the SSJC of Section 5.2 86 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes allow substantial simpliﬁcations of the computations. This technique can also be applied to Bonnor’s vacuole construction [Bon00] in Lemaˆıtre–Tolman–Bondi (LTB) spacetimes. Notice that the matching condition (5.36) implies, in particular, that the pressure must be continuous through the matching hypersurface. Since the interior is a vacuum spacetime, it follows that the pressure must vanishes also on the exterior part of the matching hypersurface and hence everywhere on the FLRW spacetime. That is the reason one has to restrict to dust FLRW spacetimes. Since we leave the cosmological constant Λ arbitrary (it may be positive, negative, or zero) the inner region is given, respectively, by a Schwarzschild–de Sitter, Schwarzschild–anti-de Sitter6, or Schwarzschild spacetime (all abbreviated henceforth by SdS). Recall that the SdS spacetime is given by the vacuum solution to Einstein equation with cosmological term

SdS 2 −1 2 2 g = V (R) dT V (R) dR R g 2 , (6.33a) − − S where 2m Λ V (R)=1 R2 . (6.33b) − R − 3

Above, gS2 denotes the metric on the unit two-sphere and m is a constant which represents the central mass (this is, as we will see below, just the Weyl part of the MS energy). A dust FLRW spacetime is given by (6.7) together with the dust equation of state: p = 0. The question is now: how shall we cut hypersurfaces Γ = γ S2 and SdS SdS × Γ = γ S2 in the spacetimes SdS and, respectively, FLRW in order that FLRW FLRW × the resulting pieces can be matched? In order to apply the SSJC we have to compute the MS energy for both spacetimes. For the FLRW spacetimes this was done in Section 6.2.1 (see (6.20)). For the SdS spacetime we have R = R, dR, dR = SdS h iSdS V (R) and with the MS energy expression (4.6): − 4π E = m + R3 ̺ . (6.34) SdS 3 SdS Λ In passing, we notice that, by (5.20), the last term on the r.h.s. of (6.34) is just the Ricci-part of the MS energy and thus the ﬁrst term is the Weyl part of the MS energy. Now, the last two conditions of the SSJC, that is the continuity of areal radius and MS energy across the matching hypersurface (yet to be determined), are equivalent to the continuity of the areal radius RFLRW = RSdS =: R, together with the suggestive relation 4π m = R3̺ . (6.35) 3 This two conditions already determine the matching hypersurface. Indeed, in- 3 serting R = RFLRW = a(t)r in (6.35) and using the relation ̺(t)a (t) = const. (see (6.12)) valid for dust FLRW models, one obtains the matching radius in terms of the FLRW comoving radial coordinate:

m 1/3 r = r := = constant . (6.36) 0 (4π/3)a3̺ 0 0 6 The Schwarzschild–(anti-) de Sitter metric (6.33a) is often called the Kottler solution, after Friedrich Kottler, who was the ﬁrst to write down this metric in [Kot18]. More details on its analytic and global structure may be found in [Gey80]. 6.3. The McVittie models 87

Here a0 := a(t0), and similarly for ̺, where t0 is some ﬁxed ‘initial’ time. This means that the matching observer moves, in the FLRW spacetime, along the integral curve of u = ∂/∂t with initial condition (t0, r0) and hence is comoving with the cosmological matter. So far we used the last two of the SSJC. As discussed above, the continuity of the areal radius and the arc-length (the proper time, in the timelike case) of the matching curve are equivalent to the equality of the functional dependencies R(τ) (up to a possible trivial constant translation in τ) which describe the matching curves in the two spacetimes to be matched. Now, because of (6.36), the matching curve (worldline) in the FLRW spacetime is simply

R(τ)= a(τ)r0 , (6.37) where a is the (unique) solution of the Friedmann equation (6.13) with initial condition a0 at τ0 = t0. (Recall that in FLRW the proper time of an observer moving along an integral line of u equals the cosmological time, hence τ = t.) From what we said above, the same functional relation R(τ) given in (6.37) must hold also in the SdS—provided we identify R with the areal radius and τ with the matching observer’s proper time, both referred to the SdS spacetime. This determines the matching curve in the SdS spacetime. Finally we need to show that the junction condition (ii) is satisfied, hence that the matching observer’s accelerations coincide. Looking at the matching worldline from the FLRW spacetime, it is immediately clear that it is geodesic, hence its acceleration vanishes. To conclude the matching procedure, we just have to check that this is also true for the matching worldline in the SdS spacetime. For this, one has just to check that the function defined in (6.37) satisfies the geodesic equation for a radial motion. The latter is given by

R˙ 2 + V (R)= e2 , (6.38) where e := gSdS(∂/∂T, v) = constant and v = T˙ ∂/∂T + R˙ ∂/∂R is the matching observer in the SdS spacetime. (Equation (6.38) can be quickly derived from the fact that v(gSdS(∂/∂T, v)) = 0, since ∂/∂T is Killing and v geodesic. Inserting e := gSdS(∂/∂T, v)= V (R)T˙ in the normalization condition 1 = gSdS(v, v)= V (R)T˙ 2 − R˙ 2/V (R) one arrives immediately at (6.38).) Now, inserting (6.37) with (6.36) in (6.38) and using the Friedmann equation (6.13), one gets R˙ 2 + V (R)=1 kr2. − 0 Hence, the geodesic equation (6.38) is satisﬁed (with e2 =1 kr2) and herewith all − 0 the four junction conditions.

6.3 The McVittie models

In this section we ﬁrst review a model, due to McVittie [McV33], for a spherically symmetric inhomogeneity in a cosmological spacetime. This consists in a metric ansatz (cf. (6.39)) together with the choice of a matter model. In Section 6.3.2 we then study in detail the physical and geometric properties of the metric ansatz. 88 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

6.3.1 The original McVittie model

The characterization of the McVittie model is made through two sets of a priori speciﬁcations. The ﬁrst set concerns the metric (left side of Einstein’s equations) and the second set the matter (right side of Einstein’s equations). The former consists in an ansatz for the metric, which can formally be described as follows: write down the Schwarzschild metric for the mass parameter m in isotropic coordinates, add a conformal factor a2(t) to the spatial part, and allow the mass parameter m to depend on time. Hence the metric ansatz reads

2 4 1 m(t)/2r 2 2 m(t) 2 2 g = − dt a (t) 1+ (dr + r g 2 ) , (6.39) 1+ m(t)/2r − 2r S where gS2 is the standard metric on the unit 2-sphere. Here we restricted attention to the asymptotically spatially flat (i.e. k = 0) FLRW metric, which is compatible with current cosmological data [K+09]. For simplicity we shall refer to (6.39) simply as McVittie’s ansatz, though this is not quite correct since McVittie started from a general spherically symmetric form and arrived at (6.39) with m(t)a(t) = const. after imposing a condition that he interpreted as the absence of matter accretion onto the central object (‘no-infall’ condition). The ansatz (6.39) is obviously spherically symmetric: the SO(3) orbits are just the spheres of constant coordinate-radius r and the two-dimensional basis manifold (see Section 4.1) is coordinatized by B (t, r). In the next section we will discuss in more detail the geometric implications of this ansatz, independent of whether Einstein’s equation holds. As already discussed in the introduction, the model here is meant to interpolate between the spherically symmetric gravitational field of a compact object and the environment. It is not to be taken too seriously in the region very close to the central object, where the basic assumptions on the behavior of matter definitely turn unphysical. However, as discussed below and in [CG10b], at radii much larger than (in geometric units) the central mass (to be defined below) the k = 0 McVittie solution seems to provide a viable approximation for the envisaged situation. The second set of specifications, concerning the matter, is as follows: the matter is a perfect fluid with density ̺ and isotropic pressure p. Hence its energy- momentum tensor is given by

T = ̺ u u + p (u u g) . (6.40) ⊗ ⊗ − Furthermore, and this is where the two sets of speciﬁcations make contact, the motion of the matter (i.e. its velocity ﬁeld) is given by

u = e0 , (6.41) where e0 is the normalization of ∂/∂t (compare (6.48)). Finally, the explicit cosmological constant on the left-hand side of Einstein’s equation is assumed to be zero, which implies no loss of generality, since a non-zero cosmological constant can always be regarded as special part of the matter’s energy-momentum tensor (compare Section 6.4). No further assumptions are made. In particular, an equation of state, like p = p(̺), is not assumed. The reason for this will become clear soon. Later generalizations will mainly concern (6.40) and (6.41). 6.3. The McVittie models 89

The Einstein equation now links the speciﬁcations of geometry with that of matter. From the expression (6.51) for the Einstein tensor given below one immediately sees that Einstein’s equation is equivalent to the following three relations between the four functions m(t),a(t),̺(t, r), and p(t, r):

(am)˙ = 0 , (6.42a)

a˙ 2 8π̺ = 3 , (6.42b) a a˙ 2 a˙ ˙ 1+ m/2r 8πp = 3 2 . (6.42c) − a − a 1 m/2r − Note that here Einstein’s equation has only three independent components (as opposed to four for a general spherically symmetric metric), which is a consequence of the fact that the Einstein tensor for the McVittie ansatz (6.39) is spatially isotropic (see (6.51)). This will be discussed in more detail Section 6.3.2.2. Equation (6.42a) can be immediately integrated:

m m(t)= 0 , (6.43) a(t) where m0 is an integration constant. Below we will show that this integration constant just the Weyl part of the MS energy and is thus to be interpreted as the mass of the central body. Clearly, the system (6.42) is under-determining. This is expected since no equation of state has yet been imposed. The reason why we did not impose such a condition can now be easily inferred from (6.42): whereas (6.42b) implies that ̺ only depends on t, (6.42c) implies that p depends on t and r iff (˙a/a)˙ = 0. Hence 6 a non-trivial relation p = p(̺) is simply incompatible with the assumptions made so far. The only possible ways to specify p are p = 0 or ̺ + p = 0. In the first case (6.42c) implies thata/a ˙ = 0 if m = 0 (since then the second term on 0 6 the right-hand side is r dependent, whereas the first is not, so that both must vanish separately), which corresponds to the exterior Schwarzschild solution, or a(t) t2/3 if m = 0, which leads to the flat FLRW solution with dust. In the ∝ 0 second case the fluid just acts like a cosmological constant Λ = 8π̺ (using the equation of state ̺ + p = 0 in the divergence equation div T = 0 it implies dp =0 and this, in turn, using again the equation of state, implies d̺ = 0) so that this case reduces to the Schwarzschild–de Sitter solution. To see this explicitly, notice first that (6.42b,6.42c) imply the constancy of H =a/a ˙ = Λ/3 and hence one has a(t) = a0 exp t Λ/3 . With such a scale-factor the McVittiep metric (6.39) with (6.43) turns into p the Schwarzschild–de Sitter metric in spatially isotropic coordinates. The explicit formulae for the coordinate transformation which brings the latter in the familiar form can be found in Sect. 5 of [Rob28] and also in Sect. 7 of [KS05]. Finally, note from (6.42a) that constancy of one of the functions m and a implies constancy of the other. In this case (6.42b,6.42c) imply p = ̺ = 0, so that we are dealing with the exterior Schwarzschild spacetime. A specific McVittie solution can be obtained by choosing a function a(t), corresponding to the scale function of the FLRW spacetime which the McVittie model 90 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

is required to approach at spatial inﬁnity, and the constant m0, corresponding to the central mass. Relations (6.42b,6.42c), and (6.43) are then used to determine ̺, p, and m, respectively. Clearly this ‘poor man’s way’ to solve Einstein’s equation holds the danger of arriving at unrealistic spacetime dependent relations between ̺ and p. This must be kept in mind when proceeding in this fashion. For further discussion of this point we refer to [Nol98, Nol99a]. In Section 6.3.2.3 below we compute the MS energy and its Ricci- and Weyl- part for the McVittie ansatz (see (6.58) and (6.59)). Using Einstein’s equations (6.42a,6.42b) and also taking into account (6.43), we get 4π E = R3̺ , (6.44a) R 3

EW = m0 . (6.44b)

Recall that R denotes the areal radius, which for the McVittie metric is given explicitly in (6.52). In Section 5.1.2 we interpreted the Weyl part of the MS energy as integrated mass-density contrast. Since this for the metrics (6.39) is spatially constant (the functions a and m in (6.59) only depend on time) so that EW is independent of the radius, it gives a reasonable measure for the mass of the central object. The constancy of EW is then interpreted as saying that no energy is accreted from the ambient matter onto the central object. We now briefly discuss the basic properties of the motion of cosmological matter. Being spherically symmetric, the velocity field u specified in (6.41) is automatically irrotational (Corollary 4.3). The last property can also be deduced from the hypersurface orthogonality of u, which is immediate from (6.39). Moreover, u is also shear free. This, too, can be immediately read off (6.39) once one takes into account the following theorem which we prove in Appendix B Theorem 6.2. A spherically symmetric observer field u in a spherically symmetric spacetime ( , g) is shear free iff the u-spatial metric hu (see (2.7)) on the u-spatial M subbundle u is conformally flat. R The metric (6.39) obviously is spatially conformally flat w.r.t. the choice (6.41) made here. Moreover, the expansion (i.e. divergence) of u is

θu =3H , (6.45) where H :=a/a ˙ , just as in the FLRW case. In particular, the expansion of the cosmological ﬂuid is homogeneous in space. Exactly as in the FLRW case is also the expression for the variation of the areal radius along the integral lines of u (that is the velocity of cosmological matter measured in terms of its proper time and the areal radius): u(R)= HR , (6.46) which is nothing but Hubble’s law. The acceleration of u, which in contrast to the FLRW case does not vanish here, is given by

m 1+ m/2r ∇ u = 0 e , (6.47) u R2 1 m/2r 1 − where e1 is the normalized vector ﬁeld in radial direction as deﬁned in (6.48). In particular, this implies (via Euler equation) that the pressure cannot be spatially 6.3. The McVittie models 91

constant. In leading order in m0/R this corresponds to the acceleration of the observers moving along the timelike Killing field in Schwarzschild spacetime. It is also important to note that the central gravitational mass in McVittie’s spacetime may be modeled by a shear-free perfect-fluid star of positive homogeneous energy density [Nol93]. The matching, which can be conveniently carried out with the SSJC, is performed along a world-tube comoving with the cosmological fluid, across which the energy density jumps discontinuously. This means that the star’s surface is comoving with the cosmological fluid and hence, in view of (6.45), that it geometrically expands (or contracts). This feature, however, should be merely seen as an artifact of the McVittie model (in which the relation (6.45) holds), rather than a general property of compact objects in any cosmological spacetimes. Positive pressure within the star seems to be only possible if 2aa¨ +a ˙ 2 < 0 (see eq. (3.27) in [Nol93] with a = exp(β/2)), that is, for deceleration parameters q> 1/2.

6.3.2 Geometry of the McVittie ansatz

In this section we will discuss the geometry of the metric (6.39) independent of the later restriction that it will have to satisfy Einstein’s equation for some reasonable energy-momentum tensor. This means that at this point we shall not assume any relation between the two functions m(t) and a(t), apart from the ﬁrst being non negative and the second being strictly positive. We will discuss the metric’s ‘spatial Ricci-isotropy’ (a term explained below), its singularities and trapped regions, and also compute its MS energy, decomposed into the Ricci and Weyl parts, as well as the kinematical properties of the Kodama observer ﬁeld. We shall start, however, by answering the question of what the overlap is between the geometries represented by (6.39) and the conformal equivalence class of the exterior Schwarzschild geometry.

6.3.2.1 Relation to conformal Schwarzschild class This question is an obvious one in view of the way in which (6.39) is obtained from the exterior Schwarzschild metric. It is clear that for m = m0 = const. the metric (6.39) is conformally equivalent to the exterior Schwarzschild metric, since upon using a new time coordinate T with dT = dt/a(t) we can pull out a2(t) as a common conformal factor. The following proposition, whose proof we shall give in Appendix B.2, states that a constant m is in fact also necessary condition:

Proposition 6.3. Let denote the set of metrics in the form of the McVittie SMcV ansatz (6.39) (parametrized by the two positive functions a and m) and the set of ScS metrics conformally equivalent to an exterior Schwarzschild metric (parametrized by a positive conformal factor and a constant positive Schwarzschild mass M0). Then the intersection between and is given by the subset of metrics in with SMcV ScS SMcV constant m or, equivalently, by the subset of metrics in whose conformal factor ScS has a gradient proportional to the Killing ﬁeld ∂/∂T of the Schwarzschild metric (see (B.6b) for notation).

Note that we excluded the ‘trivial’ cases in which m or M0 (or both) vanish for the following reason: comparing the expressions for the Weyl part of the MS energy of the two types of metrics (see (B.15) in Appendix B.2) it follows that m vanishes iﬀ M0 does and this, in turn, leads to a metric conformally related to the Minkowski 92 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes metric where the conformal factor depends only on time, that is, a FLRW metric. But such a spacetime, being homogeneous, is not of interest to us here. In particular, Proposition 6.3 implies that the metric of Sultana and Dyer [SD05] are not of type (6.39), as suggested in Sect. IVA of [FJ07] and allegedly shown in Sect. II of [Far09] (cf. Footnote 9 below). This immediately follows from the observation that the conformal factor, expressed as function of the standard Schwarzschild coordinates that appear in (B.6b), is given by Ω(T, R) = (T +2M ln(R/2M 1))2 0 0 − (compare eqs. (8) and (9) of [SD05]), which also depends on R and hence does not satisfy the condition of Proposition 6.3. We will have to say more about this at the beginning of Section 6.4 and in Section 6.4.4.

6.3.2.2 Spatial Ricci-isotropy

Deﬁnition 6.4 (Spatial Ricci-isotropy). Let v denote a timelike vector ﬁeld on a Lorentzian manifold ( , g). We say that ( , g) is spatially Ricci-isotropic w.r.t. v M M if the v-spatial7 restriction of the Ricci tensor is proportional to the v-spatial restriction of the metric.

As we will show below, a crucial feature of any metric that is covered by the ansatz (6.39) is the spatial Ricci-isotropy w.r.t. ∂/∂t. We make first the obvious remark that, in general, the v-spatial restriction of the Ricci tensor is not the same as the Ricci tensor of the v-spatial distribution endowed with the induced metric (assuming for this that v is hypersurface orthogonal). Hence the notion of spatial Ricci-isotropy is not the same as saying that the induced metric of the v-spatial slices is an Einstein metric. We note also that, since Ein = Ric (1/2) Scal g, the Ricci tensor is spatially isotropic iff the same holds for the − Einstein tensor. It is then obvious that Einstein’s equation will impose a severe restriction upon the matter’s energy-momentum tensor, saying that it, too, must be spatially isotropic. The degree of specialization implied by this for the McVittie ansatz (6.39) will be discussed in more detail below. Here we only remark that this observation already answers in the negative a question addressed, and left open, in the last paragraph of [FGCS09], of whether (6.39) is the most general spherically symmetric solution describing a black hole embedded in a spatially flat FLRW background: it clearly is not. Now, a way to actually show spatial Ricci-isotropy is to compute the components of the Einstein tensor with respect to the orthonormal tetrad e of (6.39) { µ}µ∈{0,··· ,3} defined by e := ∂/∂xµ −1 ∂/∂xµ , (6.48) µ k k where xµ = t,r,θ,ϕ (recall that v := g(v, v) ). Note that { } { } k k | | p A ∂ 1 ∂ e = and e = (6.49) 0 B ∂t 1 A2a ∂r are orthogonal to and e2, e3 tangent to the 2-spheres of constant radius r. Here and in the following we set:

A(t, r):=1+ m(t)/2r, B(t, r):=1 m(t)/2r . (6.50) − 7 Recall that ‘v-spatial’ refers to the directions orthogonal to v (v being a timelike vector). 6.3. The McVittie models 93

The non-vanishing independent components of the Einstein tensor with respect to the orthonormal basis (6.48) are: 2 Ein(e0, e0)=3F , (6.51a) 2 A 2 ˙ Ein(e0, e1)= R2 B (am) , (6.51b) Ein(e , e ) = 3F2 +2 A F˙ δ , (6.51c) i j − B ij where an overdot denotes diﬀerentiation along ∂/∂t. Expression (6.51c) now clearly shows that the McVittie ansatz (6.39) is spatially Ricci-isotropic w.r.t. ∂/∂t. We note that (6.51) can be conveniently computed via (A.98) together with the expression for the curvature of a two-dimensional Lorentzian manifold given in Ap- pendix A.6.3. In (6.51) and in the following we set:

m(t) 2 R(t, r) := 1+ a(t) r , (6.52) 2r which is the areal radius for the McVittie ansatz (6.39), and a˙ 1 (am)· F := + . (6.53) a rB a In passing we note that, beside R, also the quantities F and am, that appear in the components of the Einstein tensor, have a geometrical interpretation: the former is one third the expansion (i.e. divergence) of the vector ﬁeld e0, that is,

F = div(e0)/3, and the latter is the Weyl part of the MS energy of the metric (6.39) (see (6.59), below). Moreover the observer ﬁeld e0 is shear-free (by The- orem 6.2) and irrotational (because of spherical symmetry). Hence, taking into account the relation (B.4) between the expansion θ and the shear scalar σ of an arbitrary spherically-symmetric observer ﬁeld, the expansion of e0 can be simply written as 3dR(e0)/R so that F may be expressed as

F = dR(e0)/R . (6.54) We also make the obvious remark that this equation may be rewritten as

e0(R)= F R . (6.55) Clearly, if (am)· = 0, one has F = H (see (6.53)) and thus the above equation reduces to Hubble’s law (6.46). In order to estimate the degree of specialization implied by spatial Ricci-isotropy, we ask for the most general spherically symmetric metric for which this is the case. To answer this, we ﬁrst note that any spherically symmetric metric can always be written in the form 2 B(t, r) 2 2 4 2 2 g = dt a (t)A (t, r)(dr + r g 2 ) . (6.56) A(t, r) − S This reduces to McVittie’s ansatz (6.39) if A, B are given by (6.50). For the general spherically symmetric metric (6.56), spatial Ricci-isotropy can be shown to be equivalent to δ2(AB) 8(δA)(δB)=0 , (6.57) − where δ := r−1∂/∂r =2∂/∂r2. It is obvious that there are many more solutions to this diﬀerential equation than just (6.50). 94 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

6.3.2.3 Misner–Sharp energy and Kodama observer ﬁeld

In order to be able to interpret (6.39) as an ansatz for an inhomogeneity in a FLRW universe, it is useful to compute the MS energy and, in particular, its Ricci and Weyl parts. For the Ricci part of the MS energy of (6.39) we get

R E = 1 R3 Ein(e , e )= 1 R3F 2 = (dR(e ))2 . (6.58) R 6 0 0 2 2 0 The ﬁrst equality in (6.58) can be derived by merely using the spatial Ricci-isotropy w.r.t. e0 in the expression for the Ricci part of the Riemann tensor (see (A.50b)). The second and the third equalities follow then with (6.51a) and (6.54), respectively. The Weyl part can now be obtained as the diﬀerence between the full MS energy and (6.58). We use the expression (4.6) for the former and write g(∇R, ∇R) = 2 2 e (R) e (R) . The part involving e (R) equals the Ricci part (6.58) and 0 − 1 0 hence the Weyl part is given by (R/2) 1 (e (R))2 . From (6.52) we calculate − 1 e1(R) and hence obtain for the Weyl part of the MS energy:

EW = a m . (6.59)

Since the functions a and m depend only on time the Weyl part EW is independent of the radius and hence a reasonable measure for the mass of the central object. A possible time variation of EW is then interpreted as an exchange of energy between the central object and the ambient matter (compare (6.80a) and (6.79d)). To compute the Kodama vector field for the ansatz (6.39) we first compute the differential of the areal radius (6.52):

B dR = F R θ0 + θ1 , (6.60) A where θµ denotes the dual basis to (6.48). In particular one has { } B θ0 = dt and θ1 = A2a dr . (6.61) A

B 0 1 The Kodama vector field (as one-form) is then k = ⋆dR = A θ + F R θ and thus B k = e F R e . (6.62) A 0 − 1 The Kodama basis (4.25), consisting of the Kodama observer field and the to it orthogonal outwardly pointing normalized vector field eR (see (4.24)), is thus

K 1 e = uK = (e0 v e1) , (6.63a) 0 √1 v2 − − K 1 e = eR = ( v e0 + e1) , (6.63b) 1 √1 v2 − − where A v := B F R . (6.64) Note that the absolute value of v is just the modulus of the relative velocity of the

Kodama observer ﬁeld uK w.r.t. u,

βu(uK)= v e , (6.65) − 1 6.3. The McVittie models 95 which follows at once from (6.63a). The acceleration and expansion of the Kodama observer ﬁeld are now computed with (4.23) and (4.26). For the special case EW = m0 = const. of interest to us, a straightforward computation (with (6.51) and (6.63)) shows that: −3/2 ∇ A A 2 m0 A a¨ A 2 4 3 u uK = 1 HR R + H R K B − B R2 − B a B (6.66) 4 2 A (RH)2 m0 e , − 2B2R R R and −3/2 3 A 2 A 2 θu = 1 HR R HH˙ . (6.67) K − B B These expressions are to be compared with their counterparts for the FLRW spacetime (see Section 6.2.1). If one is interested in the situation in which

2E =: R R R := 1/H , (6.68) W S ≪ ≪ H that is far enough away from the marginally trapped spheres situated approxi- matively at the ‘Schwarzschild radius’ RS := 2EW and at the ‘Hubble radius’ RH := 1/H (see Section 6.3.2.4), the above expressions simplify as follows. There are now two small parameters: RS/R and R/RH (assuming in the ﬁrst place that η := R /R 1). The leading order terms of the acceleration are linear in R /R S H ≪ S and quadratic in R/RH . To this order of approximation we have thus:

∇ m0 a¨ u uK R e (6.69) K ≈ R2 − a R and θu 0 . (6.70) K ≈ That is, to this order of approximation, the Kodama observer ﬁeld is non-expanding like in the FLRW case. These expressions will be useful in Chapter 7 when writing down the spatial equation of motion w.r.t. the Kodama observer ﬁeld.

6.3.2.4 Singularities and trapped surfaces

Next we comment on the singularity properties of the McVittie ansatz (6.39). From (6.51c) one suspect, because of the term proportional to 1/B, a singularity in the Ricci part of the curvature at r = m/2 (that is at R = 2am = 2EW). In fact, this corresponds to a genuine curvature singularity, as one can see from looking, for example, at the following expression for the scalar curvature (i.e. the Ricci scalar), Scal = 12F 2 6 A F˙ , (6.71) − − B which can be quickly computed from (6.51). In Appendix B.3 we insert into this expression the deﬁnition (6.53) of F and expand this in powers of 1/(rB). This allows us to prove Proposition 6.5. The Ricci scalar for a metric of the form (6.39) becomes singular in the limit r m/2 for any functions a and m, except for the following three special cases: → 96 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

(i) m =0 and a arbitrary (FLRW), (ii) a and m are constant (Schwarzschild), and (iii) (am)· =0 and (˙a/a)· =0 (Schwarzschild–de Sitter).

This means that, as long as we stick to the ansatz (6.39), at r = m/2 there will always (with the only exceptions listed above) be a singularity in the Ricci part of the curvature and thus, assuming Einstein’s equation is satisfied, also in the energy momentum tensor, irrespective of the details of the underlying matter model. Hence any attempt to eliminate this singularity by maintaining the ansatz (6.39) and merely modifying the matter model is doomed to fail. In particular, this is true for the generalizations presented in [FJ07], contrary to what is claimed in that work and its follow ups [GCFS08, FGCS09]. We also remark that it makes no sense to absorb the singular factors 1/B in front of the time derivatives by writing (A/B)∂/∂t as e0 and then argue, as was done in [FJ07], that this eliminates the singularity. The point is simply that then e0 applied to any continuously differentiable function diverges as r m/2. Below we will argue that → this singularity lies within a trapped region. Specializing to the McVittie model, recall that in this case it is assumed that the fluid moves along the integral curves of ∂/∂t, which become lightlike in the limit as r tends to m/2. Their acceleration is given by the gradient of the pressure, which necessarily diverges in the limit r m/2, as one explicitly sees from (6.47). For a → more detailed study of the geometric singularity at r = m/2, see [Nol99a, Nol99b]. For spherically symmetric spacetimes the Weyl part of the curvature has only a single independent component, which is 2/R3 times the Weyl part of the MS − energy, by the very definition of the latter (see (5.14)). The square of the Weyl tensor for the ansatz (6.39) may then be conveniently expressed as

(am)2 Weyl, Weyl = 48 . (6.72) h i R6 This shows that R = 0 also corresponds to a genuine curvature singularity, though this is not part of the region covered by our original coordinate system, for which r >m/2 (that is R> 2EW). It is instructive to also determine the trapped regions of McVittie spacetime. Re- call that a spacelike 2-sphere S is said to be trapped, marginally trapped, or untrapped if the product θ+θ− of the expansions (for the deﬁnitions of θ± see Section 5.1.2.1) for the ingoing and outgoing future-pointing null vector ﬁelds normal to S is positive, zero, or negative, respectively. Taking S to be SR, that is, an SO(3) orbit with areal radius R, it immediately follows from the relation 2 θ+θ− = g(∇R, ∇R)/R2

(see (5.9)) that SR is trapped, marginally trapped, or untrapped iﬀ g(∇R, ∇R) is positive, zero, or negative, respectively. This corresponds to timelike, lightlike, or spacelike dR, or equivalently, in view of (4.6), to 2E R being positive, zero, or − negative, respectively. Using (6.58) together with (6.51a), the MS energy for the 3 2 McVittie ansatz can be written as E = EW + R F /2, so that

2E R = F 2R3 R + R . (6.73) − − S

Here we deﬁned the ‘Schwarzschild radius’ as RS := 2EW, which generally will depend on time. We wish to determine the values of the radial coordinate (r or R) 6.3. The McVittie models 97 at which the expression (6.73) assumes the value zero. We shall continue to work with R rather than r since R has the proper geometric meaning of areal radius.

In the region we are considering (that is r > m/2 or, equivalently, R > RS) the inversion of (6.52) reads r(R) = R 1 R /2R + 1 R /R /2a, so that (6.73) − S − S divided by RS can be written in the form p

2 2E R ε − = η + x3 x +1 . (6.74) RS x 1+ x(x 1)! − − − p Here we introduced the dimensionless radial coordinate x := R/RS and the (small) parameters ε := R˙S and η := RS/RH , where RH := 1/H denotes the ‘Hubble radius’. Recall that since R > RS we have x> 1.

Consider ﬁrst the McVittie case, in which ε = R˙ S = 0. Then (6.74) turns into a cubic polynomial in x which is positive for x = 0 and tends to for x . ±∞ → ±∞ Hence it always has a negative zero (which does not interest us) and two positive zeros iﬀ R /R < 2/3√3 0.38 . (6.75) S H ≈ This clearly corresponds to the physical relevant case where the Schwarzschild radius is much smaller than the Hubble radius. One zero lies in the vicinity of the Schwarzschild radius and one in the vicinity of the Hubble radius, corresponding to two marginally trapped spheres. The exact expressions for the zeros can be easily written down, but are not very illuminating. In leading order in the small parameter

η = RS/RH , they are approximated by

R = R 1+ η2 + (η4) , (6.76a) 1 S O R = R 1 η/2+ (η2) . (6.76b) 2 H − O From this one sees that for the McVittie ansatz the radius of the marginally trapped sphere of Schwarzschild spacetime (RS ) increases and that of the FLRW spacetime

(RH ) decreases. The first feature can, for the McVittie model, be understood as an effect of the cosmological environment, whereas the latter is an effect of the inhomogeneity in form of a central mass abundance. All the spheres with R < R1 or R > R2 are trapped and those with R1

In the case in which ε = R˙S is non-zero and ‘small’ (see below in which sense), we expect that the zeros (6.76) vary smoothly in ε so that, in particular, the singularity at R = RS still remains within the inner trapped region. An expansion in ε gives, for the zero in the vicinity of the Schwarzschild radius:

R (ε)= R 1+ η2 + (2 2η + 13η2)ε + (η3,ε2) , (6.77) 1 S − O which clearly reduces to (6.76a) for ε = 0. From this expression one sees that, according to the physical expectation, in case of accretion (ε> 0) the inner marginally trapped sphere becomes larger in area, whereas in the opposite case (ε < 0) it shrinks. In our approximation (6.77), the singularity R = RS continues to lie inside the trapped region for ‘accretion rates’ ε = R˙ > η2/2 or, in terms of physical S − 98 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes quantities and re-introducing the factors of c, for R˙ /c > (R /R )2/2. However, S − S H this also characterizes the region of validity of the expansion (6.77): Given a positive η, an expansion in ε around zero exists only for ε> η2/2 since there exists no − expansion on both (ε, η) around (0, 0) (this is because the partial derivative of (6.74) with respect to x does not exist at x = 1).

6.3.2.5 Other global aspects

Another aspect concerns the global behavior of the McVittie ansatz (6.39). We note that each hypersurface of constant time t is a complete Riemannian manifold, which, besides the rotational symmetry, admits a discrete isometry given in (r, θ, ϕ) coordinates by φ(r, θ, ϕ)= (m/2)2 r−1 ,θ,ϕ . (6.78)

This corresponds to an inversion at the 2-sphere r = m/ 2, which shows that the hypersurfaces of constant t can be thought of as two isometric asymptotically-flat pieces joined together at the 2-sphere r = m/2. This 2-sphere is totally geodesic since it is a fixed-point set of an isometry; in particular, it is a minimal surface. Except for the time-dependent factor m(t), this is just like for the slices of constant Killing time in the Schwarzschild metric (the difference being that (6.78) does not extend to an isometry of the spacetime metric unlessm ˙ = 0). Now, the fact that r 0 corresponds to an asymptotically flat end of each of the 3-manifolds t = const. → implies that the McVittie metric cannot literally be interpreted as corresponding to a point particle sitting at r = 0 (r = 0 is in infinite metric distance) in an otherwise spatially flat FLRW universe, just like the Schwarzschild metric does not correspond to a point particle sitting at r = 0 in Minkowski space. Unfortunately, McVittie seems to have interpreted his solution in this fashion [McV33] which even until recently gave rise to some confusion in the literature (e.g. [Gau84, Sus88, FFS96]). A clarification was given in [Nol99a].

6.4 Attempts to generalize McVittie’s model

The first obvious generalization consists in allowing for a non-vanishing cosmological constant. However, as was already indicated before, this is rather trivial since it merely corresponds to the substitutions ̺ ̺ + ̺ and p p + p in (6.42), where → Λ → Λ ̺ := Λ/8π and p := Λ/8π are the energy-density and pressure associated to Λ Λ − the cosmological constant Λ. The attempts to non-trivially generalize the McVittie solution have focused so far on keeping the ansatz (6.39) and relaxing the conditions on the matter in various ways. In [FJ07] generalization were presented allowing radial fluid motions relative to the observer vector field ∂/∂t (that is relaxing condition (6.41)) as well as including heat conduction. Below we will critically review these attempts, taking due care of the geometric constraints imposed by the ansatz (6.39), and also outline how to explicitly construct the respective solutions. Another exact solution that models an inhomogeneity in a cosmological spacetime was presented in [SD05] by Sultana and Dyer and was recently analyzed in [Far09]. Here the metric is conformally equivalent to the exterior Schwarzschild 6.4. Attempts to generalize McVittie’s model 99 metric and the cosmological matter is composed of two non-interacting perfect fluids, one being pressureless dust, the other being a null fluid. One might ask if this solution fits into the class of McVittie models, as was suggested in [FJ07]8 and allegedly confirmed explicitly in [Far09]9. However, as we already noted at the end of Section 6.3.2.1 above in view of Proposition 6.3, this is not the case. Two further way to see this are as follows: first, the Sultana–Dyer metric is not spatially Ricci-isotropic (see the next paragraph) and, second, the McVittie metric is not compatible with the matter model used by Sultana and Dyer, with the sole exception of trivial or exotic cases, as will be shown in Section 6.4.4 below. Remark 6.6 (The Sultana–Dyer metric is not spatially Ricci-isotropic). To show this, one has to show that there exists no timelike direction with respect to which the Ricci tensor (or, equivalently, the Einstein tensor) is spatially isotropic. This can be shown as follows: First note that the Einstein tensor of the Sultana–Dyer metric has the form Ein = µu u + τk k (see [SD05]), where u is a normal- ⊗ ⊗ ized future-pointing spherically-symmetric timelike vector field and k the in-going future-pointing lightlike vector field orthogonal to the SO(3)-orbits normalized such that g(u, k) = 1. In particular, the spherical part of the Einstein tensor vanishes: Hence, the Einstein tensor is spatially isotropic iff there exists a non-vanishing spacelike spherically-symmetric (i.e. orthogonal to the SO(3)-orbits) vector field s with Ein(s, s) = 0. Without loss of generality one can chose s to be normalized: s = sinh χu+cosh χe, where e is the normalized vector field orthogonal to u and to the SO(3)-orbits pointing in positive radial direction. Hence one has k = u e and − thus: Ein(s, s) = µ sinh2(χ)+ τ exp(2χ). Clearly, the latter expression vanishes nowhere in the physically interesting region (cf. eq. (26) in [SD05]), where both µ and τ are positive.

6.4.1 Einstein’s equation for the McVittie ansatz

In the following we will restrict to those generalizations of the McVittie model which keep the metric ansatz (6.39) and thus generalize only the matter model. For this purpose it is convenient to write down the Einstein’s equation for an arbitrary spherically symmetric energy-momentum tensor T . Recall that spherical symmetry implies for the component of T with respect to the orthonormal basis (6.48) that T (e , e ) = 0 and T (e , e ) δ , where a 0, 1 and A, B 2, 3 . Hence, a A A B ∝ AB ∈ { } ∈ { } 8 In Sect. IV A of [FJ07] it is incorrectly suggested that the Sultana–Dyer metric is equal to the 2/3 McVittie metric (6.39) in which a(t) = a0t and m(t) = m0, for some constants a0 and m0 (see eq. (62) in [FJ07]). What might have led to this suggestion is the fact that both metrics are conformally related to the Schwarzschild metric (the first by its very definition and the second because of Proposition 6.3) and both have vanishing spherical part of the Einstein tensor (for the latter metric this can be easily checked with our eq. (6.51c)). Moreover, imposing the zero-pressure condition to the McVittie metric, that is requiring that the right-hand side of our eq. (6.51c) vanishes, gives a differential equation for the time-dependent functions a and m of the McVittie ansatz (see our (6.39)). This (r-dependent) equation has two solutions: a and m both constant, and a(t) ∝ t2/3 and m constant. However, as we argument in the text, none of them leads to the Sultana–Dyer metric. 9 The problem with the reasoning in Sect. II of [Far09] is the following (numbers refer to equations in [Far09]): it is true by construction that the Sultana–Dyer metric (2.1) is conformally related to the Schwarzschild metric, as expressed in the second line of (2.3) (the first line in (2.3) does not follow), but the conformal function a depends non-trivially on the Schwarzschild coordinates for time and radius (denoted byη ¯ andr ˜ in [Far09]: cf. our discussion in the last paragraph of Section 6.3.2.1). Hence it is not possible to introduce a new time coordinate t¯ that satisfies dt¯ = adη¯ (the right-hand side is not a closed 1-form), as pretended in the transition to (2.5). 100 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes the only independent, non-vanishing components of T are

S := T (e0, e0) , (6.79a)

Q := T (e1, e1) , (6.79b)

P := T (e2, e2) , (6.79c) J := T (e , e ) , (6.79d) − 0 1 and these are functions which do not depend on the angular coordinates. Note that S is the energy density, Q and P the radial and spherical pressure, and J the energy flow—all referred to the observer field e0. The sing in (6.79d) is chosen such that a positive J means a flow of energy in an outward-pointing radial direction. Taking (6.79) into account, the Einstein equation for the McVittie ansatz (6.39) and an arbitrary spherically symmetric energy-momentum tensor T reduces to the following four equations:

· 2 (am) = 4πR2 B J , (6.80a) − A 8πS =3F 2 , (6.80b) 8πQ = 3F 2 2F˙ 2 A , (6.80c) − − B P = Q . (6.80d)

In view of (6.59), the first equation relates the time variation of the Weyl part of the MS energy contained in the sphere of radius R with the energy flow out of it. The last equation is nothing but spatial Ricci-isotropy. In the following subsections we will consider three models for the cosmological matter which generalize the original McVittie model: perfect fluid, perfect fluid plus heat flow, and perfect fluid plus null fluid.

6.4.2 Perfect fluid Perhaps the simplest step one can take in trying to generalize the McVittie model is to stick to a single perfect fluid for the matter, but dropping the condition (6.41) of ‘no-infall’ by allowing for radial motions relative to the ∂/∂t observer field. In this way one could hope to avoid a particular singular behavior in the pressure that may be due to the ‘no-infall’ condition, though it is clear that the persisting geometric singularity must show up somehow in the matter variables as already discussed in Section 6.3.2.4. Unfortunately, as already shown in [FJ07], the relaxation of (6.41) does not lead to any new solutions. What we want to stress here is that the reason for this, as shown in more detail below, lies precisely in the restriction imposed by spatial Ricci-isotropy. We take thus the perfect-fluid energy-momentum tensor (6.40) for the matter and an arbitrary spherically symmetric four-velocity u. The latter is given in terms of the orthonormal basis for the metric (6.39) by

u = cosh χ e0 + sinh χ e1 , (6.81) where χ is the rapidity of u with respect to the observer ﬁeld e0 (a positive χ corresponds here to a boost in an outward-pointing radial direction). The non- 6.4. Attempts to generalize McVittie’s model 101 vanishing components of the matter energy-momentum tensor (6.40) with four- velocity (6.81) are: 2 T (e0, e0)= ̺ + (̺ + p) sinh χ (6.82a) T (e , e )= (̺ + p) sinh χ cosh χ (6.82b) 0 1 − 2 T (e1, e1)= p + (̺ + p) sinh χ (6.82c)

T (e2, e2)= T (e3, e3)= p . (6.82d)

Clearly, the case of vanishing rapidity must lead to the original McVittie model. In this case, in fact, the matter energy-momentum tensor (6.82) is already spatially isotropic so that (6.80d) is identically satisﬁed. Moreover, (6.80a) implies (am)˙ = 0 and hence, in view of (6.53), F =a/a ˙ . Herewith Einstein’s equation reduces to (6.42) and thus one gets back the original McVittie model. In case of non-vanishing rapidity, spatial Ricci-isotropy (6.80d) implies the following constraint: ̺ + p =0 . (6.83) This means that the energy momentum tensor (6.40) has the form of a cosmological constant (using (6.83) in div T = 0 it implies dp = 0 and this, in turn, using again (6.83), implies d̺ = 0) so that this case reduces to the Schwarzschild–de Sitter solution and hence does not provide the physical generalization originally hoped for.

6.4.3 Perfect ﬂuid plus heat ﬂow

In a next step one may keep (6.81) and drop the condition that the fluid be perfect, in the sense of allowing for radial heat conduction. This is described by a spatial vector field q that represents the current density of heat, which here corresponds to the current density of energy in the rest frame of the fluid. Hence q is everywhere orthogonal to u.10 The fluid’s energy momentum tensor then reads

T = ̺ u u + p (u u g)+ u q + q u . (6.84) ⊗ ⊗ − ⊗ ⊗ Taking (6.81) as ﬂuid velocity and imposing the heat ﬂow-vector q to be spherically symmetric, we have

q = q e := q (sinh χ e0 + cosh χ e1) , (6.85) where q is a function of (t, r). Note that a positive q corresponds to heat ﬂowing in an outward-pointing radial direction. The independent non-vanishing components of the energy-momentum tensor are now as follows:

T (e0, e0)= ̺ + tanh χ (̺ + p) tanh χ +2 q (6.86a) T (e , e )= q cosh2 χ (̺ + p) tanh χ +2 q (6.86b) 0 1 − 1 T (e1, e1)= p + 2 sinh(2χ) (̺ + p) tanh χ +2 q (6.86c)

T (e2, e2)= T (e3, e3)= p . (6.86d)

10 We note that the parametrization of the energy-momentum tensor given in [FJ07] is manifestly different. Whereas we parametrized it in the usual fashion in terms of quantities (energy density, pressure, current density of heat) that refer to the fluid’s rest system, the authors of [FJ07] also write down (6.84) (their eq. (79)), but with q orthogonal to e0 (compare their eq. (93)) rather than u, which affects also the definition of ̺. In fact, marking their quantities with a prime, their expression (79) is equivalent to our (6.84) iff p = p′, q = q′ cosh χ, and ̺ = ̺′ − 2q′ sinh χ. 102 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

Consider ﬁrst the case of vanishing rapidity. Then the energy-momentum tensor is already spatially isotropic and Einstein’s equation (6.80) reduces to

2 (am)˙ = 4πR2 q B , (6.87a) − A 8π̺ =3F 2 , (6.87b) 8πp = 3F 2 2F˙ A . (6.87c) − − B These are three partial differential equations (though only time derivatives occur) for the five functions a,m,̺,p, and q so that the system (6.87) is clearly under- determining. However, it is not possible to freely specify any two of these five functions and then determine the other three via (6.87). For example, since the left-hand side of (6.87a) depends only on t, the same must hold for the right-hand side, which implies that q = f(t)/r2(1 (m/2r)2)2, where f(t)= (am)˙/4πa2. In − − particular, the heat flow must fall-off as 1/r2. The easiest way to generate a solution in the case of zero rapidity is to specify the two functions a(t) and m(t), then let A,B,R,F be determined by the definitions (6.50,6.52,6.53), and finally let the Einstein equations (6.87a,6.87b,6.87c) determine q,̺, and p, respectively. Notice that if we happen to specify a and m such that am is a constant, this immediately implies q = 0 and F =a/a ˙ , which leads to the standard McVittie solutions. From (6.87a) the following is evident: if q> 0 (q < 0), that is for outwardly (inwardly) pointing heat flow, the Weyl part of the MS energy decreases (increases), as one would expect. Now we turn to the general case with non-vanishing rapidity: As it was the case for the perfect fluid in the previous subsection, the condition (6.80d) of spatial Ricci-isotropy implies a constraint on the matter:

(̺ + p) tanh χ +2 q =0 . (6.88a)

Using this, the other components of the Einstein’s equation reduce to:

˙ 2 B 2 (am) = +4πR q A , (6.88b) 8π̺ =3F 2 , (6.88c) 8πp = 3F 2 2F˙ A . (6.88d) − − B These are almost the same as in the case of vanishing rapidity (see (6.87)), except for the opposite sign on the right-hand side of (6.88b). This simply results from the fact that, according to (6.86b), J = T (e , e ) = q for vanishing rapidity, − 0 1 whereas, due to the constraint (6.88a), J = T (e , e ) = q for non-vanishing − 0 1 − rapidity. This will be further interpreted below. Notice that for the equation of state ̺ + p = 0 (cosmological term) (6.88a) implies q = 0, thus leading once more to the Schwarzschild–de Sitter solution (see comment below eq. (6.83)). Henceforth we assume ̺ + p = 0, which implies that one can solve the constraint (6.88a) for 6 the rapidity: 2 q tanh χ = (6.89) −̺ + p provided that 2q/(̺ + p) < 1. | | 6.4. Attempts to generalize McVittie’s model 103

The Einstein equation gives now four equations for the six functions a,m,̺,p,q, and χ. As in the case of vanishing rapidity, this system is under-determining and it is not possible to freely specify any two of these six functions and then determine the other four. In a similar fashion as before, the easiest way to generate a solution is to specify the two functions a(t) and m(t), to let then the deﬁnitions (6.50,6.52,6.53) determine A,B,R,F , and ﬁnally use the Einstein equations (6.88b,6.88c,6.88d) and (6.89) to determine q,̺,p, and χ, respectively. Again, choosing a and m such that their product is constant implies q = 0 and F =a/a ˙ , which leads to the standard McVittie solutions. In passing we remark that the condition ̺ + p > 0 can be expressed geometrically in terms of the second time-derivative of the areal radius. Indeed, adding either (6.87b) to (6.87c) or (6.88c) to (6.88d) we obtain, taking into account e0 = (A/B)∂/∂t and (6.54):

e (R) 4π(̺ + p)= e 0 , (6.90) − 0 R which is positive iff the rate of change e0(R)/R is a decreasing function along the integral lines of the observer e0. In other words, ̺ + p is positive iff ln(R) is a concave function on the worldline of the observer e0, which is implied by, but not equivalent to, the function R being concave. From (6.89) and (6.88b), and assuming ̺ + p > 0, one sees the following: If χ > 0 (χ < 0), that is for an outwardly (inwardly) moving fluid with respect to e0, we have q < 0 (q > 0), that is an inwardly (outwardly) pointing heat flow, and the Weyl part of the MS energy decreases (increases). This means that the heat

flow’s contribution to the change of EW never compensates that of the fluid motion, quite in accord with naive expectation. Below we show that for small rapidities the contribution due to the heat flow is minus one-half that of the cosmological matter. Let us now return to the sign-difference of the right-hand sides of (6.87a) and (6.88b). From (6.86) one infers that J is the sum of the two contributions coming from the heat flow

2 Jh := q(1 + 2 sinh χ) (6.91) and from cosmological matter

Jm := (̺ + p) sinh χ cosh χ , (6.92) respectively. The constraint (6.88a) can be written in the form

2 2 2 cosh χJh + (1 + 2 sinh χ)Jm =0, (6.93) which, for small rapidities χ (that is neglecting quadratic terms in χ), implies 2Jh + J 0. In this approximation the spatial energy-momentum flow due to heat is m ≈ minus one-half that due to the cosmological matter. For the total flow this implies J = J + J J /2 J . Now the sign difference between (6.87a) and (6.88b) m h ≈ m ≈− h is understood as follows: In case of vanishing rapidity one has Jm = 0, Jh = q and hence J = q (leading to (6.87a)), whereas a short calculation reveals that in case of non-vanishing rapidity the constraint (6.93) implies J = J + J = q, leading m h − thus to (6.88b). 104 Chapter 6. Spherically symmetric inhomogeneities in cosmological spacetimes

6.4.4 Perfect fluid plus null fluid The last tentative generalization we consider is taking for matter the incoherent sum (meaning that the respective energy-momentum tensors adds) of a perfect fluid (possibly with non-vanishing pressure) and a null fluid (eventually representing electromagnetic radiation). This clearly contains as special case the matter model considered by Sultana and Dyer [SD05] in which the pressure vanishes. We already stressed in Section 6.3.2.1 that the metric ansatz of [SD05] is different from (6.39). Here we show that the matter model of [SD05] is essentially incompatible with (6.39) except for trivial or exotic cases. The matter model consists of an ordinary perfect fluid and a null fluid (e.g. electromagnetic radiation) without mutual interaction. Hence the matter’s energy- momentum tensor is just the sum of (6.40) and

T ± = λ2 l± l± , (6.94) nf ⊗ where λ is some non-negative function of t and r and l+ and l− are, respectively, the outgoing and ingoing future-pointing null vector ﬁelds orthogonal to the spheres of constant radius r partially normalized such that g(l+, l−) = 1. (It remains a freedom l± α±1l±, where α is a positive function). Without loss of generality 7→ we make use of this freedom and choose: l± = (e e )/√2 , (6.95) 0 ± 1 where e0 and e1 are the vectors of the orthonormal frame (6.48). The components of the whole energy-momentum tensor with respect to this frame are then: 2 1 2 T (e0, e0)= ̺ + (̺ + p) sinh χ + 2 λ (6.96a) T (e , e )= (̺ + p) sinh χ cosh χ 1 λ2 (6.96b) 0 1 − ∓ 2 2 1 2 T (e1, e1)= p + (̺ + p) sinh χ + 2 λ (6.96c)

T (e2, e2)= T (e3, e3)= p . (6.96d)

Here and below the upper (lower) sign corresponds to the outgoing (ingoing) null ﬁeld. In the present case, the condition (6.80d) of spatial Ricci-isotropy is equivalent to the constraint: 2 1 2 (̺ + p) sinh χ + 2 λ =0 . (6.97) In the physically relevant case in which ̺ + p> 0 this equation has only the trivial solution χ = 0 and λ = 0, which leads to the original McVittie model. In the case ̺ + p = 0 (6.97) implies λ = 0, leading thus to the Schwarzschild–de Sitter spacetime (see comment below (6.83)). Hence, a new solution is only possible if the matter is of an exotic type that satisﬁes ̺ + p < 0, which either violates the weak energy-condition (̺ > 0), or, less catastrophically, the dominant-energy condition (̺> p ). In particular, for the matter model considered by Sultana and Dyer, one | | would need to violate the weak energy-condition.

6.4.5 Conclusion We conclude by commenting on the main differences between these generalizations and the original McVittie model. First we stress once more that neither allowing 6.4. Attempts to generalize McVittie’s model 105 for a nonzero rapidity nor a nonzero heat flow can eliminate the singularity at r = m/2 (R = 2am) (as erroneously stated in [FJ07]). The only substantial new feature of these generalizations is that the Weyl part of the MS energy EW = am is not constant anymore. In fact, in this general case, one has to substitute m0 with EW in the expression (6.69) for the acceleration of the Kodama observer field. Accordingly, the same substitution has to be done in the slow-motion and weak- field approximation of the spatial equation of motion (7.14). This means that the strength of the central attraction varies in time according to (6.80a), leading to an in- or out-spiraling of the orbits if dEW(e0) > 0 or dEW(e0) < 0, respectively. We identified the origin of why we could not vary the rapidity and the heat flow independently in the condition (6.88a) of spatial Ricci-isotropy, which is built into the ansatz (6.39). We saw that this geometric feature renders this ansatz special, so that it would be improper to call it a general ansatz for spherical inhomogeneities in a flat FLRW universe. Nevertheless, at the time, the McVittie model remains the better understood exact model describing such situation.

Chapter 7

Cosmological eﬀects on local dynamics

The aim of this chapter is to study the eﬀect of the cosmological expansion on the dynamics of bounded system. We are interested in systems where the binding is of electromagnetic nature (as, e.g., atoms) as well as in systems which are gravitationally bounded (as a star-planet system). The former case is considered in Section 7.1, where we specialize to the restricted two-body problem for two charges in a FLRW spacetime. The gravitational case is studied in Section 7.2, where we consider the motion of a test particle in a McVittie spacetime. In both cases we derive the spatial version (in the sense of Section 3.1.3) of the general relativistic equation of motion, which, in an appropriate slow-motion and weak-ﬁeld approximation, gives the cosmological corrections to the Newtonian equation of motion. This improved Newtonian equation is then studied in Section 7.3. Moreover, we also analyze the exact condition for the existence of non-expanding circular orbits, which gives a restriction on the scale-factor of the spacetime (see Section 7.1.2 and 7.2.2).

7.1 Electromagnetically-bounded systems

In this section we derive the equation of motion for an electromagnetically-bounded restricted two-body problem in an expanding (spatially flat) universe. By ‘restricted’ we mean that we specialize to the case in which the mass of one particle is much bigger than the mass of the other, as is the case for a proton-electron system. This implies solving Maxwell’s equations in the cosmological background (7.3) for an electric point charge (the proton) and then write down the Lorentz equation for the motion of a particle (electron) in a bound orbit (cf. [Bon99]). This we do in Section 7.1.1 where we then derive the spatial version of Lorentz equation (in the sense of Section 3.1.3) w.r.t. the Kodama observer field. Its slow-motion and weak-field approximation gives then a Newtonian-like equation improved with a correction term due to cosmological expansion. In [CG10b] we reach the same result following an alternative approach based on a careful analysis of the famous argument of Dicke & Peebles [DP64]. In [DP64] Dicke & Peebles presented an apparently very general and elegant argument, that purports to show the insignificance of any dynamical effect of cosmological expan-

107 108 Chapter 7. Cosmological eﬀects on local dynamics sion on a local system that is either bound by electromagnetic or gravitational forces which should hold true at any scale. Their argument involves a rescaling of spacetime coordinates, (t, ~x) (λt, λ~x) and certain assumptions on how other 7→ physical quantities, most prominently mass, behave under such scaling transformations. However, as we show in [CG10b], their argument is really independent of such assumptions since it merely rely on the conformally invariance of the free Maxwell’s equations in four spacetime dimensions. In the following we adopt the former approach. This because of its striking simplicity and also because it allow to directly apply what we have learned in the previous chapters.

7.1.1 Equation of motion in FLRW spacetimes

We recall that Maxwell’s equations are given by (see e.g. Sect 1.4.3 in [Str84]):

δF =4πJ , (7.1a) dF =0 , (7.1b) where J is the electric-charge current (one-form). We recall that the electric charge Q is given by the charge to the current J (which is conserved by (7.1a)). Using (7.1a) and Stokes’ Theorem one gets

1 Q = ∗F , (7.2) −4π 2 ZS∞ ∗ 1 2 where denotes the spacetime’s Hodge-dual and S∞ the two-sphere at spacelike infinity. We solve now the Maxwell equations in a spatially-flat (k = 0) FLRW spacetime, 2 2 2 2 g = dt a (t) dr + r g 2 , (7.3) − S for a current given by a point charge, Q, moving along an integral curve of the cosmological vector field u = ∂/∂t, and which, w.l.o.g., we shall assume being situated at r = 0. Since the domain of F is simply connected (being topologically R (R3 0 )) PoincaréLemma ensures the existence of a potential A with F = dA, × \{ } determined up to a gauge transformation A A + dχ for some smooth function χ. 7→ Spherical symmetry now imposes that A = Φ(t, r)dt+Ψ(t, r)dr, which, making use of a gauge transformation, can always be brought to the form A = φ(t, r)dt. For r = 0 we have δF = 0, which implies φ(t, r) = C/(a(t)r)+ f(t) for some constant 6 C and some function f. W.l.o.g. we set f to zero, since it does not contribute to the field-strength F . The constant C is then fixed by (7.2), which gives C = Q. The potential is hence given by Q A = θ0 (7.4) R and the field-strength by

Q 0 1 Q F = dA = θ θ = µB . (7.5) R2 ∧ R2 1 This is in order to distinguish it from the Hodge-dual in the two-dimensional base manifold B, denoted by ⋆ (see Section 4.2). 7.1. Electromagnetically-bounded systems 109

Here we make use of the objects defined in Section 6.2.1: R denotes the areal µ 0 1 radius (6.14), θ the orthonormal one-forms (6.23), and µB = θ θ the volume ∧ form on the basis manifold (recall that we omit lifts). Note that in the spatially- B flat case considered here, the areal radius R is also equal to the proper distance (measured on the slices of constant cosmological time) to the point r = 0. Now, the equation of motion for a (test) particle of charge q and mass m in a given electromagnetic field is given by the Lorentz equation

q ∇γ γ˙ = (iγ F ) , (7.6) ˙ −m ˙ where γ denotes the particle’s worldline. For the electromagnetic field (7.5) this reads ∇ qQ 1 γ˙ γ˙ = eR + hu (βu (γ˙ ), eR)uK , (7.7) mR2 2 K K 1 βuK (γ˙ ) − p where we use the decomposition (3.7) of γ˙ w.r.t. the Kodama observer uK. Recall also that eR is the outwardly-pointing vector field orthogonal to uK which, for the FLRW case, is given by (6.26b). In the slow-motion and weak-field approximation, that is neglecting quadratic

(and higher) terms in βuK (γ˙ ) and keeping only leading (hence quadratic) terms in HR = R/RH , the spatial equation of motion (3.28) to (7.7) w.r.t. the Kodama observer ﬁeld is qQ a¨ αu (γ) + R e , (7.8) K ≈ mR2 a R where we have used (6.31) and (6.32).

7.1.2 Exact condition for non-expanding circular orbits

In [Bon99] a necessary and sufficient condition for the existence of non-expanding orbits is derived for the electron-proton system in a spatially flat FLRW spacetime. Here ‘non-expanding’ is defined as of constant areal radius. This condition follows directly from the Lorentz equation of motion for the electron in the external electric field of the proton, the normalization condition of the electron’s four-velocity, and the condition of constancy of the areal radius. In our notation, introducing the dimensionless quantities h(t) := RH(t)/c, l := L/Rc, and µ := Re/2R, the conditions for the existence of non-expanding circular orbits reads as follows:

2 3/2 2 2 R ˙ (1 h ) l + h h = − 2 1/2 µ . (7.9) c (1 + l ) − (1 + l2)(1 h2) ! − p Recall that Re is defined in (7.24) and H(t) and L denote the Hubble function and, respectively, the (conserved) electron’s angular momentum per unit mass. The above condition is a first-order autonomous ODE for the function h(t), and hence for the Hubble function H(t). This is the constraint on the spacetime (more precisely, on the scale factor a(t)) that one gets by imposing the existence of non-expanding circular orbits for two oppositely charged point masses. If such orbits exist, (7.9) amounts to the generalization of Kepler’s third law to FLRW spacetimes, which here gives a relation between the scale function on one hand and the orbital parameters 110 Chapter 7. Cosmological effects on local dynamics

R and L as well as the ﬁeld-strength parameter Re on the other. Recall that in Newtonian physics the third Kepler law is, in our notation, simply given by l2 = µ. The easiest solutions of (7.9) are of course the stationary ones, that is with h(t) h , for some constant h . This means that the scale factors is exponentially ≡ 0 0 expanding,

a(t)= a0 exp(H0t) , (7.10) where H0 := h0c/R and a0 is some positive constant. In other words, the spacetime is given by the de Sitter solution (Λ-dominated universe). In this case (7.9) reduces to l2 + h2 0 = µ . (7.11) (1 + l2)(1 h2) − 0

Notice that a larger Hubble parameter,p hence a larger h0, makes the l.h.s. larger. Consequently, (7.11) tells us that with a larger Hubble parameter we must give to the electron a smaller angular velocity (smaller l) in order to keep it on a non- expanding circular orbit with the same radius. This, according to intuition, is in order to compensate the extra cosmological pull with a reduced centrifugal term. 2 2 In case of Minkowski spacetime (h0 = 0) the above relation reads l /√1+ l = µ, hence one can interpret the factor 1/√1+ l2 as a special-relativistic correction to the Newtonian relation l2 = µ. The largest radius at which, in an FLRW spacetime with exponentially-growing scale factor, there is a non-expanding orbit follows from (7.11) in the limit l 0. In this limit the condition reduces to → h2/ 1 h2 = µ, which, for small parameters h , simpliﬁes to h2 µ. Solving 0 − 0 0 0 ≈ for R this gives the radius (R R2 /2)1/3, which, taking into account that q = 1 p e H 0 − because of (7.10), exactly corresponds to the critical radius (7.25). The other (non-stationary) solutions of (7.9) can also be found. After separation of variables and an elementary integration one gets t as function of h in terms of trigonometric functions composed with inverse hyperbolic functions. This exact expression is again not very illuminating and cannot generally be explicitly inverted so as to obtain h in terms of elementary functions of t. However, if we make use of the smallness of the parameters µ, l2, and h2, a leading-order expansion in these quantities gives a much simpler expression for t(h) which can be explicitly inverted. In fact, this approximate solution h(t) is obtained much quicker by solving (7.9) with the right-hand side being replaced with its leading-order expansion in the mentioned quantities, that is, by solving

R h˙ = µ l2 h2 . (7.12) c − −

Here µ l2 is a constant which depends on the orbit parameters. One must now − distinguish between three cases: (a) µ l2 =: κ2 > 0 for some positive κ, (b) µ l2 =: − − ν2 < 0 for some positive ν, and (c) µ l2 = 0. Recalling the Newtonian relation − − l2 = µ, orbits in the three cases have an angular momentum which is, respectively, smaller, bigger, and equal to the Newtonian one. Integrating (7.12) we get, putting w.l.o.g. t = 0, h(t) = κ tanh(κct/R), h(t) = ν tan(νct/R), and h(t) = R/ct, for 0 − the cases (a), (b), and (c), respectively. Then, integrating once and exponentiating the result, we get the corresponding scale functions: 7.2. Gravitationally-bounded systems 111

(a) Case µ l2 =: κ2 > 0 (non-expanding orbits have sub-Newtonian angular − momentum) κct a(t)= a cosh , t [0, ) . (7.13a) 0 R ∈ ∞

(b) Case µ l2 =: ν2 < 0 (non-expanding orbits have super-Newtonian angular momentum)− −

νct πR a(t)= a cos , t 0, . (7.13b) 0 R ∈ 2νc

In all three cases (a), (b), and (c) a0 is a positive constant and the acceleration terma/a ¨ is a constant which is positive, negative, and zero, respectively. Hence, as one would intuitively expect, the non-expanding orbits have an angular momentum which is smaller, larger, or equal the Newtonian one, depending on whether the acceleration factora/a ¨ is positive, negative, or zero.

7.2 Gravitationally-bounded systems

In the following, according to the discussion of Section 6.1, we take the McVittie solution as model for the external ﬁeld of a non-rotating quasi-isolated body (we think thereby at the Sun, as a title of example) immersed in a cosmological spacetime and we set up, in an appropriate approximation scheme, the spatial equation of motion for a test particle (idealizing a planet or a spacecraft) in this spacetime.

7.2.1 Equation of motion in McVittie spacetime

In [McV33] McVittie concluded, within a slow-motion and weak-field approximation, that Keplerian orbits do not expand as measured with the ‘cosmological geodesic radius’ r∗ = a(t)r. Later Pachner [Pac63] and Noerdlinger & Pet- rosian [NP71] argued for the presence of a radial acceleration term (ä/a)r∗ within this approximation scheme, hence arriving at (7.18a). In [CG10b] we show how to arrive at the cosmologically-improved Newtonian equation (7.18a) from the exact geodesic equation for the McVittie metric by making clear the approximations involved. Here we arrive at (7.18a) from the spatial version of the geodesic equation w.r.t. the Kodama observer field uK. Denoting with γ the geodesic worldline of the particle (planet or spacecraft) and taking into account (6.69) and (6.70), the spatial equation of motion (3.28) w.r.t. uK can be immediately written down:

m0 a¨ αu (γ) + R e . (7.14) K ≈ − R2 a R We recall that the validity region of the above equation is given (6.68). The latter condition clearly covers all situations of practical applicability in the Solar System, 112 Chapter 7. Cosmological eﬀects on local dynamics since the Schwarzschild radius R of the Sun is about 3 km = 2 10−8AU and the S · ‘Hubble radius’ R is about 13.7 109ly=8.7 1014AU. Hence the small parameter H · · η = R /R of the Sections 6.3.2.3 and 6.3.2.4 is about 2.3 10−23. S H ·

7.2.2 Exact condition for non-expanding circular orbits

Analogously to Section 7.1.2, where we ask whether there exist non-expanding circular orbits (i.e. of constant areal radius) of the electron-proton system in a FLRW spacetime, we now ask whether there exist non-expanding circular orbits of an (uncharged) test particle around the central mass. The necessary and suﬃcient condition for this to happen follows from inserting R = const in the radial part of the geodesic equation and using the normalization condition g(γ˙ , γ˙ )= c2 (reintroducing c) of the four-velocity in order to eliminate t˙ (see [CG10b]). In terms of the dimensionless quantities h(t) := RH(t)/c, l := L/Rc, and µ := m0/R, the condition for the existence of non-expanding circular orbits can be given the following form:

R 1 2µ h2 µ(1+3l2) l2 h2 h˙ = − − − − . (7.15) c (1 + l2)√1 2µ −

As for the electron-proton system in an FLRW spacetime (see (7.9)) this is a ﬁrst- order autonomous ODE for h(t) and therefore the Hubble function. In the present case the ODE is even simpler since it has the elementary form h˙ = p(h2), where p is a polynomial of degree two with constant coeﬃcients. From (7.15), to leading order in the small quantities µ, l2, and h2, we get the same approximate ODE (7.12) and hence the same approximate solutions (7.13). Hence, the same conclusions as drawn for the electron-proton system in FLRW apply here.

From (7.15) it follows that stationary solutions h(t) = const =: h0, corresponding to an exponentially-growing scale factor (7.10) (and hence leading to a

Schwarzschild–de Sitter spacetime), are those where h0 satisﬁes

l2 + h2 0 = µ , (7.16) (1+3l2) where we used that the ﬁrst factor on the numerator of the right-hand side of (7.15) is nonzero, as can be immediately inferred from the normalization condition 2 g(γ˙ , γ˙ ) = c . Notice that for a vanishing Hubble parameter (that is for h0 = 0) the above condition reduces to the third Kepler law in Schwarzschild spacetime, as expected. The eﬀect of a non-vanishing Hubble parameter is again that we must provide the orbiting particle with a smaller angular velocity (smaller l) in order to keep it on a non-expanding circular orbit with the same radius. The largest radius at which in a McVittie spacetime with exponentially-expanding scale factor (that is a Schwarzschild–de Sitter spacetime) there is a non-expanding circular orbit follows from (7.16) in the limit l 0. Then the condition reduces to h2 = µ which, solving → 0 2 1/3 for R, gives (RSRH /2) . This, exactly corresponds to the critical radius (7.23), taking into account that q = 1 for an exponentially-growing scale factor. 0 − 7.3. Next-to-Newtonian analysis 113

7.3 Next-to-Newtonian analysis

The spatial version of the equation of motion for an electron in the electromagnetic field of a proton in a spatially flat FLRW spacetime as well as the geodesic equation in the McVittie spacetime have, in the slow-motion (neglecting quadratic or higher order in β = v/c) and weak-field approximation, the form (see (7.8) and (7.14), respectively): C a¨ αu (γ)= + R e . (7.17) K −R2 a R

Here, in the gravitational case C = m0, where m0 is the mass of the central body, and in the electromagnetic case, for the electron-proton system, C = e2/m (Gaus- sian unit), where e and m are the electron’s charge and mass, respectively. The acceleration in (7.17) has thus the familiar (Newtonian) attractive one-over-R term and, in addition, there is an outwardly-pointing acceleration due to the cosmological expansion. The latter term is determined after speciﬁcation of a scale function a(t). The vanishing of the spherical part of the r.h.s. is equivalent to the conservation of angular momentum, which, as usual, we can employ to restrict, w.l.o.g., the motion to the ‘equatorial plane’ θ = π/2. Equation (7.17) is then equivalent to a system of two ODEs for the variables (R, ϕ), which describe the position of the orbiting body with respect to the central one:

L2 C a¨ R¨ = + R (7.18a) R3 − R2 a R2ϕ˙ = L . (7.18b)

These are the (¨a/a)–improved Newtonian equations of motion for the restricted two- body problem, where L represents the (conserved) angular momentum of the planet (or electron) per unit mass and C the strength of the attractive 1/R2 force. We now wish to study the eﬀect the correction term due to the cosmological expansion,

a¨ R¨ = R = q H2 R , (7.19) | cosm.acc. a − has on the unperturbed Kepler orbits. We start with the obvious remark that this term results from the acceleration and not just the expansion of the universe and, moreover, it is of quadratic order in H. Next we point out that in the concrete physical cases of interest, the time dependence of this term is negligible to a very good approximation, since this would be a contribution of third order in H. Indeed, putting f :=a/a ¨ , the relative time variation of the coefficient of R in (7.19) is f/f˙ . For an exponential scale function a(t) exp(λt) (Λ-dominated universe) this ∝ vanishes, and for a power law a(t) tλ (for example matter-, or radiation-dominated ∝ universes) this is 2H/λ, and hence of the order of the inverse age of the universe. − If we consider a planet in the Solar System, the relevant time scale of the problem is the period of its orbit around the Sun. The relative error in the disturbance, when treating the factora/a ¨ as constant during an orbit, is hence smaller than 10−9. For atoms it is much smaller, of course. In principle, a time varyinga/a ¨ causes changes in the semi-major axis and eccentricity of Kepler orbits [SJ07]. But here we shall neglect the time-dependence of (7.19) and seta/a ¨ equal to a constant A. Because 114 Chapter 7. Cosmological effects on local dynamics of (6.30) we have A := q H2. Then (7.18a) can be immediately integrated: − 0 0 1 R˙ 2 + U(R)= E , (7.20) 2 where the effective potential is

L2 C A U(R)= R2 . (7.21) 2R2 − R − 2

We will see below that the three parameters (L,C,A) can be eﬀectively reduced to two.

7.3.1 Specifying the initial-value problem

Solutions of (7.20) and (7.18b) are speciﬁed by initial conditions (R, R,˙ ϕ, ϕ˙)(t0)=

(R0, V0, ϕ0,ω0) at the initial time t0. The discussion of the dynamical behavior of R is most effectively done in terms of the effective potential. Moreover, since perturbations are best discussed in terms of dimensionless parameters, we also introduce a length scale and a time scale that appropriately characterize the dynamical perturbation and the solution to be perturbed. The length scale is defined as the radius at which the acceleration due to the cosmological expansion has the same magnitude as the two-body attraction. This happens precisely at the critical radius

C 1/3 R := . (7.22) c A | |

For R < Rc the two-body attraction dominates, whereas for R > Rc the effect of the cosmological expansion is the dominant one. In order to gain an understanding of the length scales of the critical radius it is instructive to express it in terms of the physical parameters. In the case of gravitational interaction we have C/ A = GM/( q H2) and thus | | | 0| 0 1/3 R R2 R = S H . (7.23) c 2 q | 0| Inserting the approximate value q 1/2 of the present epoch, this reduces to the 0 ≈− Schücking vacuole radius (6.2). In the electromagnetic case, e.g. for an electron-proton system, we have C/ A = | | (e2/m)/( q H2). Defining, in analogy with R , the length scale | 0| 0 S 2e2 R := 5.64 10−15 m , (7.24) e mc2 ≈ · the critical radius (7.22) becomes

R R2 1/3 R = e H 30 AU , (7.25) c 2 q ≈ | 0| where in the last step we inserted q 1/2. This is about as big as the Neptune 0 ≈− orbit! 7.3. Next-to-Newtonian analysis 115

From (7.23) and (7.25) one sees that, in both cases, a larger (smaller) q implies | 0| a smaller (larger) critical radius, according to expectations. So much for the length scale. The time scale is deﬁned to be the period of the unperturbed Kepler orbit (a solution to the above problem for A = 0) of semi-major axis R0. By Kepler’s third law it is given by

R3 1/2 T := 2π 0 . (7.26) K C It is convenient to introduce two dimensionless parameters which essentially encode the initial conditions R0 and ω0.

ω 2 L2 λ := 0 = , (7.27) 2π/T CR K 0 R 3 R3 α := sign(A) 0 = A 0 . (7.28) R C c For close to Keplerian orbits λ is close to one. For reasonably sized orbits α is close to zero. For example, in the Solar System, where R < 100 AU, one has α < 10−16. 0 | | For an atom whose radius is smaller than 104 Bohr-radii we have α < 10−57. | | Now, deﬁning

x(t) := R(t)/R0 , (7.29) equations (7.20) and (7.18b) can be written as 1 x˙ 2 + (2π/T )2 u (x)= e (7.30) 2 K λ,α 2 x ϕ˙ = ω0 , (7.31)

2 where e := E/r0 now plays the role of the energy-constant and where the reduced two-parameter eﬀective potential uλ,α is given by

λ 1 α u (x) := x2 . (7.32) λ,α 2x2 − x − 2

The initial conditions now read

(x, x,˙ ϕ, ϕ˙)(t0)=(1, V0/R0, ϕ0,ω0) . (7.33)

The point of introducing the dimensionless variables is that the three initial parameters (L,C,A) of the eﬀective potential could be reduced to two: λ and α. This will be convenient in the discussion of the potential.

7.3.2 Discussion of the reduced eﬀective potential

Circular orbits correspond to extrema of the effective potential (7.21). Expressed in terms of the dimensionless variables this is equivalent to u′ (1) = λ +1 α = λ,α − − 0. By its very definition (7.27), λ is always non-negative, implying α 1. For ≤ negative α (decelerating case) this is always satisfied. On the contrary, for positive α (accelerating case), this implies, in view of (7.28), the existence of a critical radius, given by Rc, beyond which no circular orbit exists. These orbits are stable if the 116 Chapter 7. Cosmological effects on local dynamics

uλ,α 3 4 1 / 1 − − = = 2 α α

0 1 2 3 4 5 6 x α = 0

-1

-2 α α α = 1 = 1 = 1

/ / 2 4 -3

Figure 7.1: The figure shows the effective potential uλ,α for circular orbits (for which λ =1 α) for some values of α. The initial conditions are x = 1 andx ˙ =0 (see (7.29)).− At x = 1 the potential has an extremum, which for α< 1/4 is a local minimum corresponding to stable circular orbits. For 1/4 α < 1 these become unstable. The value α = 0 corresponds to the Newtonian case.≤ considered extremum is a true minimum, i.e. if the second derivative of the potential evaluated at the critical value is positive. Now, u′′ (1) = 3λ 2 α = 1 4α, λ,α − − − showing stability for α< 1/4 and instability for α 1/4. For the accelerating case, ≥ in view of (7.28), this implies that the circular orbits are stable iff R0 is smaller than the critical value R := (1/4)1/3R 0.63 R , (7.34) sco c ≈ c where ‘sco’ stands for ‘stable circular orbits’. Summarizing, we have the following situation: in the decelerating case (i.e. for negative α or, equivalently, for negative A) stable circular orbits exist for every radius R0; one just has to increase the angular velocity by some amount stated below in (7.35). On the contrary, in the accelerating case (i.e. for positive α, or, equivalently, for positive A), we have three regions: R < R , where circular orbits exist and are stable, • 0 sco R R R , where circular orbits exist but are unstable, and • sco ≤ 0 ≤ c R > R , where no circular orbits exist. • 0 c Generally, there exist no bounded orbits that extend beyond the critical radius Rc, ′ the reason being simply that there is no R > Rc where U (R) > 0. Bigger systems will just be slowly pulled apart by the cosmological acceleration and approximately move with the Hubble flow at later times.2 Modifications of this strict qualitative distinction implied by time dependencies of A in (7.21) were discussed in [FJ07].

2 This genuine non-perturbative behavior was not seen in the perturbation analysis performed in [CFV98]. 7.3. Next-to-Newtonian analysis 117

Turning back to the case of circular orbits, we now express the condition for an extrema derived above, λ =1 α, in terms of the physical quantities, which leads to − ω = (2π/T ) 1 sign(A)(R /R )3 . (7.35) 0 K − 0 c This equation says that, in order top get a circular orbit, our planet, or electron, must have a smaller or bigger angular velocity according to the universe expanding in an accelerating or decelerating fashion, respectively. This is just what one would expect, since the effect of a cosmological ‘pulling apart’ or ‘pushing together’ must be compensated by a smaller or larger centrifugal forces respectively, as compared to the Keplerian case. Equation (7.35) represents a modification of the third Kepler law due to the cosmological expansion. In principle this is measurable, but it is an 3 −17 effect of order (R0/Rc) and hence very small indeed; e.g. smaller than 10 for a planet in the Solar System. Instead of adjusting the initial angular velocity as in (7.35), we can ask how one has to modify r0 in order to get a circular orbit with the angular velocity ω0 = 2π/TK. This is equivalent to searching the minimum of the effective potential (7.32) for λ = 1. This condition leads to the fourth order equation αx4 x + 1 = 0 with − respect to x. Its solutions can be exactly written down using Ferrari’s formula, though this is not very illuminating. For our purposes it is more convenient to solve it approximately, treating α as a small perturbation. Inserting the ansatz x = c + c α + (α2) we get c = c = 1. This is really a minimum since min 0 1 O 0 1 u′′ (x )=1+ (α) > 0. Hence we have 1,α min O

3 R0 6 Rmin = R0 1 + sign(A) + (R0/Rc) . (7.36) Rc O ! This tells us that in the accelerating (decelerating) case the radii of the circular orbits with ω0 =2π/TK becomes bigger (smaller), again according to physical expectation. As an example, the deviation in the radius for an hypothetical spacecraft orbiting around the Sun at 100 AU would be just of the order of 1 mm. Since it grows with the fourth power of the distance, the deviation at 1000 AU would be of the order of 10 meters.

Chapter 8

Cosmological eﬀects on kinematics

In this chapter we give a rigorous derivation of the general-relativistic formula for the two-way Doppler tracking of a spacecraft in FLRW and in McVittie spacetimes (this results have been published in [CG06]). The leading order corrections of the so-determined acceleration to the Newtonian acceleration are due to special- relativistic eﬀects and cosmological expansion. The latter, although linear in the Hubble constant, is negligible in typical applications within the Solar System.

8.1 Inﬂuence on Doppler tracking

Doppler measurements are very well studied in the case of weak-field approximation of the gravitational field of (arbitrary moving) isolated sources [KS99]. However, a similarly careful analysis of the cosmological effects on Doppler measurements is, in our opinion, still lacking. Considerations in this direction are contained in [LPD06] for FLRW universes and in [KKL06] for the Schwarzschild-de Sitter case. Our aim is to give a rigorous derivation of the two-way Doppler formula in relevant cosmological spacetimes (FLRW and McVittie). The analysis of the radio Doppler tracking data from the Pioneer10 and Pio- neer 11 spacecrafts yield an anomalous inward pointing acceleration of magnitude 8.5 10−10 m/s2; see [ALL+02] and [Mar02]. This magnitude is comparable to × Hc 7 10−10 m/s2, where we set H 70km/s Mpc (see [K+09] for recent ≈ × ≈ · figures). This somewhat surprising coincidence invited speculations as to a possible cosmological origin of the ‘Pioneer Anomaly’. Whereas there seems to be no disagreement over the absence of a genuine dynamical influence of cosmological expansion on Solar System dynamics, opinions on possible kinematical effects are less unanimous. Some even seem to claim that the Pioneer Anomaly can be fully accounted for by such effects [RS98] (for a critical discussion, see [CG05]). Note that all these speculations rest on the assumption that the cosmological expansion extends into the small region occupied by our Solar System, i.e. that the expansion is not screened by local inhomogeneities of mass abundance, like that given by our Galaxy and further our Solar System. Our results here imply that even in the unlikely case that such a screening does not take place, there is no effect due to cosmological expansion of the required order of magnitude.

119 120 Chapter 8. Cosmological eﬀects on kinematics

We recall that a monochromatic electromagnetic wave in the geometric-optics approximation (i.e. for wave-lengths negligibly small w.r.t. a typical radius of curvature of the spacetime and w.r.t. a typical length over which amplitude, polarization, and frequency vary) propagates along a lightlike geodesic, along which the wave- vector ﬁeld, k, obeys g(k, k) = 0 and ∇kk = 0. An observer u at p who receives a light signal with wave vector k measures the frequency (cf. Section 3.2)

(u) ωp (k)= g(u, k)p . (8.1)

(v) Likewise, a second observer v at p will measure the frequency ωp (k) = g(v, k)p. The relation between the two measurements is easily computed using the deﬁnition of relative velocity (3.2) and its projections (3.3) and that of the gamma-factor (3.4):

(v) ω (k) 1 ˆ p = 1 βk(v) , (8.2) (u) 2 u ω (k) 1 βu(v) − p p −

p kˆ where βu(v) is the modulus of the relative velocity of v w.r.t. u and βu(v) is −1 the relative velocity of v w.r.t. u in direction of kˆ := Puk Puk, the normalized k k vector in the rest space of u at p that points in the direction of the light propagation (the terminology is that of the Chapters 2 and 3). This is nothing but the general Doppler formula. Let the observer v at p carry a mirror (meaning that the mirror is at rest w.r.t. v) that is used to reﬂect back the light ray. This means that the frequency, as measured by v, does not change whereas the spatial projection of the wave vector k changes sign. In short, the process of reﬂection is given by

(after) (before) (before) k = Qvk Pvk . (8.3) |p |p − |p Another observer, say u, at p will see a frequency shift according to

(u) (after) kˆ ωp (k ) 1 βu(v)p (u) =2 − 2 1 , (8.4) k(before) 1 βu(v)p − ωp ( ) − as one can compute from (8.3), (8.2), and (3.2). Notice that here and henceforth the wave vector k that defines kˆ at the reflection point p refers to k just before reflection.

8.1.1 Two-way Doppler tracking in a FLRW spacetime

We consider a FLRW cosmological model, given by (6.7). Let k be the wave vector field along a light ray (worldline). The frequency, as measured by the cosmological observers u, will change along the ray according to the well-known cosmological redshift relation (3.50). The two-way Doppler tracking (see Figure 8.1) now consists in the exchange of light (radar) signals between us, the observers on Earth who hypothetically moves along the cosmological flow of u, and another observer γ (spacecraft). Schematically the tracking involves the following five processes:

(i) emission of the signal at p0, 8.1. Inﬂuence on Doppler tracking 121

us spacecraft

p2 γ

u v kˆ

Figure 8.1: Sketch of the two-way Doppler tracking: The observer (us) moves on an integral curve of the cosmological vector field u. At p0 he sends an electromagnetic signal to the spacecraft which moves along the worldline γ. The signal is then reflected back from the spacecraft at p1 and, finally, received at p2 by the observer. −1 kˆ := Puk Puk is the normalized spatial (w.r.t. u) wave vector of the infalling lightk ray. k

(ii) propagation from p0 to p1,

(iii) reﬂection at p1,

(iv) propagation from p1 to p2, and, ﬁnally,

(v) reception at p2.

Accordingly, using the cosmological redshift relation (3.50) between p0 and p1, the reﬂection shift (8.4) at p1, and again the cosmological redshift relation (3.50) between p1 and p2, one gets

(u) kˆ ωp (k) a(t(p )) 1 β (v) 2 = 0 2 − u p1 1 . (8.5) (u) v 2 ω (k) a(t(p2)) 1 βu( )p1 − p0 − ! This is the formula for the two-way Doppler tracking in a FLRW spacetime. It relates the spacecraft’s velocity (relative to u) with the observable quantities (emitted and received frequencies, emission and reception times) if the scale function a is known. To get a better feeling of the above relation we note that to linear order in the two quantities H∆t20 and β it reads

(u) ω (k) ˆ p2 1 2βk(v) H∆t , (8.6) (u) ≈ − u p1 − 20 ωp0 (k) where ∆t := t(p ) t(p ) is the coordinate-time interval between the emission 20 2 − 0 and the reception events, which, since we are moving along ∂/∂t, equals the proper time interval measured by us between the two events. H :=a/a ˙ denotes the Hubble parameter, which in (8.6) can be evaluated at any of the times in the interval [t0,t2]. 122 Chapter 8. Cosmological eﬀects on kinematics

Now, while tracking a spacecraft by continuously emitting a radar signal with constant frequency ω0 the whole construction is pushed forward in time. One should think of the worldlines of the spacecraft (γ) and ours as given. The latter is taken to be an integral curve of the cosmological observer-field u = ∂/∂t and the former is given by some equation of motion (plus initial conditions), which we do not need to specify here. Thus the three points (events) p0, p1, and p2 are uniquely determined by any one of them. The same holds for the respective times ti := t(pi). We can then choose to express the events pi as functions of the reception time t2. Doing this, the reflection time t1 and the emission time t0 become functions of t2 as well. One of the major tasks for the two-way Doppler tracking is to determine the spacecraft’s (spatial) acceleration. Since (8.5) relates the frequency shift with the velocity, differentiation of (8.5) w.r.t. the reception time gives a relation between the frequency-shift rate on one hand, and the acceleration and velocity on the other. In the differentiation w.r.t. t2 one has to take care of the different time dependencies of the quantities in question. For example, the received frequency is to be thought of as function of the reception time t2 and the spacecraft’s velocity as a function of the reflection time t1, which, in turn, is to be thought of as function of t2. In order to differentiate (8.5) w.r.t. t2 we thus need to know the dependencies of the emission and reflection times, t0 and t1, on the reception time t2. These dependencies can be obtained from the fact that the pairs of events (p0,p1) and

(p1,p2) are lightlike separated. For simplicity we will specialize to the spatially ﬂat case, which is compatible with current observational data [K+09]. Introducing spherical coordinates, (r, θ, ϕ), on the slices of constant t the spatial metric reads h = dr2 + r2(dθ2 + sin2 θ dϕ2). Without loss of generality we may assume our worldline to be given by r = 0. The spacecraft’s worldline is described by some functions (t1, r1,θ1, ϕ1) of the reﬂection time t1. Hence one obtains

t2 dt 1 r2 = dr , (8.7) a(t) − c Zt1(t2) Zr1(t1(t2)) where the minus sign is because the light ray is ‘inward’ pointing. Differentiating kˆ ˙ ˆ (8.7) w.r.t. t2 and noting that βu(v)p1 = a(t1)r ˙1/t1 = a(t1)dr1/dt1 (where k = ∂/∂r −1∂/∂r and the overdot denotes differentiation w.r.t. the proper time of γ) onek getsk −1 dt1 a(t1) kˆ = 1+ βu(v)p1 . (8.8) dt2 a(t2) Similarly, differentiating

t2 dt 1 r1(t1(t2)) 1 r2 =+ dr dr (8.9) a(t) c − c Zt0(t2) Zr0 Zr1(t1(t2)) w.r.t. t2 and using (8.8) one gets

kˆ dt0 a(t0) 1 βu(v)p1 = − ˆ . (8.10) dt2 a(t2) 1+ βk(v) u p1 In passing we remark that (8.10) can be used to compute the variation of the light round-trip-time ∆t20 as measured by the receiver (∆t20 is just twice the radar 8.1. Inﬂuence on Doppler tracking 123 distance between observer and spacecraft):

kˆ d a(t0) 1 βu(v)p1 kˆ ∆t20 =1 − ˆ 2 βu(v)p1 + H∆t20 , (8.11) dt2 − a(t2) 1+ βk(v) ≈ u p1 which clearly displays the expected contributions due to the spacecraft’s motion and the cosmological expansion respectively.

Returning to the derivation of the frequency-shift rate, we notice that d/dt1 = −1 ∇u (dt1/dτ1) d/dτ1 = γ˙ on functions along the spacecraft’s worldline γ, where τ1 ∇u is its arc-length and γ˙ denotes the observer derivative (3.8). Taking into account

(3.16), (3.17), (8.8), and (8.10), diﬀerentiation of (8.5) w.r.t. t2 gives

−1 −1 1 dω2 a(t0) ˆ ∇u ˆ a(t1) kˆ 2 = 2 h(α, k)+ h(β, γ˙ k) 1+ β 1 β ω0 dt2 − a(t2) ( a(t2) − h i a(t ) 1 βkˆ −2 1 2 (8.12) 4 h(α, β) − ˆ 1 β − a(t2) 1+ βk − kˆ kˆ 2 a˙(t2) a˙ (t0) 1 β 1 2β + β + − ˆ − 2 . a(t2) − a(t2) 1+ βk 1 β ! − ) Here we suppressed the arguments and the indices u for the sake of readability and (u) put ω2 := ωp2 (k). That is, for example, α stands for αu(γ), the relative acceleration of γ w.r.t. u (see (3.14)). This formula gives the exact relation between the observable frequency-shift rate (measured ‘here’) and the local kinematical properties of the spacecraft (defined ‘there’) in a spatially flat FLRW spacetime—provided the scale function a is known. In the special, but relevant, case where the spacecraft’s motion is in direction of the line-of-sight (‘radial’ for short), meaning that both β and α are collinear to kˆ, we have: β = βkˆ kˆ, and α = αkˆ kˆ, where αkˆ := h(α, kˆ). Moreover, it is ∇u ˆ straightforward to check that γ˙ k = 0. The two-way Doppler-tracking formula (8.12) now simplifies to

kˆ kˆ 1 dω a(t ) a(t ) ˆ ˆ a˙ (t ) a˙(t ) 1 β 1 β 2 = 0 1 2 αk (1+βk)−3 + 2 0 − − . ω dt − a(t ) a(t ) a(t ) − a(t ) kˆ kˆ 0 2 2 2 ( 1 1 1+ β !1+ β ) (8.13) This formula is still exact if one restricts to radially moving spacecrafts. In a linear approximation in H∆t20 it becomes

kˆ 1 dω ∆t ˆ ˆ ˆ 1 β 2 2 1 3H 20 αk(1 + βk)−3 + Hβk − . (8.14) ω dt ≈− − 2 kˆ 2 0 2 ( (1 + β ) ) In all cases of interest one is also interested in a slow-motion approximation involving an expansion in β. Retaining only terms linear in β and H∆t we simply get (dω /dt )/ω 2 αkˆ. This is exactly the result one obtains from Newtonian 2 2 0 ≈ − physics. Leading order corrections are obtained by going to quadratic order in β and also keeping mixed terms β H∆t. The result is ·

1 dω2 kˆ kˆ kˆ 2 α 1 3β 3H∆t20/2 + Hβ . (8.15) ω0 dt2 ≈− − − n o 124 Chapter 8. Cosmological eﬀects on kinematics

At this point it is crucial to realize that α is the acceleration which appears on the l.h.s. of our spatial equation of motion (see Section 3.1.3), which is to be seen as the generalization of Newton’s equation. Recall that ‘spatial’ refers to the choice of the observer field, from which the establishment of a Newtonian equation always depends. As argumented before, in a spherically symmetric spacetime a preferred observer field is given by the Kodama observer field, which for the FLRW spacetime is given by (6.26a). studied in Section 6.2.1. In view of the expressions (6.31) and (6.32) for the acceleration and, respectively, the expansion of the Kodama observer field (valid if we neglect terms of third and higher order in HR = R/RH ), the spatial projection w.r.t. uK of the geodesic equation in a slow-motion and weak-

field approximation (i.e. within quadratic order in HR and linear order in βuK (γ˙ )) is simply a¨ αu (γ)= R e γ . (8.16) K a R ◦ This is the form of the equation of motion in which dynamical considerations are most conveniently addressed (see Chapter 7). Hence in the above formulae for the Doppler tracking we need to express the relative accelerations and velocities defined w.r.t. u in terms of relative accelerations and velocities w.r.t. uK. This can be done with the general ‘addition formulas’ for relative velocity and relative acceleration which can be found, for example, in [BCJ95]. However, for our discussion it is sufficient to consider the approximated Doppler tracking formula (8.15) and within its level of approximation it makes no difference whether we refer spatial acceleration and velocities w.r.t. u or w.r.t. uK. Hence (8.15) gives the sought-after relation between the two-way Doppler measurement and the kinematical quantities that enter the Newtonian equation. Looking at (8.15) we see, among other corrections, an additional acceleration term Hcβkˆ (reintroducing the speed of light c). Hence the sometimes alleged Hc acceleration term [RS98] is actually suppressed by a factor β, as already pointed out in [LPD06], though we cannot agree with the derivation given there. In particular we conclude that, even if the universe did expand down to scales of the Solar System, there would be no effect of the same order of magnitude as the Pioneer Anomaly. Besides, notice that the above mentioned correction points in outer direction and hence opposite to the Pioneer anomalous acceleration. One expects that this conclusion remains true in the presence of an isolated local inhomogeneity, such as a single star, since the contributions coming from the ‘gravitational’ redshift of the star cancels out in the two-way process. Insofar as such an inhomogeneity can be modeled by the McVittie metric this is explicitly shown in the next section.

8.1.2 Two-way Doppler tracking in a McVittie spacetime

The McVittie model for an inhomogeneity of mass m0 in a cosmological spacetime with scale-factor a(t) is presented and extensively studied in Section 6.3. We recall that the metric of the McVittie model is given by (6.39) with the condition (6.43) together with the specification of a scale-factor a(t). Moreover, the cosmological observer field, that is the velocity vector field of the cosmological matter is given by (6.41). The redshift formula for a radial light ray in a McVittie spacetime is easily 8.1. Influence on Doppler tracking 125

computed up to linear order in H∆t10 and m0/ar to be

∂/∂t ω(u) p1 k k 1 H∆t10 , (8.17) ∂/∂t ω(u) ≈ − k k p0 where ∆t := t(p ) t(p ) and H :=a/a ˙ as previously. In case H = 0 this 10 1 − 0 reduces to the familiar gravitational redshift relation in Schwarzschild spacetime, as it should. By (8.17) one sees that the main formula (8.5) for the two-way Doppler frequency-shift we derived in the FLRW case still remains valid for the McVittie case in linear order in H∆t and m0/ar. The reason is simply that the gravitational contribution to the redshift (coming from the factors ∂/∂t 1 m /ar in (8.17)) k k≈ − 0 vanishes in the two-way process within the considered approximation. As in the previous section we wish to compute the Doppler frequency-shift rate measured by an observer (us) along u who exchanges light (radar) signals with a spacecraft moving along an arbitrary worldline γ (see figure 8.1). For this we need the generally valid relations (3.16) and (3.17) and the relations (8.8) and (8.10) which, fortunately, hold exactly also in the McVittie case, provided that central mass, observer, and spacecraft are aligned. The latter will be assumed henceforth. A difference to the FLRW case is that here t is not anymore the proper time along the observer’s worldline. Rather, to get the measured frequency-shift rate, we have to differentiate (8.5) with u = d/dτ = ∂/∂t −1d/dt . When differentiating 2 k k 2 the kinematical factor on the r.h.s. of (8.5) we proceed as before but now take into account that d/dt = (dt /dτ )−1d/dτ = ∂/∂t ∇u on functions along the 1 1 1 1 k k γ˙ spacecraft’s worldline γ, where τ1 is its arc-length. This leads to the formula

1 dω ˆ ˆ ∆τ m ∆τ ˆ 2 2 αk 1 3βk 3H 20 + 0 20 + Hβk (8.18) ω dτ ≈− − − 2 R2 2 0 2 which is valid in linear order in H∆t20 and m0/ar, and quadratic order in β and differs from (8.15) merely in the term containing m0 (note that in (8.15) ∆t20 = ∆τ20). Formula (8.18) shows that in the McVittie case, too, the acceleration correction term Hc is suppressed by a factor β. As for the FLRW case, we express the relative acceleration and velocity deﬁned w.r.t. u in terms of the relative acceleration and velocity w.r.t. the Kodama observer

ﬁeld uK, which for the McVittie spacetime is given by (6.63a). These are also the quantities which enter in the spatial projection of the equation of motion of the spacecraft. Assuming the latter moves along a geodesic, the slow-motion (i.e. in linear order in βuK (γ˙ )) and weak-ﬁeld (i.e. in leading order in HR and m0/R) approximation of the spatial (w.r.t. uK) equation of motion is given by

m0 a¨ αu (γ) + R e γ . (8.19) K ≈ − R2 a R ◦ This follows immediately inserting the approximate expressions for the acceleration and expansion of the Kodama observer ﬁeld for the McVittie spacetime (see (6.69) and (6.70)) in the general form of the spatial equation of motion (3.28). Formula (8.19) is the form of the equation of motion in the next-to-Newtonian approximation in which dynamical considerations are most conveniently addressed (see Chapter 7). 126 Chapter 8. Cosmological eﬀects on kinematics

In the special case of purely radial motion, insertion of (8.19) into (8.18) gives a formula predicting the two-way Doppler-shift rate in linear order in H∆τ20 and m0/R and quadratic order in βuK (γ˙ ):

1 dω2 m0 kˆ kˆ 2 2 1 3β + Hβ . (8.20) ω0 dτ2 ≈− − R − Hence there are two corrections to the Newtonian contribution. One is proportional to H and stems from the cosmological expansion, the other is independent of H and of purely special-relativistic origin, as one might check explicitly (see also Sect. VI.B.1 at p. 193 in [CG10b]). The ratio of the cosmological cor- 2 2 rection to the Newtonian contribution is given by Hβ/(m0/R ) = 2(R/R⋆) β, where R⋆ := √RS RH is the geometric mean of the Schwarzschild radius and Hub- ble’s radius. The latter is of order of 1023 km so that its geometric mean with a Schwarzschild radius of one kilometer is approximately given by R 2400AU. For ⋆ ≈ spacecrafts within the Solar System (i.e. at a maximal distance of, say, 100 AU from the Sun) whose velocities can reach the order of some 10 km/s (corresponding to a β of the order of 10−4) the cosmological correction is thus at most of the order of 10−7. This can be compared with the Pioneer anomalous acceleration, which is here taken as reference for the actual limit of measurements. As we already noted above, the PA is surprisingly close in magnitude to Hc so that its ratio with the 2 2 Newtonian acceleration is H/(m0/R ) = 2(R/R⋆) . Hence the cosmological correction is ‘only’ a factor of β away from the order of magnitude of the Pioneer anomalous acceleration, which for Solar System spacecrafts can amount to a factor of 10−4. This means that the cosmological correction is negligible but ‘only’ four order of magnitude smaller than the present measurements precision (which is here taken to be of order of the PA). Finally, we note that the ratio of the cosmological correction to the special- 2 relativistic one is of order (R/R⋆) , implying that the latter dominates for radii smaller than R 2400 AU. Surprisingly, in the literature we did not found trace ⋆ ≈ of any discussion on this 3β correction-term, although this lies within the measurable range.

8.2 Conclusion

It is often heard that eﬀects of cosmological expansion are at most of order H2. We have seen in Chapter 7 (cf. also eq. (8.19)) that this is true as far as deviations form the Newtonian equation of motion are concerned. However, there is a term linear in H that enters the formula for the two-way Doppler tracking, as seen in (8.20), even though the correction it gives is negligible as compared with the special relativistic one, due to its suppression with a factor of β. Furthermore, it should be stressed that the correction in the two-way Doppler tracking formula depends only on the expansion (H) of spacetime, whereas the deviation from the Newtonian equation of motion depends on the acceleration of that expansion. One should say that for direct comparison with actual measurements there are other corrections to our Doppler formulae that can easily be taken care of. For example, one needs to incorporate the motion of the Earth relative to the cosmological substrate with an additional Doppler factor (8.2) at the emission and 8.2. Conclusion 127

reception points p0 and p2. Also, one needs to take into account that in the actual tracking procedure signals are sent back with a fixed frequency translation factor (see e.g. formula (1) in [Mar02]). Finally, we also point out that another aspect which influences velocities and accelerations is given by the choice of simultaneity. Such an effect has e.g. been suggested in [RS98] and again in [Ros02] to be able to account for the PA. Their argument says that in a spatially flat FLRW universe the mismatch between adapted cosmological coordinates on the one hand and radar coordinates on the other just amount to an apparent difference in radial acceleration of magnitude Hc. We agree on the existence and conceptual importance of such an effect but we disagree on its magnitude, which seems to have been grossly overestimated. In Sect. VI.A of [CG10b] we show that this alleged Hc term is suppressed by a kinematical factor (see eq. (121c) in [CG10b]) which for the Pioneer spacecrafts is of order of 10−13, making this effect safely negligible.

Part III

Massive gravitational theories

129

Chapter 9

Linear massive theories

In the present chapter we study, in the setting of linear Poincaré-covariant theories, the possibility of modifying the field equations for gravity with the addition of a mass term. According to an old and well known argument of van Dam and Veltman [vDV70, vDV72] and, independently, from Zakharov [Zak70], which we will sketch below, the theory of massive spin 2 fields in flat Minkowski spacetime does not approximate the strictly massless theory (linearized General Relativity (GR)), in the limit as the mass tends to zero. (See also [vD74] and the more comprehensive account in [BD72].) This means that there exist corresponding observables in both theories which distinguish the massless limit of the massive theory from linearized GR. One such observable, so it is claimed, is the amount of deflection of a light ray in the background of a, say, spherically symmetric, static gravitational field. More precisely, in the limit the mass tends to zero, the deflection angle predicted by the massive theory tends to 3/4 of the angle predicted by linearized GR. As we shall see below, the argument given by van Dam & Veltman and Zakharov (vDVZ) is entirely based on the properties of free propagators, whose structure is determined by Poincaréinvariance. The aim in this chapter is to improve our understanding of the classical aspects of this limit, which is known to be non trivial for several reasons, but has not been explored in all details in the mentioned references. In particular, we wish to understand how certain observables can have a smooth limit as the mass tends to zero, even though they refer to solutions which diverge in that limit. The deflection of light is an example of such a case, whose derivation from first principles we shall consider in the following. We start in Section 9.1 recalling the geometric background of massive and massless spin-2 theory and then sketch the original argument of van Dam & Veltman and Zakharov in Section 9.2. In Section 9.3 we then give an alternative, classical and geometric approach to this argument. For this we first recall the free massive (Pauli–Fierz) spin-2 field theory and found the static spherically symmetric vacuum solution to these. We then show how this field couple to matter via the Lagrangian formulation and compute the deflection of light rays. We then conclude discussing the massless limit and its problems. This material has appeared in [CG01]. In Section 9.4 we try to by-pass these difficulties by modifying the Paul–Fierz mass term. We show that this give rise to a scalar-tensor theory where the field is a mixture between a spin-2 and a spin-0 field. In Section 9.4.2 we give an explicit decomposition (see Lemma 9.3) which allows to identify the spin-2 and the spin-0 components and prove that these propagate independently. By means of this de-

131 132 Chapter 9. Linear massive theories composition we then prove that the scalar ﬁeld is a ghost. This problem shows up explicitly when considering the emission of gravitational waves, where we illustrate that a spherically symmetric pulsating source would produce gravitational waves with negative energy density.

9.1 Geometric background

Let us now turn to the argument proper of van Dam & Veltman and Zakharov. To fully appreciate the generality of this argument, we need to recall the geometry that underlies the construction of unitary irreducible representations of the Poincaré group. For this we consider the fields in momentum space1. We identify momentum space with R4 endowed with the Minkowski metric η. The so-called ‘mass shell’ is given by the mass hyperboloid H := p R4 p2 = m2 , which for m = 0 m { ∈ | } degenerates to the ‘null-cone’ in momentum space. The tangent space of Hm at p is given by T H := q R4 q p =0 . The tangent spaces inherit a metric from p m { ∈ | · } the Minkowski metric. For m> 0 it is given by the positive definite form

p p π˜m (p)= µ ν η , (9.1) µν m2 − µν where pµ = ( ~p 2 + m2, ~p), and for m = 0 by the degenerate, positive semi- ± k k definite form p p(µ p¯ π˜0 (p)= ν) η , (9.2) µν ~p 2 − µν k k where pµ = ( ~p , ~p) andp ¯µ = ( ~p , ~p). Recall that round bracket on indices ±k k ±k k − denotes full symmetrization, hence, for example, a(µbν) = (1/2)(aµbν + aν bµ). The null space of (9.2) is span(p) T H . ⊂ p 0 Each TpHm carries a linear isometric (w.r.t.. (9.1,9.2)) representation of that subgroup in the Lorentz group that stabilizes p. For m> 0 these groups are isomorphic to SO(3), for m = 0 to E2, the group of Euclidean motions in 2-dimensions. 2 E2 = R ⋊ SO(2) acts linearly on the 3-dimensional tangent spaces via its natural 0 1 2 0 embedding in GL(3, R): Let (x , x , x )=:(x , x) be coordinates of TpHm w.r.t. an adapted orthogonal frame (~e , ~e , ~e ), where ~e p spans the degeneracy eigenspace 0 1 2 0 ∝ ofπ ˜0, and normalized spacelike ~e ; then the action of (a, R) R2⋊SO(2) on (x0, x) 1,2 ∈ is (x0, Rx + x0a). Leaving span(p) invariant this action is reducible, but irreducible after projection to the 2-dimensional quotient Tp(Hm)/span(p) (where only SO(2) acts non-trivially). Tensor fields which carry spin-2, mass m > 0 representation of the Poincaré group are given by maps from Hm to the symmetric, traceless 2-tensors over its tangent spaces, which carry the unitary irreducible angular momentum 2 representation of SO(3) with respect to the tensor product extension of (9.1). For m = 0 we also consider the 5-dimensional space of symmetric trace free 2-tensors over

Tp(H0). However, it contains the 3-dimensional subspace of tensors of the form p(µλν) with λ T H , which form the null space (perpendicular to all other ten- ∈ p 0 sors) of the tensor-product extension of (9.2). Taking the quotient with respect to

1 Fields, propagators etc. in momentum space carry a tilde. 9.2. The van Dam–Veltman–Zakharov argument 133 this null space leads to the two-dimensional space of helicity-two states with four momentum p. The projectors onto these irreducible subspaces can now be written down. For mass m> 0 it is given by

m αβ m (α m β) m m αβ Π = Π Π 1 Π Π (9.3) µν µ ν − 3 µν and for m =0 by e e e e e

0 αβ 0 (α 0 β) 0 0 αβ Π = Π Π 1 Π Π , (9.4) µν µ ν − 2 µν where e e e e e ν m ν p p Π = δν µ = π˜m ν (9.5) µ µ − m2 − µ and e

ν ν 0 ν pµp¯ +¯p p Π = δν µ = π˜0 ν (9.6) µ µ − 2 ~p 2 − µ k k are the projectors onto thee tangential spaces TpHm and TpH0, respectively. Notice that the minus sign on the r.h.s. of the last two equations is just a consequence of rising the indices with η. Since the projectors (9.5,9.6) enters quadratically m ν m ν in (9.3,9.4), these equations remains valid if one substitute there Π µ withπ ˜ µ 0 ν in the massive case, and Π withπ ˜0 ν in the massless one. As one may explicitly µ µ e check, for on-shell p, the vector and tensor projectors (9.5,9.6) and (9.3,9.4) are indeed projectors, that is:e idempotent and self-adjoint w.r.t. the scalar product (9.1) (respectively (9.2)) in the vector case and their tensor product extensions in the tensor case. The essential diﬀerence between the massive and the massless tensor projectors (9.3,9.4), is that, in the latter, the trace term has a factor of 1/2 due to the degenerate nature ofπ ˜0 with one-dimensional null space.

9.2 The van Dam–Veltman–Zakharov argument

After these preliminaries we can write down the propagators. Their form simply results from the requirement that they propagate the projected modes according to the free relativistic wave-equation:

m m Π (p) P (p) := µναβ massive case , (9.7) µν αβ p2 m2 + iε − 0e e0 Π (p) P (p) := µναβ massless case . (9.8) µν αβ p2 + iε e We now considere two systems with (Fourier transformed) energy-momentum µν µν µν tensors T1 and T2 . If we assume these tensors to be conserved, 0 = pµT1 (p)= µν µ pµT2 (p), the one-graviton interaction takes the form (we write T := T ) e e µ e µν 1 e m T1 (p)T2 ( p) T1(p)eT2( pe) κ2 T µν (p) P (p) T αβ( p)= κ2 µν − − 3 − (9.9) 1 µναβ 2 − p2 m2 + iε e e − e e e e e 134 Chapter 9. Linear massive theories in the massive case, and

µν 1 0 T (p)T ( p) T (p)T ( p) κ2 T µν (p) P (p) T αβ( p)= κ2 1 2µν − − 2 1 2 − (9.10) 0 1 µναβ 2 − 0 p2 + iε e e e e in the masslesse case,e withe gravitational constants κ and κ0 respectively. For slowly- moving massive objects the leading component of the energy-momentum tensor is T 00 and we get 2 κ2 T 00(p)T 00( p) in the massive and 1 κ2 T 00(p)T 00( p) in the 3 1 2 − 2 0 1 2 − massless case. To identify κ and κ0 one requires that both cases lead to Newton’s lawe of attraction withe the samee Newtonian constant G (takinge the limite m 0 in → the massive case). Of course we have

2 κ0 = 16πG (9.11) and thus 3 κ2 = κ2 = 12πG . (9.12) 4 0 On the other hand, if we consider the interaction of light and slowly moving matter, there are no trace-terms in (9.9) and (9.10) due to the tracelessness of the electromagnetic energy momentum tensor. But now (9.12) implies that the massive theory, in the limit m 0, leads to an interaction of light and matter → which is weaker by a factor 3/4 as compared to the massless theory. Accordingly, in that limit, light deflection comes out smaller by the factor 3/4 as compared to the massless theory, i.e., linearized General Relativity. This is often taken as sufficient justification to rigorously argue that present-day observations exclude a finite graviton mass. Here one should stress that the flatness of spacetime enters in an essential way. In fact, it was recently argued that there is no discontinuous behavior at m = 0 if the cosmological constant Λ is non-zero. For negative Λ (anti de Sitter spacetime) a smooth limit of propagator residues was shown in [Por01]. For positive Λ (de Sitter spacetime) the situation is again different, since there is no quantum theory of spin 2 2 2 fields in the mass range 0 < m < 3 Λ obeying unitarity and certain locality 2 2 requirements [Hig87, Hig89]. Here a discontinuity occurs at m = 3 Λ.

9.3 Massive tensor theory for gravity

Having reviewed the original vDVZ argument, carried out in momentum space, we present now an alternative derivation, in position space,

9.3.1 Static spherically symmetric vacuum solution As we saw in Section 9.1, a massive spin-2 field can be described, in momentum space, by a symmetric, traceless (0, 2)-tensor h˜µν living in the tangent space to the mass hyperboloid. In the position space these conditions traduce in the following partial differential equations for the field hµν : 2 ( + m )hµν =0 , (9.13a) µ ∂ hµν =0 , (9.13b) µ hµ =0 , (9.13c) 9.3. Massive tensor theory for gravity 135

To see this just recall that p in momentum space corresponds to i∂ in position µ − µ space. Notice that here, and henceforth, we assume the symmetry hµν = hνµ from the very beginning. Equations (9.13) are referred to as the vacuum massive spin-2 ﬁeld equations. We seek a solution of (9.13) which is static, i.e.

∂0hµν = 0 (9.14) and spherically symmetric in the following sense:

D(R)αD(R)β h (D(R)λxσ)= h (xλ) R SO(3) , (9.15) µ ν αβ σ µν ∀ ∈ where, in a 1+ 3-split matrix notation,

1 ~0⊤ D(R)= . (9.16) ~0 R

For physical reasons (finite energy around spatial infinity) we also require the asymp- totic fall-off condition limr→∞ hµν (x) = 0, where r is the spatial radial coordinate i 2 defined by r := i(x ) . The 00-component of (9.13a) has the one-parameter (b) family of solutions pP exp( mr) h (~x)= bf(r) := b − , (9.17) 00 − − r where the minus sign is introduced for later convenience. The 0-component of i 3 (9.13b) and (9.15) imply h0i = ax /r , for some constant a, which contradicts

(9.13a) unless a = 0; hence h0i = 0. This reﬂects exactly the fact that the spatial part of a static, spherically symmetric solution of the massive spin-1 equations (Proca equations) necessarily vanishes. To determine the spatial components of hµν , we ﬁrst remark that any spherically symmetric two-tensor in space is of the form xixj h (~x)= f (r)δ + f (r) . (9.18) ij 1 ij 2 r2

Equation (9.13a) now reduces to two coupled ODEs for f1 and f2, which may be 2 decoupled by introducing the new function f˜1 := 3f1 + f2. One obtains

(∆ m2)f˜ =0 , (9.19) − 1 (∆ m2 6 )f =0 , (9.20) − − r2 2 which, under the given fall-oﬀ conditions, have the unique 2-parameter set of solutions

exp( mr) f˜ (r) = c − , (9.21) 1 1 r exp( mr) 3 3 f (r) = c − 1+ + . (9.22) 2 2 r mr (mr)2

2 d2 2 d The Laplace operator here is just an abbreviation for dr2 + r dr . 136 Chapter 9. Linear massive theories

Condition (9.13b) and (9.13c) now imply c = 1 c and c = b respectively, 2 − 2 1 1 − thereby projecting out a unique one-parameter set of solutions, which, in terms of the function f deﬁned in (9.17), can be written in the simple form

b 2 ~0⊤ hµν (~x)= 1 f(r) . (9.23) −2 ~0 (δ 2 ∂ ∂ ) ij − m i j

We note that the ∂i∂j f - part is of a form that would be pure gauge in the massless theory. However, there is no gauge freedom in the massive theory and the prefactor, m−2, causes this term to diverge as m 0. → A slightly more geometric way to write the solution (9.23) is as follows: Let

~n = ~x/r be the radial unit vector; we deﬁne the spatial projection tensors ρij := ninj in radial direction and τ := δ ρ in the orthogonal direction, tangential to the ij ij − ij spheres of constant r. In terms of these, the spatial part of (9.23) takes the form

bf(r) 1+ mr h (~x)= τ + (τ 2ρ ) , (9.24) ij − 2 ij (mr)2 ij − ij which clearly separates the trace part τ , which stays ﬁnite for m 0, and the ∝ → trace free part τ 2ρ, which diverges as m tends to zero. ∝ − 9.3.2 Lagrangian formulation and matter couplings

As we will show below, the previous equations (9.13) are equivalent to the Euler– Lagrange equations of the following action:

S = d4x ( + ) (9.25) g L0 LPF Z where 1 = h hµν,λ 2hµh +2hµh h h,µ (9.26) L0 4 µν,λ − µ ,µ − ,µ is the Lagrangian of the massless spin-2 theory (that is linearized GR, see Ap- pendix C.3) and m2 = (h hµν h2) (9.27) LPF − 4 µν − is the so-called Pauli–Fierz mass term. Here and in the following we use the abbre- ν µν µ viations h := ∂µh and h := h ν for the divergence and trace of hµν , respectively. The Pauli–Fierz mass term is, as we will show below, the only mass term choice which implies positivity of energy (for the free ﬁeld) and a pure spin 2 theory. The coupling to some speciﬁc form of matter is described by an interaction term

1 Sint = d4xκh T µν , (9.28) −2 µν Z where T µν is the energy-momentum tensor of the matter. We remark that the most general nonderivative coupling would be (9.28) with the modiﬁed integrand

µν ′ κhµν T + κ h T , (9.29) 9.3. Massive tensor theory for gravity 137 which is not ruled out by the Equivalence Principle (see below) as sometime stated [Thi61]. However, the extra term would produce no changes (outside the source region), nor in the ﬁeld hµν , nor in the matter equation of motion. This can be seen writing (9.29) as

µν κ′ µν κ′ µν κhµν (T + κ η T )= κ(hµν + κ ηµν h)T . (9.30)

From (9.30), on the one hand it is clear that, in order to get the field equation for hµν , the coupling modification is equivalent to a redefinition of the source energy- ′ momentum tensor, namely T µν T µν + κ ηµν T . The additional source term is → κ proportional to ηµν so, as it gets multiplied by the projector in the r.h.s. of (9.42a), it becomes proportional to the ’inverse’ operator ( + m2) (see (9.44)) and hence does not propagate outside the source region (the support of Tµν ). On the other hand, for the matter equation of motion, the coupling modification is equivalent to a ′ redefinition of the gravitational field, namely h h + κ η h. This, again, give µν → µν κ µν rise to no changes outside the source region, since there the trace h vanishes. Thus, for the shake of simplicity, we can consider henceforth the easier coupling (9.28).

9.3.3 The Equivalence Principle

We show here that the massive spin 2 theory satisfies the Equivalence Principle (in linear approximation in the coupling strength) for the general coupling term (9.29). By this we mean that the motion of a spatially localized amount of matter depends on the initial conditions (position and velocity) only. Since the following argument relies only on the form of the general coupling term (9.29) and does not make any use of the field equations for the gravitational field, it applies as well to the scalar-tensor theories treated in Section 9.4. For point particles the situation is clear, since we will show that its equation of motion is given by the geodesic equation (9.50). More generally, we show in the following that the Equivalence Principle is true for any matter field which is localized in a little region of space and is initially static. We need for this the well-known von Laue theorem Theorem 9.1 (von Laue). In the Minkowskian spacetime consider a (symmetric) µν µν two-tensor T . Assume that this (i) is conserved, i.e. ∂µT = 0, (ii) static, µν i.e. there exists a frame in which ∂0T =0, and (iii) has spatially compact support. Then it holds: d3xT kν =0 , (9.31) Z that is, the spatial integrals of its components (performed in the static frame) all vanish, with the only exception of the 00-component. Applied to an energy-momentum-tensor von Laue’s Theorem says that for a static (condition (ii)) and isolated (conditions (i) and (iii)) system, the total 3-momentum (P k = d3xT 0k) and the space integral of the stress components ( d3xT ik) vanish in theR static frame. R µ λν µν Proof. First note that, because of (i), it holds ∂λ(x T ) = T . Thus, going kν k λν in the static frame and putting µ = k we get, using (ii), T = ∂λ(x T ) = 138 Chapter 9. Linear massive theories

k lν ∂l(x T ). Since this is a pure divergence, it vanishes integrating over space because of assumption (iii) giving the desired result.

Now, consider a matter field, which is localized in a spatially small region and µν is initially static (its energy-momentum tensor satisfies ∂0T = 0 on some frame), in a given external gravitational field hµν . With ‘small’ we mean that the support of the matter field is small w.r.t. the typical length over which the gravitational field varies (radius of curvature). Due to the presence of the gravitational field, the µν ν ν energy-momentum tensor satisfies ∂µT = f , where the four-force density f is of order of the coupling strength κ (it is assumed that the possibly nonzero κ′ is also of order κ). Hence, T µν satisfies the assumptions (ii) and (iii), and brake (i) in order κ. To compute the three-force of the gravitational field on the matter distribution, we compute their interaction energy

E = d3x . (9.32) − Lint Z Inserting the general coupling term (9.29) one gets, up to linear order in κ:

1 3 µν ′ E = 2 d x (κhµν T + κ hT ) Z 1 κh (t, ~z) d3xT µν + 1 κ′ h(t, ~z) d3xT ≈ 2 µν 2 Z Z 1 3 00 1 ′ 3 00 = 2 κh00(t, ~z) d xT + 2 κ h(t, ~z) d xT Z Z 1 ′ = 2 κh00(t, ~z)+ κ h(t, ~z) Mi ,

3 00 where ~z denotes the center of mass of the matter distribution and Mi := d xT its inertial mass. Above, in the ﬁrst step we use that hµν is approximatelyR constant in space over the support of the matter ﬁeld. In the second step we use Laue’s Theorem noticing that the failure of conservation of the energy-momentum tensor is of order κ and hence, in linear approximation, does not contribute to the interaction energy, being this already linear in κ. This shows that the interaction energy is proportional to the inertial mass. Then, since the equation of motion for the center of mass of the matter distribution is d2 M ~z = ~ E(~z) , i dt2 −∇ it follows that the inertial mass cancels out from it and the motion is determined by the initial conditions (position and velocity) alone as we had to show.

9.3.4 Field equations with matter and their Cauchy problem

We turn now back to the action (9.25) with the interaction term (9.28) and, in order µν to get the ﬁeld equations for hµν , we compute the variational derivative S := int δ(Sg + S )/δhµν , which is given by

Sµν = ( + m2)hµν − +2∂(µhν) ηµν hα + [ηµν ( + m2) ∂µ∂ν ]h (9.33) − ,α − κT µν . − 9.3. Massive tensor theory for gravity 139

Notice that, when computing the variational derivative, we take into account µν that hµν is symmetric and hence, for any constant two-tensor c , we have αβ µν νµ (µν) ∂(hαβc )/∂hµν = c + c = 2 c . This can be summarized in the useful formula ∂hαβ µ ν ν µ = δαδβ + δαδβ (9.34) ∂hµν which is of course only valid if contracted with any two-tensor tαβ (which may also depend on hµν ). Now, divergence and trace of (9.33) are given, respectively, by

∂ Sµν = m2(hν ∂ν h) κ∂ T µν , (9.35) µ − − − µ η Sµν = 2(h ∂ hµ)+3m2h κT . (9.36) µν − µ − Imposing the equations of motion, Sµν = 0, the last two equations can be cast into the form of ﬁve constraints (i.e. involving no second or higher derivatives):

0 = Cν := hν + κm−2∂ (T µν 1 ηµν Σ T αβ) , (9.37) 1 µ − 3 αβ 0 = C := h κ m−2Σ T αβ , (9.38) 2 − 3 αβ where, just for notational convenience, we put

2 Σ = η ∂ ∂ . (9.39) µν µν − m2 µ ν

µν (Notice that the scalar constraint C2 is just Σµν S .) Using the constraints to eliminate h and hν from the expression (9.33), the equation of motion, Sµν = 0, takes the form ( + m2) h = κ Πm αβ T , µν − µν αβ

m αβ where Π µν is the massive projector (9.3) in position space:

Πm αβ = Πm (α Πm β) 1 Πm Πm αβ , (9.40) µν µ ν − 3 µν with 1 Πm ν = δν + ∂ ∂ν . (9.41) µ µ m2 µ Summarizing, so far we have showed that the Euler–Lagrange equations, Sµν = 0, are equivalent to

( + m2) h = κ Πm αβ T (9.42a) µν − µν αβ κ ∂µh = ∂µ(T 1 η ΣαβT ) (9.42b) µν −m2 µν − 3 µν αβ κ h = ΣαβT . (9.42c) 3m2 αβ

We refer to (9.42) as to the massive spin-2 ﬁeld equations with source Tµν . Clearly, in the vacuum case, they reduce to (9.13). Notice that, for a conserved energy- momentum tensor, the source term Πm αβ T simpliﬁes to T 1 η T , which µν αβ µν − 3 µν 140 Chapter 9. Linear massive theories

0 αβ 1 diﬀers from the massless case in which the source Π µν T reduces to T η T αβ µν − 2 µν (see (9.4)), as in linearized GR. Equation (9.42a) has the straightforward interpretation of a propagation equa- m αβ tion, where the source Tµν enters only through its multiplication with Π µν . Its Fourier transformed form, (9.3), projects onto the transverse (meaning orthogonal to p) traceless modes. Hence only the transverse-traceless modes of Tµν will be propagated by (9.42a). In the position-space representation this is expressed by m α αβ the fact that contractions of Πµν αβ with ∂ or η is proportional to the ‘inverse’ propagator ( + m2):

Πm ∂α = m−2( + m2) 1 (∂ πm + ∂ πm ) 1 ∂ πm , (9.43) µν αβ − 2 µ νβ ν µβ − 3 β µν Πm ηαβ = 1 m−2( + m2)Σ (9.44) µν αβ − 3 µν

µν m µν m µν Contracting (9.43) and (9.44) with T and eliminating ΠµναβT = Παβµν T via (9.42a) results in the two equations

2 ν 2 ( + m )C1 =0 , ( + m )C2 =0 , (9.45) which show that the constraints C1 = 0 and C2 = 0 propagate. This means that we may consider the equations (9.37) and (9.38) (and thus (9.42b) and (9.42c)) as restrictions on the initial data only. The propagation via (9.42a) then ensures that they will be satisﬁed throughout. Hence we get the following result on the Cauchy initial-value problem for our equation: The full set of equations is equivalent to the hyperbolic propagation equation (9.42a) and the constraints (9.42b) and (9.42c) for the initial data. The constraints will be conserved by the evolution.

9.3.5 Coupling to a point particle The action of a free point particle of mass m in Minkowski space is given by m 0 − 0 times its arc-length. Choosing any parameter λ to parametrize its world-line, zµ(λ), we have dzµ(λ) dzν(λ) S = m dλ η . (9.46) p − 0 µν dλ dλ Z r Its energy-momentum tensor is

dzµ(τ) dzν (τ) T µν(x)= m dτ δ(4) (x z(τ)) , (9.47) p 0 − dτ dτ Z where the parameter τ is the arc-length with respect to the Minkowski metric η. Since this expression is not reparametrization invariant, we are not free to specify it otherwise. However, we can play the following trick, which turns out to be of central importance: Consider a new metric on Minkowski space, given by

gµν = ηµν + κhµν . (9.48)

The arc length w.r.t. g is called s. We consistently neglect terms of higher than linear order in κ and may therefore replace τ by s in (9.47), since this term already gets multiplied by κ in (9.28). In (9.46) we choose λ = s, replace η by g κh, − 9.3. Massive tensor theory for gravity 141 and expand to linear order on κ. The κ-term then just cancels the interaction term (9.28) and we get

dzµ(s) dzν(s) S + Sint = m ds g (z(s)) = m ds , (9.49) p p − 0 µν ds ds − 0 Z r Z which says that the gravitationally interacting particle moves on geodesics in the metric g.

In order to determine the still unknown constants κ and b in gµν (see (9.23) and (9.48)) we compare now this equation of motion with the Newton equation. To ﬁrst order in κ this reads, writing a dot for diﬀerentiation with respect to s:

z¨µ = κ ηµν (∂ h ∂ h ∂ h )z ˙αz˙β . (9.50) 2 ν αβ − α νβ − β να In a Newtonian approximation, where we neglect terms of second and higher powers in the spatial velocities dzi/ds, this reduces to

z¨0 = κh z˙0z˙k , (9.51) − 00,k κ z¨i = h (z ˙0)2 . (9.52) − 2 00,i

Re-expressing the time-dependence in terms of Minkowski time t := z0 we can rewrite the second equation, using the ﬁrst one and again neglecting quadratic terms in the spatial velocities, as:

d2zi κ = ∂ h . (9.53) dt2 − 2 i 00 Comparing this with Newton’s equation of motion

d2zi = ∂ Φ (9.54) dt2 − i and recalling the fall-oﬀ condition for hµν , we get the identiﬁcation

κ 2 h00 =Φ . (9.55) between h00 and the Newtonian potential Φ. Applied to our solution (9.17) we can now determine the product of the constants κ and b to be

κb =2GM , (9.56) where G is the Newtonian gravitational constant and M is the central mass, i.e., the source of the gravitational field. With this identification our metric coefficients gµν are now entirely determined by (9.48), (9.23), and (9.56). At this point we may also determine the coupling κ separately. For this we return to the field equation (9.42a), whose 00-component for a static source and in the limit m 0 reduces to → 2 ∆h00 = 3 κT00 . (9.57) In view of (9.55), a comparison with the Poisson equation

∆Φ=4πG̺ (9.58) 142 Chapter 9. Linear massive theories

of Newtonian gravity (taking into account that T00 = ̺) leads to

κ2 = 12πG . (9.59)

Finally, we also remark on the fact that instead of (9.46) one often ﬁnds the following alternative expression for the action of a free particle (e.g. in [Thi61]):

dzµ(λ) dzν(λ) S = 1 m dλ η . (9.60) p − 2 0 µν dλ dλ Z Like (9.46) it also leads to straight lines in Minkowski space, but in addition also to the condition that λ is (a constant multiple of) τ, the arc length measured with η. Since one is only interested in the spacetime paths and not in their parametrization, one may use either (9.46) or (9.60) to determine the free paths. However, we wish to point out that in the presence of an interaction of the form (9.28,9.47) the latter choice becomes inconsistent. The reason being the following: Adding the ‘free action’ (9.60) with λ = τ to the interaction (9.28) with the energy momentum tensor being given by (9.47), we obtain an expression like (9.60), where η is replaced by g and the parameter is still τ. But as a result of the ﬁxed parametrization the Euler– µ ν Lagrange equations now imply hµν (z(τ))(dz /dτ)(dz /dτ) = 0, which in general will have no solutions. Hence we maintain that the action for the particle should be written in the square-root form (9.46).

9.3.6 Coupling to the electromagnetic ﬁeld

The action for the free electromagnetic ﬁeld in Minkowski space is given by

1 S = d4x ηαµηβν F F , (9.61) em −4 αβ µν Z where F = ∂ A ∂ A . The corresponding energy-momentum tensor is µν µ ν − ν µ

T µν = F µαF ν + 1 ηµν F F αβ , (9.62) em − α 4 αβ which we couple to the gravitational field according to (9.28). A priori, there is no reason why different types of matter must couple with the same constant κ. However, if we choose different constants for different matter types, say κ1 for type 1 and κ2 for type 2, then the ratio of gravitational to inertial mass of type 1 would differ from the corresponding ratio of type 2 by a factor κ1/κ2 and hence violate the weak equivalence principle, according to which the ratio of gravitational to inertial mass is the same universal constant for all types of matter. Hence, in order to conform with this principle, we choose the same κ for point particles and electromagnetism. Now we can play the same trick as for the point particle and write [Thi61]

1 κ S + Sint = d4x ηαµηβν F F d4x h T µν em em −4 αβ µν − 2 µν em Z Z 1 = d4x √ ggαµgβν F F + (κ2) , (9.63) −4 − αβ µν O Z 9.3. Massive tensor theory for gravity 143

µν where g and g are the inverse and determinant of the matrix gµν . To ﬁrst order in κ we have gµν = ηµν κhµν and √ g =1+ κ h, where h = hµ and indices on − − 2 µ hµν are always raised and lowered with ηµν (see Appendix C.3). We are now in the same situation as for the point particle, since (9.63) is the action for the electromagnetic ﬁeld on a spacetime with background metric g. Without sources its equations of motion are

∂ F =0 , ∂ √ ggµαgνβ F =0 , (9.64) [µ νλ] µ − αβ which, in an eikonal approximation, imply that light rays are lightlike geodesics in the metric g. Note that our decision to couple all types of matter with the universal coupling constants has the eﬀect that light rays and particle paths are geodesics in the same metric.

9.3.7 Deflection of light rays in the massive spin-2 theory In this section we calculate the deflection of a light ray by the spherically symmetric gravitational field (9.23). We are interested in the amount of deflection as m tends to zero. A priori this limit is a precarious one since the metric coefficients diverge as m 0 (see (9.23)). However, we now show that the deflection has a finite limit. → To this end we split (9.23) in a finite part,

f(r) ~0⊤ hf (~x)= b , (9.65) µν − ~0 1 δ f(r) 2 ij which has a continuous limit as m 0, and a diverging part, → d −2 hij (~x)= bm ∂i∂j f(r) . (9.66) Since we strictly stay within the linear theory and, accordingly, keep only terms linear in κ, the contributions of hf and hd to the deflection add linearly, so that we can consider them separately. Expanding f(r) = exp( mr)/r in powers of m, we − explicitly checked that no power smaller than 2 of m in (9.66) contributes to the deflection, so that in the limit as m 0 the diverging part does not contribute → at all. In fact, one can argue that for each m the whole diverging part does not contribute to light deflection. The argument is as follows: Recall that under a spatial coordinate transformation xi x′i = xi + κ ki(~x) the spatial components 7→ of the metric g change—to first order in κ—according to g g′ = g ij ij 7→ ij ij − 1 −2 d κ(ki,j + kj,i). Hence, choosing ki(~x) = 2 bm ∂if(r), we can remove h from the f d f metric g = η + κ(h + h ). The resulting coefficients gµν = ηµν + κhµν are then understood with respect to the coordinate system x′i (we drop the prime hereafter). Since ki falls of as r , the new coordinate axes asymptotically approach the → ∞ old ones. In particular, they are again asymptotically orthogonal. Hence we may calculate the light deflection δ(m) as usual, using the metric gf = η + κ hf with the identification (9.56). δ(m) is a continuous function of m through the continuous dependence of the light deflection on the background metric. Hence, in order to obtain δ(0), we may calculate the light deflection for the limit metric

2GM ~ ⊤ f 1 r 0 lim gµν (~x)= − . (9.67) m→0 ~0 δ 1+ GM − ij r 144 Chapter 9. Linear massive theories

But instead of performing an explicit calculation, we merely need to compare (9.67) with the Schwarzschild metric in General Relativity. In isotropic coordinates the latter reads

2 1− GM 2r ~ ⊤ 1+ GM 0 Schw 2r g (~x) =   (9.68) GM 4  ~0 δij 1+   − 2r    2GM ~ ⊤ 1 r 0 = − + (G2) . (9.69)   O ~0 δ 1+ 2GM − ij r   As is well known, in leading (linear) order of G, light deﬂection receives equal contributions from the time-time and space-space parts:

2GM 2GM 4GM δEinst = + = , (9.70) q q q where q is the impact parameter. Applied to our metric this ﬁnally leads to the van Dam–Veltman–Zakharov value for the light deﬂection in the zero-mass limit:

2GM GM 3GM 3 lim δ(m)= + = = δEinst . (9.71) m→0 q q q 4

9.3.8 The massless limit and its problems The above computation of the light deflection explicitly shows the fragility of the vDVZ argument. This lies in the fact that, how it is clear from (9.23), as m tends to zero some coefficients of hµν diverge. So, rigorously spoken, one can not take the limit m 0 in the framework of the linear theory (since this would be incompatible → with the assumption of linear theory that the field is small). Interestingly, these field components are not seen by the light deflection and, moreover, they have the form of a pure gauge term of the massless theory (being proportional to ∂ν ∂νφ for some function φ). However, we stress that for a non-vanishing mass there is no such gauge freedom. One then might argue that in order to compute the massless limit one has to go over to the full theory which includes self-interaction and is thus nonlinear. In [Vai72, DDGV02] is stated that if one does so the massless limit is well defined and, in addition, it is argued that the vDVZ discontinuity disappears. This point will be addressed in the next chapter. But before going over to nonlinear theories we explore how this discontinuity is sensitive to changes in the linear theory presented in this section. Looking at the action, the possibilities one has to modify the theory are essentially two: In the coupling term and/or in the mass term. For the coupling term we already showed above that an additional trace-trace coupling does dot cause any essential changes. One could also try to modify the coupling in a mass-dependent way, imposing that for zero mass this reduces to (9.28). In [FvD80] this is studied and it is showed that, in order to have a continuous limit, the source has to satisfy certain conditions. But this conditions are not satisfied even by the simple source like a point particle, perfect fluids or scalar fields. 9.4. Massive scalar-tensor theories for gravity 145

For the mass term the situation is different. In the next section we show that modifying the Pauli–Fierz mass term one gets, in addition to a massive spin 2 field, an additional scalar field. The massless limit in these theories is well defined and converges to the massless theory (linearized GR). However, the price one has to pay is that the scalar field is a ghost.

9.4 Massive scalar-tensor theories for gravity

It was already noted by Boulware and Deser [BD72], and later used by Visser [Vis98], that the just showed vDVZ discontinuity is peculiar of the Pauli–Fierz mass term and that one can avoid this discontinuity by modifying the mass term. This modification implies the presence of an extra degree of freedom which can be identified with a scalar field and which, unfortunately, is a ghost. (Recall that a field is said to be a ‘ghost’ if its free-field energy is unbounded from below.) All this is known in the literature (e.g. [BD72, Des90, Vis98, DKP03]) but, sometimes, people seems not be worried about the ghost (see e.g. [Vis98, FS02, SF02]) or even claim that this is not present [BG03]. Interesting enough, in references [FS02, SF02] the authors computes the energy emitted by a mass distribution in form of gravitational waves and gets for it an expression which is clearly not positive definite (see formula (3.14) in [FS02] and (3.7) in [SF02]) but still use this theory fitting it to the binary pulsar data to get a bound for the graviton mass. For the sake of clarity, in this chapter we present how a modified Pauli–Fierz mass term give rise to a new degree of freedom, thereby presenting a precise decomposition (Lemma 9.3) of the field in two independently propagating part, a spin-2 (tensor) and a spin-0 (scalar) part, and with this result we show that the scalar part is a ghost.

9.4.1 Field equations for a general mass term

The most general mass term one can build for a two-tensor ﬁeld hµν is a linear µν 2 combination of hµν h and h which we write as follows

m2 (h hµν αh2) (9.72) − 4 µν − where α is a free real parameter. The choice α = 1 gives obviously the Pauli–Fierz mass term and thus the massive spin-2 theory we studied in the previous chapter. Hence, in the following we are interested in the α = 1 case. The outcome is that 6 only for α = 1, i.e. for the Pauli–Fierz mass term, the ﬁeld energy is positive and the theory is a pure (massive) spin 2 one. Otherwise for α = 1 one has an admixture 6 of spin 2 and spin 0, where the latter is a ghost. As in the previous chapter we compute the variational derivative of the action (9.25) using now the mass term (9.72). One gets:

Sµν = ( + m2)hµν − +2∂(µhν) ηµν hσ + [ηµν ( + αm2) ∂µ∂ν ]h (9.73) − ,σ − κT µν . − 146 Chapter 9. Linear massive theories

Its divergence and trace are given respectively by

∂ Sµν = m2(hν α∂ν h) κ∂ T µν , (9.74) µ − − − µ η Sµν = 2(h ∂ hµ)+(4α 1)m2h κT . (9.75) µν − µ − − Imposing the equations of motion, Sµν = 0, in the case α = 1 the last two equa- 6 tions give rise to only four constraints (i.e. equations involving no second or higher derivatives of hµν ), instead of the ﬁve of the Pauli–Fierz (α = 1) case. The Euler– Lagrange equations are herewith equivalent to m2 ( + m2)h + (2α 1)Σ h = κ T 1 η T + 2 ∂ T (9.76a) µν 2 − µν − µν − 2 µν m2 (µ ν) κ hν α∂ν h = ∂ T µν , (9.76b) − −m2 µ where Σµν is deﬁned in (9.39).

Relation with linearized General Relativity

We show now that the equations (9.76), in the case α =1 and of a conserved energy 6 momentum tensor, reduce in the massless limit to the equations of linearized GR, provided one chooses for the latter the appropriate gauge. The assumption of a conserved energy momentum tensor is needed, ﬁrst, since this must hold for linear GR and, second, in order to be at all able to take the massless limit on the right- hand sides of (9.76). A we will see below, the assumption α = 1 is needed, in 6 order to be able to take the gauge choice for the equations of linearized GR which reproduces the constraint (9.76b). Starting from the Einstein equation written in the form

2R = κ2(T 1 g T ) , (9.77) µν 0 µν − 2 µν 2 where κ0 = 16πG, putting gµν = ηµν + κ0hµν and linearizing in κ0 one gets with (C.17d): h + h 2∂ h = κ (T 1 η T ) . (9.78) µν ,µν − (µ ν) − 0 µν − 2 µν The diﬀeomorphism covariance of (9.77) traduces for the linearized equation (9.78) in the gauge freedom

h h′ := h + ∂ Λ + ∂ Λ . (9.79) µν → µν µν µ ν ν µ This freedom enable us to ‘simulate’ the constraint (9.76b) of the massive scalar- tensor theory by choosing the gauge

hν α∂ν h =0 . (9.80) −

In order to do this one has to solve the following equations for Λµ

[ηµν (2α 1)∂µ∂ν ]Λ = h′ν α∂ν h′ (9.81) − − ν − and this is possible, since the diﬀerential operator on the l.h.s. is hyperbolic and non-degenerate for any α = 1, and that is exactly the case here. Now, choosing the 6 gauge (9.80), we can write the equation (9.78) of linearized GR as

h (2α 1)h = κ (T 1 η T ) (9.82) µν − − ,µν − 0 µν − 2 µν 9.4. Massive scalar-tensor theories for gravity 147

µν which is the equation one formally gets putting ∂µT = 0 and then m = 0 in the equation (9.76a) of the massive scalar-tensor theory. We notice that the equations (9.82) and (9.80) have still the residual gauge freedom of transformations (9.79) where the ﬁeld Λµ satisﬁes

[ηµν (2α 1)∂µ∂ν ]Λ =0 . − − ν 9.4.2 Scalar-tensor decomposition and Cauchy problem In order to make clear the presence, in case of α = 1, of a tensor and of a scalar field 6 we show that the above constrained and coupled equations for hµν and h can be decoupled in two set of equations for a massive spin-2 (tensor) and a spin-0 (scalar) field, respectively. First we recall that the Lagrangian for a massive scalar field with source ρ is given by

1 = (∂ φ)(∂µφ) m2φ2 κφρ (9.83) Ls 2 µ − − which implies the ﬁeld equations

( + m2)φ = κρ . (9.84) − The sign in front of κ in the scalar Lagrangian is chosen so that, in the static case, (9.84) reproduces the correct sign of Poisson’s equation of Newtonian gravity (see (9.58)).

Starting from a solution hµν of (9.76) we deﬁne a scalar ﬁeld via

φ := 2(1 α)ηαβ h (9.85) − αβ and with the help of the operators

αβ := 1 (1 α)Σ ηαβ , (9.86a) Sµν 3 − µν αβ := id αβ αβ , (9.86b) Tµν µν − Sµν which we call the scalar and tensor projector, respectively, we deﬁne two more ﬁelds as follows: h(0) := αβ h = 1 Σ φ , (9.87a) µν Sµν αβ 6 µν h(2) := αβ h . (9.87b) µν Tµν αβ

αβ α β Above, idµν := δµ δν is the identity operator on the linear space of the (0, 2)- tensors and Σµν is deﬁned in (9.39). (0) (2) As we will now show, hµν and hµν are indeed the scalar (spin 0) and tensor

(spin 2) part of hµν , respectively. Clearly, the former is uniquely determined by the scalar ﬁeld φ. First of all, the presence of a propagating scalar ﬁeld is easily shown taking the trace of (9.76a):

2(1 α) + (4α 1)m2 ηµν h = κ Σ T µν . (9.88) − − µν µν In terms of the scalar ﬁeld φ this reads as

2 µν ( + m ) φ = κ Σµν T , (9.89) 148 Chapter 9. Linear massive theories which is the equation for a scalar ﬁeld with α-dependent mass

m := m (4α 1)/2(1 α) (9.90a) − − p and source

ρ := Σ T µν . (9.90b) − µν If we exclude tachyonic fields (meaning negative mass-squared), we clearly must 1 restrict α to the interval [ 4 , 1). Doing so, m can take any value in the interval [0, ). We have m = m iff α = 1/2. Notice that from (9.88) one clearly sees ∞ that there is a propagating scalar field exactly if α = 1: in fact, in the α = 1 6 case, (9.88) becomes an algebraic constraint on h. Since the scalar part is given by h(0) = αβ h = 1 Σ φ, its equation follows immediately from (9.89): µν Sµν αβ 6 µν

( + m2) αβ h = κ Σ Σαβ T . (9.91) Sµν αβ 6 µν αβ

(2) Before we are going to derive the equations for the tensor part hµν , we warm up computing the following useful identities (meaning valid in general, without the use of the equation of motion) for the scalar and tensor operators deﬁned in (9.86). We collect this identities in the following

Lemma 9.2 (Properties of the scalar and tensor operators). The scalar and tensor operators (9.86) satisfy the following ‘projections properties’:

αβ γδ = γδ γδ , (9.92a) Sµν Sαβ Sµν − D Sµν αβ γδ = γδ γδ , (9.92b) Tµν Tαβ Tµν − D Sµν αβ γδ = αβ γδ = γδ , (9.92c) Tµν Sαβ Sµν Tαβ D Sµν where, for notational convenience, we put

2(1 α) := − ( + m2) . (9.93) D 3m2

Moreover, the tensor operator satisﬁes:

∂µ κλ = ∂κδλ α∂ ηκλ + ηκλ , (9.94a) Tµν ν − ν D ηµν κλ = ηκλ . (9.94b) Tµν D Proof. The ﬁrst identity of (9.92) can be shown by direct computation. The other two then follow using the relation = id . The identities (9.94) follow also by T − S a direct computation.

In view of (9.91), the properties (9.92) say that, on a vacuum solution of (9.76), the scalar and tensor operators and are indeed complementary projectors. In S T order to derive the equations for the tensor part, it is useful to rewrite (9.76a) as

( + m2) h + 3(2α−1) m2 κλ h = κ κλ T , (9.95) µν 2(1−α) Sµν κλ − Jµν κλ 9.4. Massive scalar-tensor theories for gravity 149 where κλ := id κλ 1 η ηκλ + 2 δ(κ∂ ∂λ) . (9.96) Jµν µν − 2 µν m2 (µ ν) At this point, it is instructive to consider the vacuum case first: here, using the projection properties (9.92) and looking at (9.94), it is immediately clear that the coupled equations (9.76) are equivalent to a massive (mass m) propagation equation for the scalar field φ and the vacuum massive (mass m) spin-2 equations for αβ h . To see this note that, since + = id, equation (9.76a) (that Tµν αβ S T is (9.95)) is equivalent to the set given by its projection with and its projection S with . Using the projection properties (9.94), the first, after a rearrangement, T gives ( + m2) αβ h = 0 (9.97) Sµν αβ and the second, since = 0, gives immediately T S ( + m2) αβ h =0 . (9.98) Tµν αβ These are the propagation equations for the scalar and the tensor part, respectively. Recall, that taking the trace of (9.76a) one gets

( + m2) φ =0 , (9.99) which already implies (9.97). Hence, in vacuum, (9.76a) is equivalent to (9.98) and (9.99). We turn now to the constraint (9.76b): looking at (9.94), it is clear that the set given by (9.76b) and (9.99) is equivalent to the divergence and trace constraints (9.13b,9.13c) for αβ h . Hence, summarizing, the coupled equa- Tµν αβ tions (9.76) are equivalent, in vacuum, to the independent set given by the massive propagation equation (9.99) for the scalar ﬁeld φ and the massive spin-2 equations (9.13) for αβ h . Tµν αβ In presence of a source, one carries out exactly the same steps, thereby keeping track of the source terms on the r.h.s. In order to get the source term of the propagation equation for the scalar part one shows the identity

κλ ρσ = 1 (1 α)Σ Σκλ , (9.100) Sµν Jκλ − 3 − µν which indeed leads to (9.91). For the tensor part, projecting (9.95) with , one gets: T

( + m2) κλ h + (2α 1)( + m2) κλ h = κ κλ σρ T (9.101) Tµν κλ − Sµν κλ − Tµν Jκλ σρ which, using equation (9.91) to eliminate the scalar part, takes the familiar form

( + m2) κλ h = κ Πm κλ T , (9.102) Tµν κλ − µν κλ and this is the propagation equation (9.42a) for a massive spin-2 ﬁeld with mass m. For the step from (9.101) to (9.102) one just uses the identity

κλ 2α−1 κλ ρσ = Πm ρσ , (9.103) Tµν − 2(1−α) Sµν Jκλ µν which can be shown substituting = id and using (9.100). T − S We collect this equivalence in the following 150 Chapter 9. Linear massive theories

Lemma 9.3 (Scalar-tensor decomposition). Let α = 1. If a symmetric field h 6 µν satisfies the field equations (9.76), then the fields κλ h and φ := 2(1 α)ηµν h Tµν κλ − µν 3 satisfy, respectively, the massive spin-2 equations with mass m and source Tµν and the massive spin-0 equation4 with mass m := m (4α 1)/2(1 α) and source − − ρ := Σ T µν. Conversely, if a symmetric field h(2) satisfies the massive spin- − µν pµν 2 equations with mass m and source Tµν and a scalar field φ the massive spin-0 equation with mass m := m (4α 1)/2(1 α) and source ρ := Σ T µν , then − − − µν the combination p (2) 1 hµν := hµν + 6 Σµν φ (9.104) satisfies the field equations (9.76). Moreover, in the vacuum case, the spin-0 and spin-2 part (9.87a) and (9.87b) are formally orthogonal w.r.t. the standard scalar product (9.105) induced by the Minkowski metric on the linear space of the (0, 2)-tensor fields. Proof. All it remains to show is the formal (meaning: neglecting boundary terms) orthogonality between the spin-0 and the spin-2 part of a vacuum solution of (9.76) w.r.t. the scalar product a, b := d4xaµν b (9.105) h i µν Z induced by the Minkowski metric on the linear space of the (0, 2)-tensor fields. To see this, note first that the formal adjoint operator to κλ w.r.t. (9.105) is Sµν † κλ = 1 (1 α)η Σκλ. Since, in vacuum, κλ h has vanishing trace and S µν 3 − µν Tµν κλ divergence, it holds † κλ ρσ h = 0, from which it follows the orthogonality S µν Tκλ ρσ between κλ h and κλ k , where h and k are any two vacuum solutions Sµν κλ Tµν κλ µν µν of (9.76).

Summarizing, the above decomposition lemma enables to identify αβ h Tµν αβ with the spin-2 and αβ h (proportional to the trace h) with the spin-0 com- Sµν αβ ponent of the h field which solves (9.76) for α = 1. It says also that, re-expressed µν 6 in these components, the equations (9.76) decouple into two independent set of equations: the massive spin-2 and spin-0 field equations. This enables to set up the Cauchy problem for (9.76) which reduces to the two separated Cauchy problems for the massive spin-2 and spin-0 components, both of which are well-posed. A note on the choice of the field decomposition (9.85) is needed. One can of course redefine φ by multiplication with a (nonzero) constant. Since the equation for φ is linear this redefinition must be accompanied by a same redefinition of the source. In (9.85) we simply choose the constant in such a way that the source does not depends on α. With this choice, the dependence on α of a solution hµν of (9.76) hides solely in the mass of its scalar part φ.

9.4.3 The scalar is a ghost

It is widely known that the non Pauli–Fierz mass terms (α = 1) lead all to theo- 6 ries containing ghosts. However, in [BG03] it was argued, considering a particular

3 See (9.42). 4 See (9.84). 9.4. Massive scalar-tensor theories for gravity 151 gravitational-wave solution, that this belief is not true. With the help of the above decomposition lemma, we will now confirm that the above theory, for any α = 1, 6 does contain ghosts. By this we mean that the energy of the free field is unbounded from below, and not that every solution must have negative energy. Hence, finding a particular solution with positive energy like in [BG03], do not implies that the theory is ghost-free. In the following we show that the ghost is given precisely by the scalar component φ. To see this one has to compute the Hamiltonian for the action (9.25) with the modified mass term (9.72). In fact, it suffices to look at the kinetic part of the Hamiltonian and show that the latter for its own is not positive definite. The kinetic part of the Hamiltonian is simply given by those terms in the Lagrangian which are quadratic in the velocities (i.e. the time derivative of hµν ), since the Legendre transformation leave unaltered these terms and kills those terms linear in the velocities. Thus, from (9.26), one gets that the kinetic energy density is

ε = 1 h˙ h˙ kl (h˙ k)2 (9.106) kin 4 kl − k for any α. This expression, integrated over some initial Cauchy hypersurface, is to be thought of as a functional on the initial data, which, by the decomposition lemma, are given by the independent initial data for the massive spin-2 component (2) hµν and for the massive scalar field φ. We choose vanishing initial data for the spin- 2 component and let for the moment unspecified those for the spin-0 field, which ˙ 1 are given by φ and its time derivative φ. Inserting hµν = 6 Σµν φ in (9.106) one gets

1 4 2 ε = 3φ˙2 + φ˙∆φ˙ + (∂ ∂ φ˙)(∂ ∂ φ˙) (∆φ˙)2 , (9.107) kin −72 m2 m4 k l k l − where one has to sum over repeated indices. The last term in (9.107) is a pure divergence and hence, assuming initial data with appropriate fall-oﬀ behavior, vanishes when integrating it over the initial Cauchy hypersurface and thus does not contribute to the total kinetic energy. Rewriting the second term with φ˙∆φ˙ = div(φ˙ φ˙) φ˙ 2 and integrating (9.107) over the initial Cauchy hyper- ∇ − k∇ k surface, again neglecting the divergence term, the contribution of the scalar part to the kinetic energy is:

4 E = 1 d3x 3φ˙2 φ˙ 2 . (9.108) kin − 72 − m2 k∇ k Z This is clearly unbounded from below: it suﬃces to choose for φ˙ a function which varies very slowly in space, so that the ﬁrst term in (9.108) dominates the second one.

9.4.4 Comment on gravitational waves The presence of a ghost in a theory is particularly dramatic when computing the energy radiated by a source. In [FS02, SF02] it is attempted to ﬁnd an upper limit of the graviton mass using a massive theory whose linearization corresponds exactly to the theory considered in this section with the choice α =1/2. Assuming a source with conserved energy-momentum tensor and periodic time-dependence, 152 Chapter 9. Linear massive theories with a Fourier analysis in time the emitted power is computed to be (see formulae (3.14) in [FS02] or (3.7) in [SF02]):

∞ m2ω4 L = L + n I˜ (ω )I˜∗ (ω ) I˜(ω ) 2 + (m4) , (9.109a) GR 3 kl n kl n − | n | O n=1 X where ∞ 2 L = ω6 I˜ (ω )I˜∗ (ω ) 1 I˜(ω ) 2 (9.109b) GR 5 n kl n kl n − 3 | n | n=1 X k l 3 is the emitted power of GR, Ikl(t) := x x T00(t, ~x)d x the quadrupole moment ˜ ˜ of the source, I(t) its trace, Ikl(ωn) theRn-th Fourier coefficient of Ikl(t), I(ωn) its trace, and ωn = 2πn/τ, where τ is the period of the source. The above formulae are better understood writing Ikl in terms of its trace I and the reduced (traceless) quadrupole moment Q := I 1 δ I. In these terms the emitted power reads as kl kl − 3 kl ∞ m2ω4 L = L + n Q˜kl(ω )Q˜∗ (ω ) 2 I˜(ω ) 2 + (m4) , (9.110a) GR 3 n kl n − 3 | n | O n=1 X with ∞ 2 L = ω6 Q˜ (ω )Q˜∗ (ω ) . (9.110b) GR 5 n kl n kl n n=1 X Whereas the general relativistic contribution (9.110b) is obviously positive, the contribution in (9.110a) coming from the graviton mass is clearly indefinite and, in principle, unboundedly negative. In [FS02, SF02] the above radiation formulae are applied to a system of binary stars: fortunately, if they are on a Keplerian orbit, the contribution due to the graviton mass turns out to be nonnegative. In the simple case of two bodies with mass on a circular orbit this is easily seen, since the trace I is a constant and thus all Fourier coefficients I˜(ω ) with n 1 vanish. For n ≥ the general situation of two bodies with arbitrary mass on a Keplerian orbit with ellipticity e, the Fourier coefficient of the quadrupole moments can be expressed in terms of Bessel’s functions [PM63] and the contribution to the emitted power originating from the graviton mass relative to the general relativistic contribution is found to be—up to leading order in m:

L L 5 mτ 2 (1 e2)3 GR = , (9.111) − 73−2 37 4 LGR 24 2π 1+ e + e 24 96 as one can show using the formulae on summation of Bessel’s functions found in [PM63]. From (9.111) it is clear that the contribution due to the graviton mass is nonnegative: it is maximal for circular orbits (e = 0), falls monotonically for e between zero and one, and vanishes in the limiting case e 1. → However, very diﬀerent is the situation in case of a spherically symmetric moving source, that is one with T (t, ~x)= ρ(t, ~x ) , (9.112) 00 k k for some positive function ρ. This can be, e.g., a pulsation or an accelerating expansion or contraction. In this case, due to the spherical symmetry, the 9.4. Massive scalar-tensor theories for gravity 153

1 ∞ 4 quadrupole moment is a pure trace: Ikl(t) = 3 δkl4π 0 dr r ρ(t, r) (just insert (9.112) in the deﬁnition of Ikl and perform the angularR integration using the k l 2 4π formula dΩ x x /r = 3 δkl) and thus:

R ∞ Qkl(t) 0, and I(t)=4π dr r4ρ(t, r) . (9.113) ≡ Z0 Hence, in the case of a spherically symmetric moving source, the general relativistic contribution identically vanishes (as it should) and the emitted power is

∞ 2m2ω4 L = n I˜(ω ) 2 + (m4) , (9.114) − 9 | n | O n=1 X which, up to leading order in m, is manifestly negative definite! Of course to have a concrete case of emission of gravitational waves with negative energy (in the setting of the present theory) one should find a source with conserved energy-momentum tensor satisfying (9.112). As example, one can think at a pulsating rubber balloon (seen as a thin shell) inflated with an ideal gas.

9.4.5 Formal solution and massless limit

Formally one can now write the particular solution of the massive equations (9.76) for a general α = 1. Using the above decomposition lemma and restricting to the 6 case of a conserved energy-momentum tensor, we can write, formally, the solution as

h = h(2) + 1 (η 2 ∂ ∂ )φ µν µν 6 µν − m2 µ ν 2 −1 1 = κ( + m ) (Tµν 3 ηµν T ) − − (9.115) κ 2 −1 + 6 ηµν ( + m ) T + κ ( + m2)−1 ( + m2)−1 ∂ ∂ T . 3m2 − µ ν Here, the inverse of the Klein–Gordon operators would mean the convolution with the fundamental solution to the Klein–Gordon equation to some appropriate boundary conditions. Here and in the following, computations are to be understood as carried out in momentum space (after Fourier transformation) where ∂µ goes over in ipµ. The limit m 0 is non singular and it gives → κ 2α 1 h = κ−1(T 1 η T )+ − ∂ ∂ (−1−1T ) . (9.116) µν − µν − 2 µν 2 1 α µ ν − The first term is exactly the same of the massless theory and the second one remains finite in the limit and is thus now a pure gauge. This shows that for all α = 1 the 6 massive (scalar-tensor) theory goes over, in the massless limit, to linearized GR. A particular case arise here for α =1/2 (the value adopted from Visser [Vis98]) where the second term vanishes. This is though nothing special, since this extra term is just a pure (and finite) gauge. 154 Chapter 9. Linear massive theories

9.4.6 Static spherically symmetric vacuum solution As we did for the massive spin-2 theory, we give here the static, spherically symmetric, vacuum solutions of (9.76) for the α = 1 case. In contrast to the pure massive 6 spin-2 theory, the massless limits of these solutions do exist and coincide with the solution of the massless theory (linearized Schwarzschild solution). By the decomposition lemma, the most general static spherically symmetric vacuum solutions is now given by a two parameter family of solutions, given by the (2) (0) (2) sum hµν + hµν , where hµν is the spin-2 part given by equation (9.23) and

h(0) = c 1 Σ f(r) (9.117a) µν − 6 µν is the spin-0 part, where exp( mr) f(r) := − . (9.117b) r The minus sign in (9.117a) is just for later convenience. In presence of a central point mass M the identiﬁcation of the two free parameter, b and c, is carried out as follows. First, since the matter coupling is the same as discussed in the previous chapter, test particles moves along geodesics in the metric gµν = ηµν + κhµν and hence, comparing the Newtonian approximation of the geodesic equation with the Newtonian equation of motion, we get the same identiﬁcation (9.55) between the h00 and the Newtonian potential. Now, h00 is given by

h = h(2) + h(0) = bf(r) c 1 f(r) . (9.118) 00 00 00 − − 6 In the massless limit this, together with the identification (9.55), this imply the relation 1 κ(b + 6 c)=2GM . (9.119) In order to identify separately the two free parameter we have first to identify the coupling strength κ. For this note that in the massless limit the field equation for the 00-components of the spin-2 and spin-0 components reduce, in the static case, to (2) 2 ∆h00 = 3 κT00 (9.120) and ∆φ = κT (9.121) − 00 respectively. Hence, the 00-component of the hµν field (9.104) satisfies, in that limit, κ ∆h00 = 2 T00 . (9.122)

Comparison with Poisson’s equation (9.58) (provided T00 = ̺) determines the coupling strength to be κ2 = 16πG (9.123) which corresponds to the coupling strength of the massless case (GR). The two parameters b and c can now be separately determined comparing (9.120) and (9.121) with the distributional equation ∆( 1/r)=4πδ and using (9.123) and − that T00 = ̺ = Mδ. This gives 8 κb = 3 GM (9.124a) 9.5. Conclusions about linear massive theories 155 and κc = 4GM (9.124b) − which conclude the identifications. Now, inserting the identifications (9.124) and the solutions (9.117) and (9.23) in (0) (2) hµν = hµν + hµν , we can write the one-parameter family of spherically symmetric, asymptotically flat, vacuum solution to (9.76) for a given α = 1: 6 2GM 1 ~0⊤ κh (~x)= f(r) µν 3 ~0 ( δ 2 ∂ ∂ ) − ij − m2 i j 4GM 2 ~0⊤ 1 f(r) . (9.125) − 3 ~0 (δ 2 ∂ ∂ ) ij − m i j The massless limit of (9.125) is then computed as follows: The part of (9.125) which does not contain negative powers of m gives:

2GM 1 ~0⊤ lim κhµν (~x)= , (9.126) m→0 − r ~0 δ ij which is just the solution of the massless theory (linearized Schwarzschild). The rest, which is proportional to m−2 and hence potentially singular in the massless limit, is 4GM 1 κhs (~x) := ∂ ∂ f(r) f(r) . (9.127) ij 3 m2 i j − In fact, an explicit computation shows that this limit exists and gives:

s 1 2α lim κhij (~x)= − GM∂i∂j r . (9.128) m→0 1 α − For the computation of this limit it is useful to rewrite the diﬀerence f f in − terms of hyperbolic functions as follows: f(r) f(r) (e−mr e−mr)/r = − ≡ − 2e−(1+p/2)mr sinh(pmr/2)/r, where p = p(α) := (4α 1)/(2(1 α)) 1. The − − − massless limit of (9.127) can be then easily computed to be 1−2α GM (δ xixj /r2). p 1−α r ij − Rewriting 1 (δ xixj /r2) as ∂ ∂ r one arrives at (9.128). r ij − i j After having taken the massless limit one is in the massless theory, which has the gauge freedom h h + ∂ Λ + ∂ Λ . Hence, the term (9.128) is clearly pure µν 7→ µν µ ν ν µ gauge. Alternatively, as explained in the previous chapter below equation (9.66), the term (9.128) can be eliminated from the metric gµν = ηµν +κhµν by the spatial coordinate transformation xi x′i = xi + 1 1−2α GM∂ r. 7→ 2 1−α i Summarizing, the massless limit of the spherically symmetric, asymptotically ﬂat, vacuum solution of (9.76) for any α [1/4, 1) exists and coincides with the ∈ solution of the massless theory (linearized Schwarzschild solution) (9.126).

9.5 Conclusions about linear massive theories

So far we have shown that, in the setting of Poincaré-covariant linear massive theories, we have the following situation. From one side we have the pure spin-2 massive theory (Section 9.3), which is satisfying from the theoretical viewpoint since it fulfills 156 Chapter 9. Linear massive theories the basic requirements (energy positivity for the free field, Equivalence Principle, and well-posedness of the Cauchy problem) but not from the phenomenological one, since it gives a prediction for the light deflection which does not agree with observations, no matter how small the graviton mass is (vDVZ discontinuity). On the other side we have the (one parameter family of) massive scalar-tensor theories (Section 9.4). These are sometimes used in the literature, since their static, spherically symmetric, asymptotically flat solution agrees, in the massless limit, with the corresponding solution of the massless theory (linearized Schwarzschild solution) and hence passes, within linear order, all static Solar-System tests. Unfortunately, however, none of the theories satisfies the energy positivity for the free field. This is particularly dramatic when dealing with radiation problems: the emitted power in form of gravitational waves of a spherically symmetric pulsating source would be negative! Hence, none of the considered massive theories gives a valuable theory for (linear) gravity which is both theoretically satisfying and in agreement with observations. At this point one is perhaps tempted to conclude that there is no valuable massive theory for gravity, at least in the realm of Poincaré-covariant linear theories in four spacetime dimensions. (We recall that the spin-1 theory (Proca) is to be excluded a priori since, if it has to satisfy energy positivity for the free field, it predicts a repulsive force (see, e.g., Chap. I.5 in [Zee03]).) However, an interesting statement has been made some time ago in [Vai72] and more recently in [DDGV02] according to which the vDVZ discontinuity is peculiar of the linear theory and, if one consider the massless limit in the full nonlinear massive theory, this discontinuity vanishes. The next chapter is devoted to this statement. Chapter 10

Nonlinear massive theories

In the present chapter we seek nonlinear generalizations of the massive theories studied in the previous two chapters. We have seen that these split up in two classes, depending on whether α = 1 or not. If α = 1 one has a pure massive spin-2 field (Pauli–Fierz theory) else, if α = 1, one has a mixture of a spin-2 and a scalar 6 field. Theories in the latter class have, from one hand, all a well-defined massless limit which agrees with linearized GR; on the other hand, however, they all contain a ghost. On the contrary, the Pauli–Fierz theory is ghost-free but not only it does not agree with linearized GR in the massless limit as recognized by vDVZ but, strictly speaking, such limit does not even exist within the linear setting. Despite the presence of ghosts, a particular theory of this second class (with α = 1/2) where used by Finn & Sutton to produce a graviton-mass bound using binary pulsar observations [FS02, SF02]. Moreover, nonlinear generalizations where considered by Visser [Vis98] (also for the case α = 1/2). Here, we maintain that energy positivity of the free field is a feature one should reasonably impose to a theory and hence we restrict from now on to the α = 1 case and seek for its possible nonlinear generalizations. Our primary goal is to address the claim of Vainshtein [Vai72] and colleagues [DDGV02], according to which the massless limit exists and the vDVZ discontinuity vanishes, provided one goes over to the (still to be specified) nonlinear generalization of the massive Pauli–Fierz theory. We proceed as follows: we first present a classification of various possible ‘natural’ nonlinear generalizations of the Pauli–Fierz theory, give their scaling property, and formulate the coupling with matter. Then we focus on static spherically symmetric situations, where we review the only known static spherically symmetric exact solutions due to Salam & Strathdee, identifying the geometrical origin of their classification of such solutions. We then give the field equations for two ansätze which we use for numerical investigations.

10.1 Free theory

According to the setting in Chapter 1 we deﬁne the nonlinear generalizations of the Pauli–Fierz massive spin-2 theory considered in the literature (see, e.g., [DKP03]) as follows. The basic objects of the theory are the ‘physical metric’ g (together with its Levi-Civita connection denoted by ∇) and a ‘background metric’ f. From g and f one can build a two-tensor h which is, essentially, their diﬀerence and is

157 158 Chapter 10. Nonlinear massive theories to be seen as the generalization of the field h of the linear theory. The background metric f, needed to define h and the mass term, is in general chosen to be the flat Minkowski metric η. This we will assume when looking for static, spherically symmetric, asymptotically flat solutions. For the moment f remains unspecified (but fixed) and is to be seen as an ‘absolute element’ (i.e. non-dynamical) of the theory. The field equations for g are required to be derived by a variational principle, that is as stationary point of an action. In analogy with the linear case one requires that the Lagrangian is given by the sum of the ‘massless’ Lagrangian—taken to be the Einstein–Hilbert Lagrangian of GR—and a ‘mass term’ generalizing the Pauli– Fierz mass term of the linear theory. Hence the action is given by (C.8) in which the Lagrangian density is taken to be

= + , (10.1) L LEH Lm where is the Einstein–Hilbert Lagrangian (C.9)1 for the metric g, LEH 1 EH = √ g Scalg (10.2) L −κ2 −

(the coupling constant κ is to be determined in an appropriate Newtonian approximation), and is a mass term generalizing the Pauli–Fierz mass term (9.27). Lm Accordingly, for we impose that: Lm (i) it is a scalar density of weight 1/2 (ii) it is an algebraic function of the metrics g and f (iii) in leading (quadratic) order in the difference g f it reduces to the Pauli–Fierz − mass term (9.27) Defining h to be a symmetric two-tensor field which, in linear approximation, reduces to the difference g f between the physical and the background metric, a − mass term which surely satisfies the above requirements has the following structure:

1 m2 = h, h (trh)2 vol , (10.3) Lm −κ2 4 h i− where vol is a scalar density of weight 1/2. Here, one still has to specify the explicit form of the tensor ﬁeld h and of the scalar density vol, as well as specify to which metric the norm and trace in (10.3) are referred. Accordingly, a possible concrete implementation of this mass term satisfying the above requirements (i)–(iii) is given by the following choices: 1. For the tensor ﬁeld h one can put

√ g β hβ := − g f , (10.4a) √ f − − where β is a real parameter, 2. for the scalar density of weight 1/2 vol one can put

vol := (√ g)1−γ ( f)γ , (10.4b) γ − − where γ is a real parameter, p

1 One specializes here to the case of vanishing cosmological constant. 10.1. Free theory 159

3. and one can refer scalar product and trace w.r.t. a convex combination of g and f: l := (1 δ)g + δf , (10.4c) δ − where δ [0, 1]. ∈ Summarizing, this concrete implementation gives the following 3-parameter family (β,γ,δ) of mass terms:

1 m2 (β,γ,δ) := h , h (tr h)2 vol , (10.5) Lm −κ2 4 h β βil(δ) − l(δ) γ (Here, we reserve the letter α for those who want to consider non-Pauli–Fierz mass term, as we did at linear level in the last chapter.) Remark 10.1. This is of course not the most general implementation of the above requirements. For example, one might define vol also as a convex combination of √ g and √ f or as √ l , where l is the metric (10.4c). What is important to − − − δ δ us, however, is that (10.5) gives a parametrization for the mass term which covers all the particular mass terms considered in the literature up to now. For this we note that some authors define the field h as a (2, 0) tensor field, that is

β √ g −1 −1 Hβ := − g f (10.6) √ f − − where we generalized the deﬁnition H := g−1 f −1 present in the literature as we − did for h in (10.4a). This is also covered by our parametrization noticing that

2 2 H , H g (trgH ) vol = h , h f (trf h ) vol (10.7) h β βi − β γ h −β −βi − −β γ−2β and, analogously,

2 2 H , H f (trf H ) vol = h , h g (trgh ) vol . (10.8) h β βi − β γ h −β −βi − −β γ−2β Hence, for example, the work [DKP03] considers the four terms corresponding to (β,γ,δ) = (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), [BDZ10, BDZ09] considers (0, 0, 0), and [SS77] (0, 1, 0). The variation of the action w.r.t. g gives the massive vacuum ﬁeld equations

2 Eing + m M(g,f) = 0 (10.9) where Eing is the Einstein tensor to g and the mass term M is the (0, 2) tensor ﬁeld deﬁned by the equation

1 2 µν δg = √ gm M δg . (10.10) Lm −κ2 − µν

In (10.9) we write M(g,f) in order to stress that M depends (in a pure algebraic way) on both metrics g and f. Alternatively, one can write the mass term on the r.h.s. and think of it as a ﬁctional matter-source with energy-momentum tensor

T := 2 m2M . (10.11) m − κ2 160 Chapter 10. Nonlinear massive theories

As a consequence of (10.9) and of the divergence-freeness of the Einstein tensor, it follows that div M =0 , (10.12) where the divergence is taken w.r.t. the Levi-Civita connection to g. At this point we note that, even without knowing the explicit expressions for the mass term M, the vacuum ﬁeld equation (10.9) has the following conformal property of under the simultaneous conformal transformation of both metrics g and f: (g, f) (g , f ) := (λ2g, λ2f) , (10.13) 7→ λ λ for some positive constant λ. It holds the following

Lemma 10.2 (Conformal property of the vacuum ﬁeld equations). Let the pair (g, f) satisfy the vacuum ﬁeld equation (10.9) for the mass parameter m. Then

(gλ, fλ) satisﬁes the vacuum ﬁeld equation for the mass parameter m/λ.

Proof. By construction, /vol is invariant under the transformation (10.13); there- Lm fore the same holds for m/√ g. This, in view of (10.10), implies that the mass L − 2 term M transforms like the metrics, that is M(gλ,fλ) = λ M(g,f). Hence one 2 2 has Eingλ +(m/λ) M(gλ,fλ) = Eing + m M(g,f). For this recall that Ein is invariant under a conformal transformation by a constant conformal factor (see Ap- pendix A.8).

In words, Lemma 10.2 says that if we diminish the mass, the ﬁeld grows in magnitude. This may be seen in relation to the fact that diminishing the mass of a ﬁeld one increases its range. As example we give the explicit form of the mass term for the case (β,γ,δ) = (0, 1, 0), which corresponds to the choice of [SS77]. One gets:

1 √ f M = − f g−1 f −1 f tr f g−1 f −1 f (10.14) 2 √ g · − · − · − − From the above expressions one immediately sees that putting overall g = f + h and taking terms up to linear order in h one gets (h trf h), which is exactly − − the mass term we get from (9.27) in the linear theory—provided one identiﬁes the background metric f with the Minkowski metric. For this notice that g−1 = f −1 − f −1 h f −1 + (h2). Since the linearization of the Einstein tensor gives exactly the · · O massless part of the Euler–Lagrange equations (9.73) we get as formal linearization of this nonlinear theory the linear Pauli–Fierz massive theory we studied in the last chapter.

10.2 Matter coupling

The matter coupling is required to be as described in Chapter 1: one takes the matter Lagrangian of Special Relativity and replaces everywhere the Minkowski metric with the ‘physical matric’ g and the partial derivative with ∇, the Levi-Civita connection to g, so that the resulting matter action is diffeomorphism invariant. This is then to be added to the free action described above in order to give the total 10.3. Static spherically-symmetric configurations 161 action. The variation of the latter w.r.t. the metric g gives then the massive field equations: 2 κ2 Eing + m M(g,η) = 2 TM , (10.15) where the matter energy-momentum-tensor is defined as usual (see Appendix C.2), whereas the variation w.r.t. the matter field(s) gives the equation of motion for the matter. Note that, because of the diffeomorphism invariance of the matter part of the action, it follows that TM is divergence-free (w.r.t. the divergence induced by the Levi-Civita connection to g). By (10.15) it then follows that the mass term M too must be divergence-free in the same sense.

10.3 Static spherically-symmetric conﬁgurations

10.3.1 Spherical symmetry and staticity A spherically symmetric configuration is one for which all the fields of the theory are invariant under the rotation group (thereby referring to the same group action for all the fields). The fields of the considered theory are given by the two metrics g and f. The definition of spherically symmetric spacetimes of Chapter 4 (Definition 4.1) extends also here —provided we require the background metric also to be spherically symmetric (in the sense of that definition). Hence, one gets again the structure of warped product space, where the spacetime manifold is (locally) the product = S2 between a two-dimensional manifold and the two-sphere. Both the M B× B physical and the background metric have the structure:

2 g = gB R g 2 , (10.16a) − g S 2 f = fB R g 2 , (10.16b) − f S where Rg and Rf are the areal radii and gB and fB the -parts of the metrics g B and f, respectively. We call this a spherically symmetric bi-metric configuration. Similarly, a configuration is called stationary if there exist a non-vanishing timelike vector field K, say, having the property that all the fields are invariant under its flow. A static configuration is then a stationary configuration where the vector field K is also required to be hypersurface-orthogonal w.r.t. a given metric. A stationary spherically symmetric bi-metric configuration is static exactly for the same reasons that a stationary spherically spacetime (with just one metric) is static: this is because the vector field K can be chosen to be spherically symmetric. Hence, both the distributions kerKg and kerKf are integrable (see Proposition 4.2) but, in general, the related integral hypersurfaces do not necessarily coincide. This is because, in general, there will be no coordinates such that both metrics are diagonal. In fact, choosing coordinates adapted to the integral hypersurface to, say, the distribution kerKf one has:

2 2 2 g = A(r)dt B(r)dr C(r)dt dr R (r)g 2 , (10.17a) − − ∨ − g S 2 2 2 f = D(r)dt E(r)dr R (r)g 2 . (10.17b) − − f S This gives the general static spherically symmetric bi-metric ansatz. (The other possibility is, of course, to choose coordinates adapted to the integral hypersurface to the distribution kerKg, which leads to an ansatz where g is diagonal, but not f.) 162 Chapter 10. Nonlinear massive theories

The only known exact solutions to the massive field equation for a static spherically symmetric situation have been found by Salam & Strathdee [SS77] and then generalized by Isham & Storey [IS78]. Their result is that the solutions can be subdivided in two classes, as we shall now see. Our main goal here is to understand the origin of this subdivision, which arise, according to [SS77], from a clever combination of the components tangent to of the field equation. To this purpose B we specialize to the mass term of [SS77], that is, to (β,γ,δ) = (0, 1, 0). We first compute the -part of the mass term assuming only spherical symmetry, hence B using the general spherically symmetric ansatz (10.16). The computations are conveniently carried out with respect to an orthonormal basis on for the metric g. B Denoting this by e , e and its dual basis by θ0, θ1 (i.e. θµ(e )= δµ) one has: { 0 1} { } ν ν g = θ0 θ0 θ1 θ1 and g−1 = e e e e . Then one easily compute the ⊗ − ⊗ 0 ⊗ 0 − 1 ⊗ 1 part of the mass term (10.14) within square brackets, which we denote by M˜ :

M˜ (e , e ) = det fB 2χfB(e , e ) (10.18a) 0 0 − 0 0

M˜ (e , e )= det fB 2χfB(e , e ) (10.18b) 1 1 − − 1 1

M˜ (e , e )= 2χfB(e , e ) . (10.18c) 0 1 − 0 1

2 Here det fB := fB(e , e )fB (e , e ) fB(e , e ) is the determinant of fB and 0 0 1 1 − 0 1 2 Rf 3 χ := 2 . (10.19) Rg − 2

We recall that until now we used just spherical symmetry and that the orthonormal basis e , e for gB is arbitrary. Now we impose staticity: this implies the existence { 0 1} of a hypersurface-orthogonal timelike Killing ﬁeld K for g, which, as consequence, is an eigenvector of the Ricci tensor (see, e.g., Sect. 18.6 of [SKM+03]) and herewith of the Einstein tensor (both regarded as endomorphisms). Adapting our basis to K, that is choosing e K, we thus have Ein(e , e ) = 0 and, by the ﬁeld equations, 0 ∝ 0 1 M(e0, e1) = 0. By (10.18c) this is equivalent to

2 Rf 3 fB e e 2 ( 0, 1)=0 , (10.20) Rg − 2!

2 that is: either the two areal radii are related by the algebraic condition (Rf /Rg) = 3/2 or the metrics f and g are simultaneously diagonalizable. These conditions give, respectively, the two classes of solutions according to [SS77], called henceforth class I and II. Moreover, [SS77, IS78] explicitly found the solutions of class I, which are given by configurations in which both g and f are Schwarzschild-de Sitter (but with different cosmological constants) or by a configuration where g is Schwarzschild- de Sitter and f is the Minkowski metric. These are the only known exact solutions to the field equation of massive gravity covered by the parametrization (10.5). Clearly, asymptotically flat solutions with g = f = η at spatial infinity are not contained in the class I solution. For this reason we concentrate in the following to the class II, where g and f are simultaneously diagonalizable. For the class II there are until now no known asymptotically flat solutions. It has been also argued by means of a numerical analysis that such solutions are not 10.3. Static spherically-symmetric configurations 163 likely to exist [DKP03]. We complemented and confirm this analysis considering also two alternative ansätze which we describe in the following. Since we are looking for asymptotically flat solutions we take the background metric to be the Minkowski metric η.

10.3.2 Spatially isotropic coordinates

The static spherically symmetric ansatz in spatially isotropic coordinates is:

2 U 2 4 2 2 g = dt V (dr + r g 2 ) (10.21a) V − S 2 ′2 2 2 f = η = dt φ dr φ g 2 (10.21b) − − S where U, V and φ are function of the radial coordinate r only. With this ansatz the vacuum massive ﬁeld equation are equivalent to:

∆U = m2F (U, V, φ/r, φ′) (10.22a) ∆V = m2G(U, V, φ/r, φ′) (10.22b)

′ ′ 1 ′ 2 ′ 2U V + r (UV ) = m H(U,V,φ/,φ ) (10.22c)

Here ∆ stands for the radial part of the Laplacian and hence is just a short form for d2/dr + (2/r)d/dr. The functions F, G and H are rational functions, have a zero at (U, V, φ/r, φ′) = (1, 1, 1, 1) (corresponding to the trivial solution g = η), and are smooth in an open neighborhood of this point. The above equations are a convenient rearrangement of the three independent component of the field equation (tt, rr, and θθ): the second and the third equations are the tt- and the rr- components, respectively, whereas the first one is the combination ‘θθ tt+rr’. The − equations (10.22) can be written as a first order ODE system for the five functions (U,U ′,V,V ′, φ).

10.3.3 Curvature-coordinates The static spherically symmetric ansatz in spatially isotropic coordinates is:

2 2 −1 2 2 g = S T dt T dR R g 2 (10.23a) − − S 2 ′2 2 2 f = η = dt φ dR φ g 2 (10.23b) − − S where S, T and φ are function of the radial coordinate R only. With this ansatz the vacuum massive ﬁeld equation are equivalent to:

RS′ = (mR)2F (S,T,φ/R,φ′) (10.24a) RT ′ + T 1 = (mR)2G(S,T,φ/R,φ′) (10.24b) − m2 H(S,T,φ/R)Rφ′′ I(S,T,φ/R,φ′) (mR)2K(S,T,φ/R,φ′) =0 (10.24c) − − The functions2 F,G,H,I,K are rational functions, have a zero at (S,T,φ/R,φ′)= (1, 1, 1, 1) and are smooth in a neighborhood around it. As in the previous case,

2 Although having the same names, the functions F,G, and H in (10.24) and in (10.22) are not the same. 164 Chapter 10. Nonlinear massive theories the field equation has three independent components. The first equation is just the tt-component and the second the sum of the RR- and the tt-components. For the third equation one substitute the θθ-component of the field equation by the divergence-freeness of the mass term (one can show that the new set of equations is equivalent to the original one). Finally, one uses the first two equations to eliminate the derivatives of the functions S and T from the last one. The equations (10.22) can be written as first order ODE system for the four functions (S,T,φ,φ′).

10.3.4 Numerical and analytical investigations We have been seeking for static, spherically symmetric, asymptotically flat solutions of the massive nonlinear theory with mass term (10.5). Thereby we special- ized to the two most popular cases: (β,γ,δ) = (0, 0, 0) [BDZ10, BDZ09, DKP03] and (0, 1, 0) [SS77]. One gets, depending on the specific form of the ansatz (see Section 10.3.2 and 10.3.3), a system of nonlinear, coupled ODEs which have all in common a singular character. This can be identified in the presence of a coefficient of the higher derivative term in the last equation of the corresponding ODEs system (the coefficient of φ′ in (10.22c) and the coefficient H of φ′′ in (10.24c)) which becomes zero as the metric g approaches the Minkowski metric η. We have studied the outward problem starting at some radius much smaller than Rg := 1/m (the ‘Compton wavelength of the graviton’) and the inward problem starting at a radius much larger than Rg. In both cases we took initial conditions on a lattice around the trivial solution g = f = η. The simulations were performed with Mathematica and they all ended up with a singularity at a finite radius. One may even conjecture that the considered class of nonlinear massive gravity does not allow for static, spherically symmetric, asymptotically flat solutions other than the Minkowski spacetime itself. This conjecture seems to be supported by our numerical analysis, whose results agree with those of Damour and collaborators [DKP03]. However, one should note that numerical analysis are much more suitable for giving an hint for the existence of a solution3 rather than giving a proof of its non-existence. Hence, we explored the possibility to establish a non-existence theorem. We did this by applying the following techniques: (i) scaling arguments (like, e.g., in [Heu96], [HS92], [Der64], and [Col75]); (ii) energy arguments (like the positive mass theorem); and (iii) divergence or integral identities. In this respect, none of the applied methods was successful and hence no such non-existence theorem could be established.

3 As classical example think at the Bartnik McKinnon solution [BM88]. Appendices

165

Appendix A

Diﬀerential geometry

For the sake of self-containedness, and as reference for the reader, we give in this appendix a collection of some deﬁnitions and mathematical formulae we used— sometime tacitly—throughout the work. In the following ( , g) denotes a semi- M Riemannian manifold of dimension n. When using indices, we adopt the following convention: for general semi-Riemannian manifolds indices are Latin and run from 1 to n. When specializing to Lorentzian manifolds Greek indices run from 0 to n 1, where the 0-th index is taken to be timelike and the other (spacelike) indices − are written in Latin and run from 1 to n 1. −

A.1 Basic notation

Recall that a semi-Riemannian manifold ( , g) consists of a differentiable manifold M together with a symmetric non-degenerate smooth (0, 2) tensor field g (called M metric) on . A tensor field 1 of rank (r, s) on , or simply an (r, s) tensor field M M on , is a smooth assignment of an (r, s) tensor over T (the tangent space of M pM at p) to each point p in . In turn, an (r, s) tensor over a vector space V is M M an element of the tensor-product space V ⊗r (V ∗)⊗s, where V ∗ is the dual space ⊗ of V . The set of the (r, s) tensor fields on is denoted by r( ) and the direct M Ts M sum of all r( ) is denoted by ( ). As example, a tensor fields of rank (1, 0) Ts M T M is a vector field, one of rank (0, 1) a differential one-form, and one of rank (1, 1) an endomorphism. Recall that a differential k-form on is a smooth assignment to M k each point p of a k-form over T , that is an element of (T ∗ )∧ , where ∈ M pM p M T ∗ := (T )∗ denotes the cotangent space of at p. The set of all differential p M pM M k-form on is denoted by Ω ( ) and the direct sum of all Ω ( ) is denoted M k M k M by Ω( ) (see Appendix A.4). For brevity, we often write simply ‘k-form’ for M ‘differential k-form’. We recall that r( ), Ω ( ), ( ), and Ω( ) have the Ts M k M T M M structure of a module over the ring of the smooth functions C∞( ) (addition and M scalar multiplication are defined pointwise). Moreover, ( ( ), ) and (Ω( ), ) T M ⊗ M ∧ have the structure of a graded algebra (associative but not commutative) over the field R of the real numbers. The metric g, being non-degenerate, provides a one- to-one map between the tangent space and its dual: given a vector X its related

1 In books of diﬀerential geometry, the name ‘section of the tensor bundle’ is used instead of ‘tensor ﬁeld’.

167 168 Appendix A. Differential geometry one-form (linear functional) is defined by X := g(X, ). Correspondingly, given a · one-form α, its related vector α is the unique solution of the equation g(α, )= α. · Of course it holds that and are the identity maps. These operations ◦ ◦ are what one also calls ‘lowering’ and ‘raising’ of indices with the metric g. When performed with the spatial metric (see (2.7)) these operations are denotes with ♭ and ♯, respectively. The metric g extends naturally to the whole tensor bundle in the standard way defining a scalar product which we denote by , . Explicitely, if h· ·i i1···ir j1 js T and S are (r, s) tensor fields, we have T = T Ai Ai α α j1···js 1 ⊗···⊗ r ⊗ ⊗··· i (and similar for S), where Ai is a basis of the tangent space and α the related i {i } { } dual basis (i.e: α (Aj )= δj ). Then

j1l1 jsls i1···ir k1···kr S, T = gi k gi k g g T S . (A.1) h i 1 1 · · · r r · · · j1···js l1···ls This scalar product, being non-degenerate, allows to define a semi-norm by T := k k T , T . A normalized vector field is one for which it holds X = 1. If |h i| k k x1,...,xn is some local coordinate chart, then the induced coordinate basis on p{ } the tangent space is denoted by ∂/∂xi . For brevity, we often write ∂ for ∂/∂xi. { } i Given a connection D on a semi-Riemannian manifold ( , g) one defines M the torsion, metricity defect, and curvature of D by TD(X, Y ) := DX Y − DY X [X, Y ], qD(X, Y , Z) := (DX g)(Y , Z), and RD(X, Y )Z := DX DY Z − − DY DX Z D Z. They are a (1, 2), a (0, 3), and a (1, 3) tensor field, re- − [X,Y ] spectively. The Levi-Civita connection, the unique metric and torsion-free connection associated to a semi-Riemannian manifold ( , g), is denoted by ∇. Its M Riemann tensor, defined by Riem(W , Z, X, Y ) := g(W , R(X, Y )Z), is antisymmetric under the exchange X Y or W Z and symmetric under the ex- ↔ ↔ change of the pairs (W , Z) (X, Y ). Moreover, the antisymmetrization over ↔ any three slots vanishes (first Bianchi identity). The related Ricci tensor is defined as Ric(Y , Z) := tr(X R(X, Y )Z), which is a symmetric (0, 2) tensor field. 7→ Alternatively, this can be expressed as Ric = rc(Riem), where rc is the Ricci- contraction, hence the contraction, with the help of the metric, of the first with the third slot (see text under formula (A.53) in Appendix A.6). The scalar curvature, or Ricci-scalar, is the contraction of the Ricci tensor with the help of the metric and it is denoted by Scal. The Einstein tensor is then Ein := Ric (1/2) Scal g. − The components of the Riemann, Ricci, and Einstein tensors with respect to some basis A of the tangential space are denoted by R := Riem(A , A , A , A ), { i} ijkl i j k l Rij := Ric(Ai, Aj ), and Gij := Ein(Ai, Aj ), respectively. Both contractions described above are between a pair of covariant slots (or, if one prefers, indices) and hence both need the metric g in order to first convert one of the slots (indices) in a contravariant one. In Part III, when dealing with nonlinear generalizations of massive gravitational theories, one has at disposal two metrics (a physical and a background metric), hence, to distinguish which metric is used for raising the in- −1 µν dices we denote by trgA := tr(g A)= g A the contraction, or trace w.r.t. g, · µν of the (0, 2)-tensor A. Hence, for example, Scal = trg Ric. Here, and henceforth, a dot between two tensors whose adjacent slots are of different type (that is covariant and contravariant, respectively) denotes the contractions between these slots. The same notation is extended also to the case where both slots are of the same type: A.2. Tensor fields along a map 169 the contraction is then taken with the help of the metric. The exterior derivative is denoted by d, the codifferential by δ (see Ap- pendix A.4), and the Lie-derivative by L. On vector fields we have LX Y = [X, Y ], where [X, Y ]f := X(Y (f)) Y (X(f)) are the Lie-bracket between X and Y . The − exterior product between a p-form α and a q-form β is defined in formula (A.9). With this definition, for one-forms we have: α β = α β β α. Given a form ∧ ⊗ − ⊗ ω, we denote by iX ω or Xyω the insertion of the vector X in the first slot and with ωxX the insertion in the last slot of ω. Given a smooth function f on gradient is ∇f := df, the Hessian the sym- M metric two-tensor Hess(f)(X, Y ) := g(∇X ∇f, Y ) and the Laplacian its trace

∆f := trg Hess(f). If X is a vector field, its divergence, div X, is defined by the equation LX µg = (div X)µg, where µg denotes the volume form induced by the metric g. The divergence can also be expressed in terms of the Levi- Civita connection: expressing in the above definition the Lie derivative in terms of the covariant one (using the torsion-freeness of ∇) and using (A.8) one gets: div X = tr(X ∇X). For a smooth map φ : between two dif- 7→ M → N ferentiable manifolds and we denote by φ its differential map, as well M N ∗ as the push-forward, and by φ∗ the pull-back. An endomorphism L of T M is called self-adjoint w.r.t. g if g(X, L(Y )) = g(L(X), Y ) and skew-adjoint if g(X, L(Y )) = g(L(X), Y ) for any vector fields X, Y . The self- and skew-adjoint − −1 part of the endomorphism L w.r.t. g are g(L) := (L) = g (g L) and S S · S · −1 g(L) := (L) = g (g L), respectively. Here, and denote the sym- A A · A · S A metric and antisymmetric projectors (A.34) and (A.10), respectively. Notice that

g(L) and g(L) do not depends on the signature choice since they involve the S A metric in a quadratic manner. If A and B are endomorphism, we denote with [A, B] := A B B A the commutator and with A, B := A B + B A the ◦ − ◦ { } ◦ ◦ anti-commutator between A and B.

A.2 Tensor ﬁelds along a map

Some words of explanation are needed when we are considering a curve γ, that is a map γ : R , and we write things like ∇γ γ˙ or ∇γT , for some tensor field T →M ˙ ˙ on . A possible way to give a meaning to this objects involves the concepts of M ‘tensor field along a map’ (see pp. 36–37 in [Poo81]) and ‘induced connection along a map’ (see Sect. 14.3 in [Str04]). Another, fully equivalent way, is to work with the pull-back of the tensor bundle and the pull-back of the connection via the considered map. However, we prefer the former approach since we like to think to our vector and tensor fields as living on (the manifold which models the spacetime). If M φ : is a smooth map from some manifold to , an (r, s) tensor field N →M N M along the map φ is a smooth assignment of an (r, s) tensor over T to each φ(q)M point q . Vector fields along φ of the form v = φ vN , where vN is a vector ∈ N ∗ field on , are called tangential vector fields along φ. An example for such a vector N field is of course the a tangential vector field along a curve γ given by γ˙ := γ∗∂/∂τ, where ∂/∂τ T R. ∈ In order to define the covariant derivative of tensor fields in direction of tangential vector fields along φ one introduces the connection induced by ∇ along φ as the 170 Appendix A. Differential geometry map φ∇ which, to a tangential vector field v along φ and a vector field X along φ φ, associates a vector field along φ denoted by ∇v X. This map is required to be C∞( )-linear in the first argument and, in the second argument, to be R-linear N φ φ and to satisfy Leibniz rule, in the sense that ∇v(fX) = f ∇v X + (vN (f))X ∞ for v = φ vN and f C ( ). Moreover, on vector fields along φ of the form ∗ ∈ N φ X = XM φ where XM is a vector field on , one requires that ∇ reduces to ◦ φ M ∇ in the sense that ∇v(XM φ) = (∇v XM) φ, where ((∇v XM) φ)(q) = ◦ ◦ ◦ ∇ XM T . Existence and uniqueness of such a map is shown locally as v(q) ∈ φ(q)M follows. Let an open set in such that φ( ) is fully contained in U ⊂ N N U ⊂ M the domain of a chart of with coordinates x1,...,xn . A vector field X along M { } φ can then be expressed as X = Xi∂ φ, for some functions Xi in C∞( ). The i ◦ U above requirements imply then, for a tangential vector field v = γ∗vN :

φ i i ∇v X = vN (X )(∂ φ)+ X (∇v∂ ) φ . (A.2) i ◦ i ◦ Clearly, the r.h.s. is uniquely determined by ∇, which shows the uniqueness of φ∇. The existence is shown defining the map φ∇ by the local expression (A.2) and then proving that all the above requirements are indeed satisfied. One then φ ∞ φ defines ∇v f := vN (f) on functions in C ( ) and extends derivatively ∇ on N tensor fields of arbitrary rank along φ in the standard way, i.e. requiring Leibniz rule w.r.t. tensor product and commutativity with contractions. Moreover, one also shows that φ∇ is metric and torsion-free (see Sect. 14.3 in [Str04]). For brevity, in the text we drop the superscript φ and write simply ∇ for φ∇.

A.3 Derivations

Definition A.1 (Derivation). A derivation of the tensor algebra ( ( ), ), called T M ⊗ also derivation of tensor fields, is a mapping D : ( ) ( ) which is: (i) R- T M → T M linear, that is D(c T + c T ) = c DT + c DT for any T , T ( ) and 1 1 2 2 1 1 2 2 1 2 ∈ T M c ,c R, (ii) rank-preserving, that is D( r( )) r( ), and (iii) it satisfies 1 2 ∈ Ts M ⊂ Ts M the Leibniz rule D(S T )= DS T + S DT . ⊗ ⊗ ⊗ Notice that the familiar Leibniz rule on vector fields, D(fX) = fDX + D(f)X, follows from (iii) with the identification fX = f X. ⊗ Lemma A.2 (Uniqueness of derivations). Let D be a derivation of ( ( ), ) T M ⊗ which commutes with contractions. Then, if D vanishes on functions and on vector fields (or, alternatively on functions and on one-forms), it follows that D vanishes on the whole ( ). T M Proof. The proof is straightforward. First, notice that if D vanishes on functions and on vector fields, then it must vanish also on one-forms; and if it vanishes on functions and on one-forms, then it must vanish also on vector fields. To see this, just use Leibniz rule and the commutativity with contractions to get the formula D(α(X)) = (Dα)(X)+ α(DX), which is valid for any vector field X and one- form α. This immediately implies the above assertion. The proof is concluded by induction using Leibniz rule and commutativity with contractions. Alternatively, A.3. Derivations 171 one can also use the fact that a general tensor field can be written as a linear combination of tensor product between vector fields and one-forms, where the coefficient of the linear combination are functions. Applying D to such an expression, using R-linearity and Leibniz rule, the statement of the lemma follows from the vanishing of D on functions, vector fields, and one-forms.

This lemma, together with the fact that the difference of two derivation is again a derivation (see the next lemma), implies that a derivation of ( ( ), ) which T M ⊗ commutes with contractions is uniquely determined by its values on functions and on vector fields (or, alternatively, on functions and on one-forms). If one has an R-linear operator D˜ defined on functions and on vector fields (or, alternatively, on functions and one-forms) which maps functions onto functions and vector fields onto vector fields and satisfies the Leibniz rule on functions and on vector fields, hence D˜ (fg) = D˜ (f)g + fD˜ (g) and D˜ (fX) = f D˜ X + D˜ (f)X, then one can extend D˜ to a derivation D on tensor fields of arbitrary rank by imposing the Leibniz rule w.r.t. the tensor product and imposing that it commutes with contractions. The above lemma then ensures the uniqueness of this extension. We call this extension procedure for derivatives simply derivative extension. It can be easily checked that the following lemma holds:

Lemma A.3 (Linear combination of derivations). Let D1, D2 be derivations of ( ( ), ) and f ,f some smooth functions. Then the linear combination D = T M ⊗ 1 2 f D + f D is also a derivation of ( ( ), ). Moreover, it holds: if D , D 1 1 2 2 T M ⊗ 1 2 commute with contractions, then also D does; if D1, D2 are metric (i.e. D1g =0=

D2g), then also D is metric. In the ﬁrst part of the present work we deal with several derivations and need to relate these to each other. For this, the next lemma is very useful:

Lemma A.4 (Diﬀerence of two derivation). Let D1, D2 be two derivations which commute with contractions and which are equal on functions. Then, their diﬀerence E := D D is an endomorphism of the tensor bundle. 1 − 2 To show that E is an endomorphism simply note that R-linearity follows directly from the R-linearity of D1 and D2 and function-linearity follows from the Leibniz rule for D1 and D2 and from their equality on functions. Now, summarizing, if one has two derivations D1, D2 which commute with contractions, are equal on functions, and

D D = E (A.3) 1 − 2 on vector fields, then Lemma A.3 and Lemma A.4 imply that E is an endomorphism and that the generalization of the above equation on arbitrary tensor field is simply given by D D = Eˇ , (A.4) 1 − 2 where Eˇ denotes the derivative extension of the endomorphism E, defined by Eˇf = 0 on functions2, EXˇ = EX on vector fields, and then derivatively extended to tensor fields of arbitrary rank (i.e. imposing the Leibniz rule w.r.t. the tensor product and commutativity with contractions). Notice that this extension

2 A derivation which vanishes on functions is also called algebraic. 172 Appendix A. Differential geometry has to be distinguished from the projective extension used for projectors (defined in (2.4) and denoted by a hat). As example, and for reference, take a one-form α, an endomorphism L, a (0, 2)-tensor field t, and a form µ of top degree (hence an n-form if the manifold is n-dimensional). Then we have, respectively:

D α D α = Eˇ(α)= α E , (A.5) 1 − 2 − ◦ D L D L = Eˇ(L)= E L L E = [E, L] , (A.6) 1 − 2 ◦ − ◦ D t D t = Eˇ(t) = t(E , ) t( , E ) , (A.7) 1 − 2 − · · − · · D µ D µ = Eˇ(µ) = tr(E)µ . (A.8) 1 − 2 − Notice that in the ‘index language’ the above formulae mean that contravariant indices get a matrix E and covariant ones a E transposed. In Section 2.2 we relate − covariant, Fermi, Lie, and comoving derivative to each other by computing their respective differences. Since all these derivatives are equal on functions Lemma A.4 applies. It suffices, therefore, to compute their relation on vector fields: one gets the respective generalized relations on an arbitrary tensor field simply by putting a check over the endomorphism on the r.h.s. of those equations.

A.4 Antisymmetric multilinear forms

Let α and β denote a p-and a q-multilinear antisymmetric form, respectively. Their exterior (or wedge) product is deﬁned by

(p + q)! α β := (α β) , (A.9) ∧ p! q! A ⊗ where is the antisymmetrization projector. For a (0,p)-tensor t this is given by A 1 ( t)(v1, ..., vp) := sgn(π) t(vπ(1), ..., vπ(p)) . (A.10) A p! S πX∈ p

Here Sp denotes the permutation group of p objects and sgn(π) the signature of the permutation π. is a function-linear mapping from (0,p)-tensors onto p-forms, it A satisﬁes the projection property = , and on forms it reduces to the identity A ◦ A A map. The combinatorial factor in (A.9) is just a convention in order to prevent an overall factor in the formula for the exterior derivative (see the comment at p. 169 of [BG68]). With this convention, for one-forms on has: α β = α β β α. ∧ ⊗ − ⊗ The wedge product is function-bilinear, associative and it satisﬁes

α β = ( 1)pqβ α . (A.11) ∧ − ∧ Denotes with Ω( ) = n Ω ( ) the graded R-algebra of the exterior dif- M p=0 p M ferential forms (called from now on simply ‘forms’) on and let in the following L M α and β be a p- and a q-form, respectively, and f a differentiable function on . M Definition A.5 (Graded derivation). A derivation of degree k Z of (Ω( ), ) ∈ M ∧ is a mapping D : Ω( ) Ω( ) which: (i) is R-linear, (ii) it shifts the rank M → M by the amount k, that is D(Ω ( )) Ω ( ) for p = 0, 1, ..., n, and (iii) it p M ⊂ p+k M A.4. Antisymmetric multilinear forms 173 satisfies the Leibniz rule D(α β) = Dα β + α Dβ. An antiderivation of ∧ ∧ ∧ degree k Z is a mapping with the same properties as above, except for the last ∈ one, where the Leibniz rule is substituted by the ‘anti-Leibniz’ rule: D(α β) = ∧ Dα β + ( 1)pα Dβ. ∧ − ∧ The analogue of Lemma A.2 for graded derivations and antiderivations is

Lemma A.6 (Uniqueness of graded derivations and antiderivations). Let D be a derivation or an antiderivation of (Ω( ), ). Then, if D vanishes on functions M ∧ and on differentials, it follows that D vanishes on the whole Ω( ). M The proof is a consequence of the fact that, locally, one can write a general form as a linear combination (where the coefficients are functions) of wedge products of differentials. One has then to show that D ‘commutes’ with the processes of localization and globalization (see e.g. [Str04], pages 30–32). Lemma A.6 implies that a graded derivation of (Ω( ), ) is uniquely determined by its values on M ∧ functions and on differentials. Moreover, similarly to the previous section, the above lemma ensures the uniqueness of the extension of an R-linear operator D˜ defined on functions and differentials and which satisfies the Leibniz rule on functions and on differentials, hence D˜ (fg)= D˜ (f)g + fD˜ (g) and D˜ (fdg) = (D˜ f) dg + fD˜ dg, ∧ to a derivation (or an antiderivation) D of (Ω( ), ). (Here dg is the differential M ∧ of the function g, hence the one-form defined by dg(X) := X(g).) As an application of this, the exterior derivative is defined as the unique operator d on forms which satisfies: (i) d is an antiderivation of degree 1 of (Ω( ), ), M ∧ (ii) d d = 0, and (iii) on functions it holds df(X) = X(f). Notice that the ◦ antiderivation property implies

d(α β)= dα β + ( 1)pα dβ . (A.12) ∧ ∧ − ∧ The above equations (A.9), (A.12) extend also to functions, which can be seen as forms of degree zero.

The interior product iX α of a vector X with a form α is the insertion of X in the ﬁrst slot of α. In formulae: iX α(...) = α(X, ...). This implies that iX is an antiderivation of degree 1 of (Ω( ), ). Hence, in particular, one has: − M ∧ p iX (α β)= iX α β + ( 1) α iX β . (A.13) ∧ ∧ − ∧

Moreover, it holds iX iY = iY iX , iX iX = 0, and i X = fiX = iX f on ◦ − ◦ ◦ f forms. Recall that the Lie-derivative LX w.r.t. a vector ﬁeld X is a derivation of ( ( ), ). The R-linearity of LX then implies that LX is a derivation of degree 0 T M ⊗ of (Ω( ), ). The following formula, valid on forms, express the Lie derivative in M ∧ terms of the inner product and the exterior derivative:

LX = d iX + iX d . (A.14) ◦ ◦ This is referred to as Cartan’s formula and it can be easily proven by showing that both sides are derivations of degree 0 of (Ω( ), ) and by checking the above M ∧ identity on functions and diﬀerentials. Lemma A.6 then ensures that the identity 174 Appendix A. Diﬀerential geometry holds on Ω( ). Cartan’s formula is often very useful. For example, it implies M immediately:

L X = fLX + df iX (A.15) f ∧ LX d = d LX (A.16) ◦ ◦ LX iX = iX LX . (A.17) ◦ ◦ A.4.1 The Hodge star

Denote with µg the volume-form induced by g. Then the Hodge star ⋆ is deﬁned as the function-linear mapping Ω ( ) Ω ( ) satisfying p M → n−p M

α ⋆β = β ⋆α = α, β µg (A.18) ∧ ∧ h i for all α, β Ω ( ). Here , denotes the scalar products induced by g on the ∈ p M h· ·i spaces Ω ( ). Explicitely, if α and β are p-forms and A i = 1 ...n is some p M { i | } basis of the tangent space, we deﬁne

1 α, β := gi1j1 . . . gipjp α β , (A.19) h i p! · · i1...ip j1...jp

ij where αi1...ip := α(Ai1 ,..., Aip ), and similar for β, g is the inverse matrix of gij := g(Ai, Aj ), and all the indices run from 1 to n. The factor 1/p! in (A.19) ensures that if θi i = 1 ...n is an orthonormal basis of Ω ( ) (i.e. θi, θj = { | } 1 M h i δij ) then θi1 . . . θip , θi1 . . . θip = 1 for 1 i < i <...

⋆ 1= µg (A.21a) s ⋆ µg = ( 1) (A.21b) − ⋆α,⋆β = ( 1)s α, β (A.21c) h i − h i ⋆⋆α = ( 1)p(n−p)+sα (A.21d) − α γ, µg = ⋆α, γ (A.21e) h ∧ i h i iX ⋆ α = ⋆(α X) (A.21f) ∧ iX µg = ⋆X . (A.21g)

The codiﬀerential is now the mapping δ :Ω ( ) Ω ( ) deﬁned by p M → p−1 M δ := ( 1)n(p+1)+s ⋆ d ⋆ . (A.22) − A.4. Antisymmetric multilinear forms 175

Since d d = 0, it holds δ δ = 0. (Note that with this definition δ is minus ◦ ◦ the formal adjoint of d w.r.t. the scalar product ( , ) := , µg.) The Laplace– · · Mh· ·i de Rham operator is then the mapping ∆ : Ω ( ) Ω ( ) defined as p M → pRM ∆= δ d + d δ , (A.23) ◦ ◦ which, on functions, equals the Laplacian. Sometimes, the following relation is useful to express the divergence in terms of the codifferential; one has div X = δX (A.24) for any vector field X. This follows immediately taking the Hodge dual of the definition of divergence in terms of the Lie derivative of the volume form µg.

A.4.2 Hodge star for two-dimensional Lorentzian manifolds Two-dimensional Lorentzian manifolds play a key role for spherically symmetric spacetimes, as we will see in Appendix A.10. Hence we specialize here to n = 2 and s = 1. In this case the above equations (A.21d) and (A.22) simplify, respectively, to ⋆⋆ = ( 1)p+1 (A.25) − δ = ⋆ d ⋆ . (A.26) − In particular, on one forms, the Hodge star is an involutive skew-adjoint endomorphism, that is: ⋆⋆α = α (A.27) α,⋆β = ⋆α, β (A.28) h i −h i for any one-forms α, β. These properties follow directly from (A.21d) and the deﬁnition (A.18), respectively. Let e be a positive oriented orthonormal basis of the tangent space { µ} 0 1 (i.e. g(eν , eν ) = ηµν , where ηµν := diag(1, 1), and µg(e0, e1) = 1) and θ , θ µ − µ { } the related dual basis of one-forms (i.e. θ (eν )= δν ). (In passing we note that the metric duals of the e ’s are, respectively, e = θ0 and e = θ1.) Then one has µ 0 1 − θµ, θν = ηµν (ηµν = diag(1, 1)) and h i − 0 1 µg = θ θ , (A.29) ∧ from which it follows, with deﬁnition (A.18):

⋆ θ0 = θ1 and ⋆ θ1 = θ0 . (A.30)

Using this, one can easily show3 that the double Hodge dual of a symmetric (0, 2) tensor τ is given by: ⋆ τ ⋆ = τ tr(τ ) g . (A.31) − Here, the ﬁrst star acts on the ﬁrst slot (index) of τ and the second on the second one. We notice that for the trace (w.r.t. the metric) of the double Hodge dual of τ it holds: tr(⋆τ ⋆)= tr(τ ) , (A.32) − which follows immediately from (A.31).

3 For this one conveniently decomposes τ in its trace- and trace-free part. 176 Appendix A. Diﬀerential geometry

A.5 Symmetric multilinear forms

Analogously to the previous section, let α and β denote a p- and a q-multilinear symmetric form, respectively. Their symmetric product is deﬁned by

(p + q)! α β := (α β) , (A.33) ∨ p! q! S ⊗ where is the symmetrization projector. For a (0,p)-tensor t this is given by S 1 ( t)(v1, ..., vp) := t(vπ(1), ..., vπ(p)) . (A.34) S p! S πX∈ p Again, the combinatorial factor is just a matter of convection: it is chosen in order to prevent a numerical factor in front of the second term on the r.h.s. of (A.36). With this convention, for p = q = 1, one has α β = α β +β α. The symmetric ∨ ⊗ ⊗ product is function-bilinear, associative, and commutative:

α β = β α . (A.35) ∨ ∨ As in the previous section we denote with ( )= ( ) the graded R- V M p≥0 Vp M algebra of the symmetric multilinear forms on . In the following, let α ( ) M L ∈Vp M and β ( ). Then LX is a derivation of degree 0 of ( ( ), ). The formula ∈Vq M V M ∨

L X = fLX + df iX (A.36) f ∨ provides the analogue to (A.15) for the symmetric case. As before, iX denotes the insertion of X in the first slot. In fact here, because of the symmetry, it does not make any difference in which slot X is inserted in. The above formula can now be proved by direct computation just using the basic properties of the Lie derivative and the definition of the symmetric product. Applied to the metric g, (A.36) immediately implies:

L X g = fLX g + df X . (A.37) f ∨ A.6 Connections and curvature

Let D be a connection on a semi-Riemannian manifold ( , g). It is easily seen M that D is uniquely determined by its torsion TD and metricity defect QD. Deﬁning

d(X, Y , Z) := Xg(Y , Z) , (A.38) l(X, Y , Z) := g(X, [Y , Z]) , (A.39)

tD(X, Y , Z) := g(X, TD(Y , Z)) , (A.40)

qD(X, Y , Z) := (DX g)(Y , Z) , (A.41) building the combination d(X, Y , Z) := d(X, Y , Z)+ d(Y , X, Z) d(Z, X, Y ), − and using the deﬁnitions of torsion one gets

2 g(Z, DX Y )= (d l tD qD)(X, Y , Z) . (A.42) − − − A.6. Connections and curvature 177

This is nothing but the Koszul formula generalized to the situation of arbitrary metricity-defect and torsion. Following the philosophy ‘first things first’ as discussed in Chapter 1, we take in this work the connection to be the Levi-Civita connection, denoted by ∇, that is the unique metric and torsion-free connection of the considered semi-Riemannian manifold. Given a vector field Z, since ∇ is function-linear in the first argument, ∇Z is an endomorphism of the tangential space. Often it is useful to see an endomorphism, say L, as a bilinear functional and hence we define

L(X, Y ) := g(L(X), Y ) . (A.43)

Accordingly, for the endomorphism ∇Z we have:

∇Z(X, Y )= g(∇X Z, Y )= ∇Z(X, Y ) , (A.44) where the last equality follows from the metricity of ∇.4 Hence, to denote the bilinear form associated to the endomorphism ∇Z, we can use both ∇Z and ∇Z. For aesthetical reasons we choose the last form. The symmetric and antisymmetric part of ∇Z are given by the (half of the) Lie derivative of the metric along Z and by the (half of the) exterior diﬀerential of Z, respectively. In formulae:

1 (∇Z)= LZ g , (A.45) S 2 (∇Z)= 1 dZ . (A.46) A 2 Here and are the symmetric and the antisymmetric projectors (A.34) S A and (A.10), respectively. The proof of these formulae is a straightforward computation, which we report here for completeness. For the ﬁrst formula one has: LZ g(X, Y ) = Z(g(X, Y )) g(LZ X, Y ) g(X, LZ Y ) = g(∇Z X − − − LZ X, Y ) + (X Y ) = g(∇X Z, Y ) + (X Y ) = ∇Z(X, Y ) + (X Y ) = ↔ ↔ ↔ 2 (∇Z)(X, Y ), where we used the metricity and torsion-freeness of the Levi- S Civita connection. In order to show (A.46) one uses, besides these properties of the

Levi-Civita connection, the Cartan’s formula: dZ(X, Y )= iY iX dZ = iY (LX Z − d iX Z) = LX Z(Y ) Y (Z(X)) = X(g(Z, Y )) g(Z, LX Y ) Y (g(Z, X)) = − − − g(Z, ∇X Y ∇Y X LX Y )+ g(∇X Z, Y ) g(∇Y Z, X)=2 (∇Z)(X, Y ). − − − A A.6.1 Decomposition of the curvature tensor

In dealing with curvature tensors it is convenient to introduce a mapping which, starting from two symmetric (0, 2)-tensors, builds up a (0, 4)-tensor with the same algebraic symmetry properties of the (covariant) Riemann tensor. This is provided by the Kulkarni–Nomizu product, deﬁned by5

1 (a b)(X, Y , W , Z) := a(X, W )b(Y , Z) a(X, Z)b(Y , W ) ⊙ 2 − (A.47) +b(X, W )a(Y , Z) b(X, Z)a(Y , W ) , − 4 Recall also that, by definition, (∇α)(X, Y ):=(∇X α)(Y ) for any one-form α. 5 Sometime in the literature (e.g. [Bes87]) the overall factor 1/2 in the definition of the Kulkarni– Nomizu product is suppressed. One reason for our choice is, for example, that no numerical factors appear in the formulae (A.57) and (A.56). 178 Appendix A. Differential geometry where a and b are symmetric (0, 2)-tensors. Notice that this product is a bilinear operation (like the tensor product) and, moreover, is commutative: a b = b ⊙ ⊙ a. This can be used to conveniently write down the orthogonal decomposition (w.r.t. the scalar product (A.1)) of the Riemann tensor:

Riem = Ricci + Weyl , (A.48) were Weyl, the Weyl tensor, is defined to be the traceless part and Ricci, the Ricci part of the Riemann tensor, the trace part. The latter can be further decomposed in orthogonal terms. For n 4 one gets: ≥ Scal 2 1 Ricci = g g + Ric Scal g g . (A.49) n(n 1) ⊙ n 2 −n ⊙ − − Inserting this into (A.48) gives the explicit expression for Weyl. We notice that in the important case n = 4, we have: Ricci = Ric 1 Scal g g (A.50a) − 6 ⊙ 1 = Ein trg(Ein) g g . (A.50b) − 3 ⊙ In dimensions n 3 the Weyl tensor vanishes identically, due to its symmetry ≤ properties and its tracelessness, hence the decomposition of the Riemann tensor reads Scal 1 Riem = Ricci = g g +2 Ric Scal g g (A.51) 6 ⊙ −3 ⊙ for n = 3, and Scal Riem = Ricci = g g (A.52) 2 ⊙ for n = 2. A quick way to derive these decompositions is to notice that the trace part of the Riemann tensor can be written as linear combination of Scal g g and Ric g ⊙ ⊙ or, equivalently, as a linear combination of Scal g g and (Ric (1/n) Scal g) g ⊙ − ⊙ (the latter two being orthogonal). The coefficient of the last linear combination are then easily computed taking the successive contractions rc and tr, thereby using the relations: 1 n 2 rc(a g)= tr(a)g + − a (A.53a) ⊙ 2 2 tr(rc(a g)) = (n 1)tr(a) (A.53b) ⊙ − which hold for any symmetric (0, 2)-tensor a. Here, rc denotes the Ricci-contraction, hence the contraction of the first and the third slot of a (0, 4)-tensor with the help of the metric, and tr denotes the trace w.r.t. g of a (0, 2)-tensor, that is the full contraction of the latter with the help of the metric. Hence one has: rc(Riem)= Ric and tr(Ric) = Scal. Explicitly, if s and a are a (0, 4)- and a (0, 2)-tensor, respectively, A some basis of the tangent space, g := g(A , A ), and gij its { i} ij i j ij ij inverse we have: rc(s)(X, Y )= g s(Ai, X, Aj , Y ) and tr(a)= g a(Ai, Aj ). For later reference we note that for any symmetric (0, 2)-tensors a and b one may easily check that: 1 rc(a b)= tr(a) b + tr(b) a (a b + b a) , (A.54a) ⊙ 2 − · · tr(rc(a b)) = tr(a)tr(b) a, b . (A.54b) ⊙ − h i A.6. Connections and curvature 179

This, in the special case b = g, imply (A.53).

A.6.2 Sectional curvature

The sectional curvature of a non-degenerate plane Π in T (that is a two- pM dimensional subspace of T for which the restriction g is non-degenerate) is pM |Π deﬁned by Riem(X, Y , X, Y ) sec(Π ) := , (A.55) Q(X, Y ) where X, Y are any two linear independent vectors in Π , and

Q(X, Y ) := g(X, X)g(Y , Y ) g(X, Y )2 = (g g)(X, Y , X, Y ) . (A.56) − ⊙ Notice that Q(X, Y ) gives the square of the area of the parallelogram spanned by | | X and Y which is non-zero iﬀ the considered plane is non-degenerate. At a point p the sectional curvature is said to be isotropic if it is indepen- ∈M dent from the plane on which it is evaluated, that is: sec(Π )p = k(p) for all planes Π T . One then refers to k(p) as to the sectional curvature at p. In this case, ⊂ pM because of (A.55) and (A.56), one has:

Riem = k g g , (A.57) ⊙ and thus, by means of (A.53a):

Ric = (n 1)k g , (A.58) − where k is the sectional curvature at p. In cosmology one needs the following

Lemma A.7 (Isotropy implies homogeneity). Let n = dim( ) 3. Then, if the M ≥ sectional curvature is isotropic in some open set of it is also constant on . U M U Proof. The contraction of (A.58) gives Scal = n(n 1)k, and thus Einstein’s tensor − is Ein = Ric (1/2) Scal g = (1/2)(n 1)(2 n)kg. From the divergence-freeness − − − of the latter, it follows immediately (provided n 3) that dk =0 on and hence ≥ U k = const on . U

Notice that in two dimensions the sectional curvature is already isotropic because the only plane that one has is the tangential space itself. Hence, imposing isotropy one gets no new constraint. In fact, as it is well-known from Gauss’ theorema egregium, in two dimensions the whole information about curvature is contained in a single scalar function: the Gaussian curvature or, which is the same, the sectional curvature. For the Ricci-tensor and -scalar in two dimension one has:

Ric = k g (A.59a) Scal = 2k , (A.59b) where k is the sectional (or the Gaussian) curvature. From (A.59) it follows immediately that, in two dimensions, the Einstein tensor vanishes identically. 180 Appendix A. Diﬀerential geometry

A.6.3 Curvature of two-dimensional Lorentzian manifolds It is a nice exercise to compute the curvature of a two-dimensional Lorentzian manifold with the use of Cartan’s structure equations (see e.g. [Str04]). These formulas are useful when dealing with spherically symmetric (four-dimensional) spacetimes: how we will see in Appendix A.10, the latter can be expressed as ‘warped’ product (to be deﬁned below) of a two-dimensional Lorentzian manifold and a two-sphere. Writing the metric w.r.t. local coordinates (t, r) as

g = A2(t, r)dt2 B2(t, r)dr2 = θ0 θ0 θ1 θ1 , (A.60) − ⊗ − ⊗ where θ0 = A(t, r)dt and θ1 = B(t, r)dr, we get for the Riemann, Ricci, and scalar curvature Riem = fg g , (A.61a) ⊙ Ric = fg , (A.61b) Scal = 2f , (A.61c) respectively, where f is given by

1 B′ A˙ f = A A′′ A′ B B¨ B˙ . (A.62) (AB)2 − B − − A !! Note that, once one computed (A.61a), the other two formulas follows immediately from (A.53).

A.7 Submanifolds

In a Lorentzian manifold ( , g) endowed with Levi-Civita connection ∇ consider M a non-null smooth submanifold Γ of codimension one and normal vector ﬁeld n. Hence Γ is either spacelike (and n is timelike) or timelike (and n spacelike). The submanifold Γ inherit from the ambient manifold ( , g) a metric and a connection M in a natural way. We introduce the orthogonal projectors

Qn := ε(n) n n (A.63a) ⊗ Pn := id Qn , (A.63b) − which are nothing but the projectors (2.1) for the general case where the vector n can be timelike, as well as spacelike. The indicator ε(n), defined for any non-null vector by g(X, X) +1 if X is timelike, ε(X) := = (A.64) g(X, X) 1 if X is spacelike, | | (− ensures that the projections properties (2.2) are satisfied. The induced metric on Γ (also called first fundamental form) is given by

g := ε(n)Png , (A.65) Γ − where the sign is chosen in order to get a positive definite metric in the case where Γ is spacelike. Given two vector field X, Y tangent to Γ , hence vector field on M A.7. Submanifolds 181

for which it holds QnX 0 and PnX X (and the same for Y ), one decomposes ≡ ≡ the covariant derivative of Y with respect to X in the orthogonal components

∇X Y = Pn(∇X Y )+ Qn(∇X Y )

Γ = ∇X Y + KΓ (X, Y )n (A.66) where Γ ∇X Y := Pn(∇X Y ) (A.67) is the induced connection on Γ and

K (X, Y ) := ε(n) g(∇X Y , n)= ε(n) g(∇X n, Y ) (A.68) Γ − is the extrinsic curvature of Γ in (also called second fundamental form). The M equal sign in (A.68) is an immediate consequence of the metricity of ∇ and the fact that X and Y are orthogonal to n. The torsion-freeness of ∇, together with the fact that [X, Y ] is tangent to Γ , implies that the extrinsic curvature KΓ is a symmetric (0, 2) tensor ﬁeld on Γ . Hence we can write:

K = ε(n) (Pn∇n) . (A.69) Γ − S We recall also that the induced connection Γ ∇ is in fact the Levi-Civita connection of (Γ, gΓ ), as one may easily check. The relation between the curvature (intrinsic and extrinsic) of Γ and that of M are now given by the Gauss and the Codazzi–Mainardi equations, respectively:

Riem(X, Y , W , Z)= ε(n) Γ Riem +K K (X, Y , W , Z) , (A.70a) − Γ ⊙ Γ Γ Γ Riem(n, Z, X, Y )= ε(n) ∇X K (Y , Z) ∇Y K (X, Z) , (A.70b) Γ − Γ h i valid for any vector ﬁelds X, Y , W , Z tangent to Γ . These equations follow straightforwardly inserting the decomposition formula (A.66) in the deﬁnition of the Riemann curvature. The equations (A.70) now imply the following relations for the Einstein tensor:

Γ 2 2 Ein(n, n)= Scal +(trg K ) K , K g , (A.71a) Γ Γ − h Γ Γ i Γ Ein(n, X)= div K trg (K ) g (X) . (A.71b) Γ Γ − Γ Γ Γ These formulas are conveniently derived making use of an adapted orthonormal basis (taking, e.g., ei tangential to Γ and e0 = n). Moreover, in the derivation of (A.71a) one can take advantage of (A.54b). Note that the normal-tangential-normal-tangential part of the Riemann curvature Riem(n, X, n, Y ) cannot be related to the (intrinsic or extrinsic) properties of Γ , since this would involve the variation in normal direction of quantities deﬁned only on Γ . This is the reason why in (A.71) there is no relation for the tangential part Ein(X, Y ) of the Einstein tensor.6

6 If one considers, instead of a single hypersurface Γ , a one-parameter family Γt of hypersurfaces which foliates M (at least locally) one can, in this case, express the normal-tangential-normal- tangential part of the Riemann curvature, as well as the tangential part of the Einstein tensor, in terms involving, among others, the normal variation LnKΓ of the extrinsic curvature and the ‘acceleration’ ∇nn of n (see, e.g., [Giu98]). 182 Appendix A. Diﬀerential geometry

A.8 Conformal transformations

Let ( , g) be an n-dimensional semi-Riemannian manifold and φ a smooth positive M function on . One can deﬁne a new metric on , called conformal related to the M M old one, putting g := e2φg . (A.72) We give here the relations between the geometric objects constructed from the ‘new’ metric g (overlined) and the ‘old’ metric g (not overlined). The relation (A.74) between the Levi-Civita connections can be easily read oﬀ the Koszul-formula:

2 g(Z, ∇X Y )= Xg(Y , Z)+ Y g(X, Z) Zg(X, Y ) − (A.73) g(X, [Y , Z]) g(Y , [X, Z]) + g(Z, [X, Y ]) , − − which follows setting metricity defect and torsion to zero in (A.42), and uniquely determines the connection. The other relations listed below follow then more or less straightforwardly from (A.74) and (A.72). (i) Levi-Civita connection

∇X Y = ∇X Y + dφ(X)Y + dφ(Y )X g(X, Y )∇φ (A.74) − (ii) Gradient ∇f = e−2φ ∇f (A.75)

(iii) Hessian Hess(f)= Hess(f) dφ df + g(∇φ, ∇f) g (A.76) − ∨ (iv) Laplacian ∆f = e−2φ ∆ f + (n 2) g(∇φ, ∇f) (A.77) − (v) Riemann tensor (as (0, 4)-tensor)

Riem = e2φ Riem 2 Hess(φ) g − ⊙ (A.78) + 2(dφ dφ) g g(∇φ, ∇φ) g g ⊗ ⊙ − ⊙ (vi) Weyl tensor (as (0, 4)-tensor)

Weyl = e2φ Weyl (A.79)

(vii) Ricci tensor

Ric = Ric (∆ φ) g + (n 2) dφ dφ Hess(φ) g(∇φ, ∇φ) g (A.80) − − ⊗ − − (viii) Ricci scalar

Scal = e−2φ Scal 2(n 1) ∆ φ (n 1)(n 2)g(∇φ, ∇φ) (A.81) − − − − − Once computed (A.78), the expressions for the Ricci-tensor and Ricci-scalar follow easily with (A.53), taking care that rc = e−2φrc and tr = e−2φtr. A.9. Warped products 183

A.9 Warped products

In GR, when dealing with spacetimes allowing some symmetry, the concept of warped product introduced in [BO69] reveals to be very useful. In this section we collect the relations which allow to express the geometry of a warped product manifold in terms of the geometry of its constituents. For reference, see [O’N83]7 and [Str04]. Some examples of warped product manifolds in GR are: spherically symmetric spacetimes (reported at the end of this section), static spacetimes, and FLRW spacetimes.

Let ( , gB) and ( , gF ) be two semi-Riemannian manifolds (with arbitrary sig- B F nature) and f a positive functions on . Then the warped product = B M B×f F between these two manifolds, by means of the warping function f, is the product manifold = (A.82) M B×F endowed with the metric

∗ 2 ∗ g = π (gB ) + (f πB) π (gF ) . (A.83) B ◦ F ∗ ∗ Here πB , πF are the projections onto and , respectively, and π , π their pull- B F B F backs. Putting nB = dim( ) and nF = dim( ) then clearly n := dim( )= nB+nF . B F M One can think of as the ‘base’ and as the ‘fiber’ of the product manifold . We B F −1 M recall that for a point p = (pB,pF ) = the ‘fibers’ pB = πB (pB) and −1 ∈M B×F ×F the ‘leaves’ pF = π (pF ) are semi-Riemannian submanifolds of . Moreover, B× F M the tangential space of at p = (pB,pF ) decomposes in the direct sum of two M subspaces: the one tangential to the leave pF and the other tangential to B × the fiber pB . This means that a vector X on the tangential space of at ×F M p = (pB ,pF ) can be uniquely decomposed as the sum

X = XkB + XkF (A.84)

of a vector X B tangent to the leave pF and a vector X F tangent to the fiber k B× k pB . ×F Arbitrary tensor fields on and on can be lifted to tensor fields on in the B F M standard way. For covariant tensor fields (and hence, in particular, for functions) this is achieved via the pull-pack of the respective projection: as an example, just look at (A.83). For contravariant tensor fields it suffices to consider the special case of vector fields. Let, for instance, XB be a vector in the tangential space of

at pB. Then the lift X at (pB,pF ) of XB is defined as the unique vector in the B tangential space of at (pB,pF ) with dπB(X) = XB and dπF (X) = 0 (that is, M with XkF = 0). Since this assignment is smooth, one gets the lifting of a vector field via the pointwise lifting just described. Henceforth, we will mainly omit lifts and projections and not explicitly distinguish between original and lifted quantities. For example, when referring to a vector ‘tangent to ’ we refer to a vector in the B tangential space of or to the lift thereof in the tangential space of . B M We give in the following the relations which express the curvature of in terms M of the warping function f and the curvature of and . The starting point are the B F relations between the Levi-Civita connection ∇ of ( , g) with those of the base M 7 Take care that the Riemann curvature tensor in [O’N83] is minus ours: see Lemma 35 at p. 74 there. 184 Appendix A. Differential geometry and the fiber, denoted by B∇ and F ∇, respectively. These relations can be derived, for example, by means of the Koszul formula (see e.g. [O’N83] Proposition 7.35). Let in the following X, Y , Z be vector fields tangent to and U, V , W tangent to B . Suppressing lifts and projections one has: F B ∇X Y = ∇X Y (A.85a)

−1 ∇X V = ∇V X = f X(f)V (A.85b) ∇ F ∇ ( V W )kF = V W (A.85c) −1 (∇V W ) B = g(V , W )f ∇ f (A.85d) k − Notice that, for a function h on , the lift of the gradient is the gradient of the B lifted function, that is (suppressing the lifts):

∇h = B∇h . (A.86)

For brevity, we write just ∇h for it. Take care that for the Hessian this is in general only true on vectors tangent to . Hence we write explicitly the superscripts ‘ ’ in B B BHess(h) and B∆ h to denote the Hessian and Laplacian of h on , respectively, B or the lifts thereof. By means of (A.85) one can compute the expressions for the Riemann tensor. Suppressing the lifts one has: R(X, Y )Z = BR(X, Y )Z (A.87a) R(X, V )Y = (1/f) BHess(f)(X, Y )V (A.87b) R(X, Y )V = R(V, W )X =0 (A.87c)

B R(X, V )W = f gF (V , W ) ∇X ∇f (A.87d) − F R(V, W )U = R(V, W )U + df, df gF (V, U)W gF (W, U)V , (A.87e) h i − B B where df, df = g(∇f, ∇f)= gB( ∇f, ∇f) (again suppressing lifts). From the h i above formulae one can easily read oﬀ the expression for the covariant version of the Riemann tensor:

B 2 F B Riem = Riem +f Riem df, df gF gF 2f gF Hess(f) , (A.88) −h i ⊙ − ⊙ where we have omitted the lifts. Clearly, the ﬁrst summand is the part tangent to the leaves pF , the second is the part tangent to the ﬁbers pB , and the last B× ×F summand is a mixed term. Applying the Ricci contraction (see (A.53a)) on the last expression, one gets:

nF Ric = BRic BHess(f) − f (A.89) F B + Ric f ∆ f + (nF 1) df, df gF , − − h i where the ﬁrst line on the r.h.s. is the part of the Ricci tensor tangent to the leaves and the second line the part tangent to the ﬁbers. Notice that there are no mixed terms. Finally, taking the trace, one gets the expression for the Ricci scalar:

1 2nF nF (nF 1) Scal = BScal+ F Scal B∆ f − df, df . (A.90) f 2 − f − f 2 h i A.10. Spherical symmetric spacetimes 185

Sometimes it is useful to consider the metric dual of the expressions (A.85). This can be condensed in the single formula:

B 2 F ∇X = ∇ XB + fXB(f) gF + f ∇ XF + f df XF . (A.91) B F ∧ F where we again drop lifts. Here, X is an arbitrary vector field on and XB := M dπB(X) and XF := dπF (X) its projections onto the tangential spaces of and B , respectively. Moreover, denotes the metric dual w.r.t. gB and w.r.t. gF . F B F The first term in (A.91) is the part tangential to the leaves, the second and the third terms are the part tangential to the fibers, and the last one the mixed part. Contracting (A.91) with the inverse of the metric we get immediately

−1 div X = divB XB + divF XF + nF f XB(f) , (A.92) which gives the decomposition for the divergence of a vector ﬁeld. Another consequence of (A.91) are the following two lemmata concerning Killing ﬁelds:

Lemma A.8 (Killing fields, I). Let K be a vector field on and let KB := dπB(K) M and KF := dπF (K) the projections of K onto the tangential spaces of and , B F respectively. Then K is Killing for g iff the following three conditions hold: KB −1 is Killing for gB, f KB(f) is constant, and KF is conformal Killing for gF with −1 (constant) conformal factor f KB(f). − Proof. We recall that the Killing equation is (∇K) = 0, where denotes the S S symmetrization projector (A.34). By (A.91) this is equivalent to the equations pair

B ( ∇ KB )=0 , (A.93a) S B F −1 ( ∇ KF )= f KB(f) gF . (A.93b) S F − 8 The ﬁrst equation means that KB is Killing for gB. The second equation means −1 that KF is conformal Killing for gF , where the conformal factor f KB(f), being − a function on , must be a constant. B Lemma A.9 (Killing ﬁelds, II). With the same notation of Lemma A.8 the following holds:

(i) A vector ﬁeld K tangent to is Killing for g iﬀ KB is Killing for gB and B KB(f)=0.

(ii) A vector ﬁeld K tangent to is Killing for g iﬀ KF is Killing for gF . F Proof. This is also a direct consequence of the Killing equation and (A.91).

A.10 Spherical symmetric spacetimes

As already discussed in Section 4.1 a spherically symmetric spacetime has the structure of a warped product between a two-dimensional Lorentzian manifold ( , gB) 2 B and the standard unit two-sphere (S , gS2 ), where the warped function is given by the areal radius R. That is, spacetime is given by the product manifold

= S2 M B× 8 Recall that the lifts are dropped! 186 Appendix A. Diﬀerential geometry endowed with the metric ∗ 2 ∗ g = π (gB ) (R π) σ (g 2 ) , (A.94) − ◦ S where π and σ are the projections of S2 onto and S2, respectively, and π∗, σ∗ B× B their pull-backs. The minus sign in (A.94) (compare this with (A.83)) arises just because of our ‘mostly minus’ signature choice. One can of course repeat all calcula- tions done in the last section keeping track of the minus sign in front of the warping function, but this is indeed not very instructive. A short-cut is just to perform the formal substitution f = iR in all the formulae of the last session. We stress that this has nothing to do with the introduction of a complex structure on the manifold, it is just a book-keeping trick for the minus sign which appears in (A.94). The sectional curvature of S2 is constant and equal to one, hence from (A.57) and (A.58) we have that the Riemann tensor, the Ricci tensor, and the Ricci scalar are simply: S2 S2 S2 Riem = g 2 g 2 , Ric = g 2 , and Scal = 2, respectively. Moreover, since S ⊙ S S the basis manifold is two-dimensional, one can express the curvature tensors in B terms of the scalar curvature (see (A.52)). From (A.88) we get (with the formal substitution f = iR) the expression for the Riemann tensor:

B Scal 2 B Riem = gB gB R 1+ dR, dR g 2 g 2 +2R g 2 Hess(R) (A.95) 2 ⊙ − h i S ⊙ S S ⊙ The expression for the Ricci tensor follows from (A.89)

B Scal 2 B B Ric = gB Hess(R)+ 1+ dR, dR + R ∆ R g 2 (A.96) 2 − R h i S and those for the Ricci scalar from (A.90): 2 4 Scal = BScal 1+ dR, dR B∆ R . (A.97) −R2 h i − R Hence, for the Einstein tensor we have

1 2 B 2 B Ein = 1+ dR, dR + ∆ R gB Hess(R) R2 h i R − R (A.98) B Scal 1 B 2 + ∆ R R g 2 . 2 − R S and for the Weyl tensor, using (A.48) and (A.50a), we get the simple expression

2 2 2 Weyl = w (gB gB + gB R g 2 + R g 2 R g 2 ) . (A.99a) ⊙ ⊙ S S ⊙ S Here we put 1 2 2 w := BScal 1+ dR, dR + B∆ R (A.99b) 6 −R2 h i R 1 1 = Scal+ B∆ R . (A.99c) 6 R

In the derivation of (A.99) we made use of the formula hB gB = (1/2)(trgB hB)gB ⊙ ⊙ 9 gB, valid for any symmetric bilinear form hB on , in order to express the only B 9 To prove this just note that the only independent component of this formula is the (e0, e1, e0, e1) one, where {eµ} is an adapted orthonormal basis of (M, g) such that e0, e1 are tangent to B 2 and e2, e3 are tangent to S . Then, the equality follows immediately using the definition (A.47) of the Kulkarni–Nomizu product. A.10. Spherical symmetric spacetimes 187 term involving BHess R in terms of the Laplacian of R. Note that from (A.99a) it is immediate that the Weyl tensor of a spherically symmetric spacetime has only one independent component, as it must be the case due to it being of Petrov-type D. Comparing expression (A.95) with the definition of sectional curvature (see (A.55), (A.56)) one can immediately read off that the sectional curvature of the tangential plane TSp at p to the (two-dimensional) SO(3)-orbit Sp through a point p is

1 sec(TS )= 1+ dR, dR . (A.100) p −R2 h i |p With (A.100) the deﬁning expression (4.5) of the MS energy simpliﬁes to:

R E = 1+ dR, dR . 2 h i Then, using the MS energy, we can write the Einstein tensor as:

2 B E 2 B Ein = ∆ R + gB Hess(R) R R2 − R B (A.101) Scal 1 B 2 + ∆ R R g 2 . 2 − R S 

The ﬁrst line is the part tangential to , which we denote by Ein B and the second B k 2 line the part tangential to S . The trace w.r.t. gB of EinkB is

2 B 2E tr (Ein B)= ∆ R + . (A.102) gB k R R2 

The double Hodge-dual of EinkB follows now with (A.31):

2 B E ⋆ Ein B ⋆ = Hess(R)+ gB . (A.103) k −R R2 

Here, ⋆ denotes the Hodge star operator in the base manifold ( , gB). B We conclude giving the decomposition for the covariant derivative and the divergence of a spherically symmetric vector ﬁeld X, that is one with X = XB and

XS2 = 0. From (A.91) we have

B ∇X = ∇ X RX(R) g 2 . (A.104) B − S Notice that the mixed term vanishes—as it should be because of spherical symmetry. For the divergence one has

2 1 2 div X = divB X + X(R)= divB(R X) , (A.105) R R2 which follows either taking the contraction of (A.104) with the inverse of the metric or directly from (A.92).

Appendix B

Proofs

In this appendix we report the proofs of some theorems stated in Section 6.3.

B.1 Proof of Theorem 6.2

To prove Theorem 6.2 we ﬁrst note that a spherically symmetric observer ﬁeld is necessarily hypersurface orthogonal (Proposition 4.2), meaning that the u-spatial distribution u is (locally) integrable. This implies that we can locally introduce R so-called isochronous comoving coordinates (t,r,θ,ϕ) on , with respect to which M u = A(t, r)−1∂/∂t and

2 2 2 2 2 g = A (t, r) dt B (t, r) dr R (t, r) g 2 . (B.1) − − S (Note the diﬀerent meanings of the functions A and B as compared to (6.56)). The shear (and expansion) of u is now computed from the u-spatial endomorphism

Pˆu∇u (see Deﬁnition 2.10 in Section 2.3). A direct computation using (B.1) yields

ˆ ∇ r S2 Pu u = u ln(B) Pu + u ln(R) Pu , (B.2)

r S2 where Pu and Pu are the projections parallel to ∂/∂r and parallel to the tangent 2 r S2 2-planes to the S -orbits of SO(3), respectively. Clearly, both Pu and Pu are u-spatial. The trace θ of Pû∇u, which gives the expansion of u, is θ = u ln(B) + ˆ 2u ln(R) , so that the trace-free part of Pu∇u, known as the shear endomorphism σ, is given by: 1 S2 r σ := Pû∇u θ id = σ (P 2P ) , (B.3) − 3 Ru u − u 1 where σ := 3 u ln(R/B) denotes the shear scalar (only defined in a spherically- r S2 symmetric setting) and id Ru = P + P the identity endomorphism in u. In u u R passing, we note that the defining equations for θ and σ just given immediately lead to the following simple relation between the shear scalar, expansion, and the variation of the areal radius along u, that we made use of in Section 5.1.3 and Section 6.3.2.2: σ + θ/3= u(ln(R)) . (B.4) Now, according to (B.3), the shear of u vanishes iff the shear scalar σ does, that is, iff u ln(R/B) vanishes. This is equivalent to R/B being independent of t or 189 190 Appendix B. Proofs to R(t, r) = µ(r)B(t, r) for some function µ, so that the line element (B.1) can be rewritten in the spatially conformally flat form

2 2 2 2 2 g = A˜ (t,ρ) dt C˜ (t,ρ) dρ + ρ g 2 , (B.5a) − S where A˜(t,ρ) := A(t, r(ρ)), C˜(t,ρ) := C(t, r(ρ)), and

B(t, r)µ(r) C(t, r)= (B.5b) ρ(r) with r dr′ ρ(r)= ρ exp . (B.5c) 0 µ(r′) Zr0 Hence we see that the vanishing of the shear of u implies conformal ﬂatness of the corresponding spatial metric. For the converse we ﬁrst note that, since u and g are spherically symmetric, the spatial metric hu is itself spherically symmetric, so that g can be written in the form (B.5a). This implies that the corresponding R/B depends only on the radial coordinate and hence that the shear of u vanishes.

B.2 Proof of Proposition 6.3

The proof of Proposition 6.3 consists in the computation of the intersection of the set of metrics of type (6.39), which we denote in the following by gMcV, SMcV a,m with the set of metrics conformally related to an exterior Schwarzschild metric. ScS Explicitly, the latter are of the form

gcS := Ω2gSchw , (B.6a) Ω,M0 M0 where −1 Schw 2M0 2 2M0 2 2 g = 1 dT 1 dR R g 2 , (B.6b) M0 − R − − R − S denotes the Schwarzschild metric with mass M0 in ‘standard’ coordinates. The question is: For which functions a and m and, respectively, for which function Ω and parameter M does the equation gMcV = Ω2gSchw hold? Such an equality 0 a,m M0 (between two tensor fields on the same manifold) can be eventually established by finding a coordinate transformation, φ say, between the coordinates1 (t, r) in (6.39) and (T, R) in (B.6) which brings (6.39) in form (B.6). This involves solving coupled, non-linear partial differential equations for φ, which depend on the four unknown parameter a,m, Ω, and M0. Needless to say that this is not really a thankful task. Alternatively, a better approach would be to compare all the independent, algebraic curvature-invariants of the two metrics: This would lead to a system of equations between scalars which involves the coordinate transformation φ in an algebraic way (i.e. non differentiated). We adopt here an approach which is somewhere in the middle: First, we use just three invariants (the areal radius and the Ricci and the Weyl part of the MS

1 Without loss of generality we can restrict the coordinate transformation to the time- and radial

coordinates, because the metrics gS2 in (B.6) and in (6.39) already coincide as tensor ﬁelds on S2 (the unit 2-sphere), which is deﬁned independently of any coordinate choices by the condition of spherical symmetry, to which both metrics are subject. B.2. Proof of Proposition 6.3 191 energy) to drastically restrict the form of the coordinate transformation (see (B.19)) and derive thereby constraints on the free parameters a,m, Ω, and M0 (see (B.15) and (B.18)). Second, we perform this restricted coordinate transformation and determine it completely. To simplify the calculation, instead of gMcV =Ω2gSchw, we a,m M0 consider the equivalent equation Ω−2gMcV = gSchw. In fact, for the Schwarzschild a,m M0 metric (B.6b) it is immediate that the above mentioned quantities are, respectively, given by:

R(gSchw) = R , (B.7) M0

Schw EW(g )= M , (B.8) M0 0

Schw ER(g ) =0 . (B.9) M0

−2 McV In order to compute the respective quantities for the metric Ω ga,m we ﬁrst give their scaling behavior under conformal transformations. Clearly, because of their very deﬁnitions, for the areal radius and the Weyl part of the MS energy it holds: R(Ψ2g)=ΨR(g) (B.10) and 2 EW(Ψ g)=ΨEW(g) , (B.11) respectively. For the whole MS energy it easily follows from (4.6) and (B.10):

2 2 ∇ ∇ 1 3 ∇ ∇ E(Ψ g)=Ψ E(g)+ R g( R, lnΨ) + 2 R g( ln Ψ, ln Ψ) , (B.12) where all the quantities on the right-hand side are referred to the metric g. Hence, taking the diﬀerence between (B.12) and (B.11) one gets that the Ricci part of the MS energy scales exactly like the whole MS energy, that is according to (B.12). Using these scaling properties together with (6.52) and (6.59) we get immediately:

−2 McV −1 2 R(Ω ga,m ) =Ω (1 + m/2r) ar , (B.13)

−2 McV −1 EW(Ω ga,m )=Ω a m . (B.14)

The equality between the Weyl part of the MS energy (B.14) and (B.8) implies

Ω(t, r) M0 = a(t) m(t) , (B.15) which gives a condition between the parameter a,m, Ω, and M0. Since we assumed that M0 is positive, (B.15) can be read as the expression for the conformal factor in the (t, r) coordinates. This, together with the equality between the areal radius (B.13) and (B.7), implies in turn

M R(t, r)= 0 (1 + m(t)/2r)2r , (B.16) m(t) which gives the ﬁrst component of the coordinate transformation φ. Now, using the scaling property (B.12) for the Ricci part of the MS energy, the expressions (6.58) and (6.52) for the Ricci part of the MS energy and, respectively, the areal radius 192 Appendix B. Proofs of the McVittie metric ansatz, and (B.15) for the conformal factor, one gets, after some computations,

2 M 3 m˙ E (Ω−2gMcV)= 0 (1 + m/2r)2ar . (B.17) R a,m 2 am m 

The equality between (B.17) and (B.9) then implies

m˙ =0 , (B.18)

that is m = m0 for some positive constant m0. This, in turns, implies that the transformation (B.16) for R depends only on r and not on t. Since the metrics are both in diagonal form, this implies that the transformation for T must depend on t only.

Summarizing, so far we have seen that a set of necessary conditions for the equality of the two metrics implies the constraints (B.15) and (B.18) and that the coordinate transformation between (t, r) and (T, R) is of the form

T (t) = f(t) (B.19a)

M0 2 R(r)= (1 + m0/2r) r , (B.19b) m0 for some diﬀerentiable function f of t. Now, explicitly expressing the metric Ω2gSchw M0 in the (t, r) coordinates according to the coordinate transformation (B.19) and the McV constraints (B.15) and (B.18), and putting the result equal to ga,m , the only new condition that one gets is

M 1 f˙ = 0 . (B.20) ± m0 a

Here, the plus can be chosen in order to exclude a time inversion. It is important to note that (B.20) (together with an initial value) determines f uniquely and do not give any constraint on the parameters a,m, Ω, and M0: The only constraints remain thus (B.15) and (B.18).

The proof is concluded noticing that (B.15) means that the only constraint on Ω is that, expressed in the (t, r) coordinates, it depends on t only and hence, in view of (B.19a) and expressed in the (T, R) coordinates, that it depends on T only. More geometrically, this can be restated saying that the gradient of Ω must be proportional to ∂/∂T , the Killing ﬁeld of the Schwarzschild metric (see (B.6b)). B.3. Proof of Proposition 6.5 193

B.3 Proof of Proposition 6.5

Inserting the deﬁnition (6.53) of F in the expression (6.71) for the Ricci scalar and organizing the result in powers of rB r m/2 we get: ≡ − a˙ 2 Scal = 12 − a 6 a˙(am)˙ a˙ ˙ 4 + rA − rB a2 a ! (B.21) 2 6 (am)˙ (am)¨ a˙ (am)˙ 2 +rA − (rB)2 a a − a2 ! 3 m˙ (am)˙ rA . − (rB)3 a

Hence, the Ricci scalar remains finite in the limit r m/2 iff all the three coeffi- → cients of (rB)−k, for k 1, 2, 3 , vanish in this limit, that is iff it holds: ∈{ } a˙ (am)˙ a˙ ˙ 4 + m = 0 (B.22a) a2 a 2 (am)˙ (am)¨ a˙(am)˙ 2 +m = 0 (B.22b) a a − a2 m˙ (am)˙ =0 . (B.22c)

These conditions are clearly understood to hold for all times t in which the functions a and m and their derivative exist. In view of (B.22c) we have to distinguish between two cases:m ˙ = 0 and (am)· = 0, respectively. In the first case the system (B.22) reduces to the set of conditions m(ä/a + 3(˙a/a)2) = 0 and m(ä/a + (˙a/a)2) = 0, which, in turn, reduces to m = 0 (and a arbitrary), corresponding to the FLRW metric, or toa ˙ = 0 (andm ˙ = 0), corresponding to the Schwarzschild metric. In the second case, in which (am)· = 0, (B.22) reduces to m(˙a/a)· = 0, which implies either m = 0 (and a arbitrary), corresponding again to the FLRW metric, or (˙a/a)· = 0. Together with (am)· = 0, the latter corresponds to a McVittie metric with exponentially-growing (or -falling) scale factor a(t), that is to a Schwarzschild– de Sitter metric.

Appendix C

General Relativity

In this appendix we collect the basic equations of GR, their variational formulation, and their linearization.

C.1 Basics equations

The Einstein equation relates the Ricci part of the spacetime’s curvature to the 1 matter energy momentum tensor TM. In geometric units this is given by

Ein Λg =8πT , (C.1) − M were Λ is the cosmological constant. Sometimes, it is useful to write the cosmological term on the r.h.s. and to interpret it like a contribution to the matter energy- momentum tensor. Putting Λ T := g (C.2) Λ 8π and

T := TM + TΛ (C.3) Einstein equation now reads Ein =8π T . (C.4) For a positive Λ the contribution of the cosmological term to the energy momentum tensor is that of a ‘matter ﬁeld’ with positive energy density and negative pressure:

Λ ̺ = T (u, u)= , (C.5a) Λ Λ 8π Λ p = T (e, e)= . (C.5b) Λ Λ −8π

Here, u denotes an arbitrary observer field and e any normalized spatial vector field. It should be noted that we put the cosmological term on the r.h.s. just for computational convenience: this does not mean that we think of it as a physical matter field. For what concerns the present work, Λ is simply a free parameter of the theory which shall be fixed by experimental measurements—and nothing more.

1 If one does not use geometric units, the factor 8π on the r.h.s. of (C.1) is to be replaced by 8πG/c4, where G is Newton’s gravitational constant and c the speed of light.

195 196 Appendix C. General Relativity

Assuming that spacetime is four-dimensional and g satisﬁes Einstein’s equation, we can express the Ricci part of the Riemann tensor in terms of T by

1 Ricci =8π T trg(T ) g g , (C.6) − 3 ⊙ as it follows from (A.50b) and (C.4). Recall that, as a consequence of the (twice contracted) second Bianchi identity and, respectively, of the metricity of the Levi-Civita connection, the divergence of the Einstein tensor and that of TΛ vanish identically. Hence, the Einstein equation requires as integrability condition the divergence-freeness of the matter energy- momentum tensor

div TM =0 . (C.7)

For some matter models (e.g. for perfect ﬂuids) this implies already the full set of matter equation of motion.

C.2 Variational formulation

Our primary interest here is to ﬁx signs and factors and not to discuss the details of the variational formulation itself. For the correct formulation with the so-called Gibbons–Hawking–York boundary term, see [Yor72], [Yor86], and [GH77]. The vacuum Einstein equation are a stationary point of the action

S = d4x , (C.8) L Z where d4x := dx0 dx1 dx2 dx3 is the 4-form arising from the considered ∧ ∧ ∧ coordinates and is given by the Einstein–Hilbert Lagrangian L 1 √ EH = 2 g (Scalg +2Λ) . (C.9) L −κ0 −

2 Notice that this is a scalar density of weight 1/2 and recall that κ0 = 16πG (reintroducing G). Standard computations (chain rule) show that, up to a divergence term, the variation of (C.9) is given by:

1 √ µν δ EH = 2 g (Gµν Λgµν)δg . (C.10) L −κ0 − −

Clearly, the ﬁrst term of the Einstein–Hilbert Lagrangian (C.9) contains second- order derivatives of the metric (see (C.16)), however it is equal, up to a divergence term, to the ﬁrst-order Lagrangian

1 √ µν τ σ τ σ EH1 = 2 gg (Γµσ Γτν Γµν Γτσ) , (C.11) L −κ0 − −

λ where the Christoﬀel symbols Γµν are deﬁned in (C.15). In presence of matter, described by a Lagrangian density , the Lagrangian LM in (C.8) becomes = + . (C.12) L LEH LM C.3. Linearization 197

Deﬁning the (0, 2)-tensor TM, to be identiﬁed with the matter energy-momentum tensor, by

1 µν δg M = √ gTM δg , (C.13) L 2 − µν one gets Einstein equation (C.1).

C.3 Linearization

We are interested here in the linearization of Einstein’s equation with vanishing cosmological constant around its trivial solution given by Minkowski spacetime. In the following, we refer all quantities to Minkowskian coordinates, hence global coordinates xµ w.r.t. which the components of the Minkowski metric satisfy ev- { } µ ν erywhere: η(∂/∂x , ∂/∂x )= ηµν = diag(1, 1, 1, 1) and hence, in particular, µ ν − − − ηµν,λ = 0. Putting gµν := g(∂/∂x , ∂/∂x ) and

gµν = ηµν + ǫhµν , (C.14)

µν µ we expand the metric determinant g, inverse g , Christoﬀel-symbols Γαβ, Ricci tensor Rµν , Ricci scalar R, Einstein tensor Gµν , and the Einstein–Hilbert Lagrangian up to leading order in ǫ. For completeness, we recall that 1 Γ µ = gµλ (g + g g ) (C.15) αβ 2 αλ,β βλ,α − αβ,λ R = Rλ = Γ λ Γ λ + Γ σ Γ λ Γ σ Γ λ (C.16) µν µλν νµ,λ − λµ,ν νµ λσ − λµ νσ In the formulae below, we take the convention to rise and lower indices with the µν µ Minkowski metric and we use the abbreviations h := η hµν and hν := ∂ hµν for the trace and divergence of hµν , respectively. One has:

g =1+ ǫh + (ǫ2) (C.17a) − O gµν = ηµν ǫhµν + (ǫ2) (C.17b) − O ǫ Γ µ = 2hµ h ,µ + (ǫ2) (C.17c) αβ 2 (α,β) − αβ O ǫ R = 2h h h + (ǫ2) (C.17d) µν 2 (µ,ν) − µν − ,µν O R = ǫ hµ h + (ǫ2) (C.17e) ,µ − O ǫ G = h h +2h (hσ h)η + (ǫ2) (C.17f) µν 2 − µν − ,µν (µ,ν) − ,σ − µν O 2 1 ǫ µν,λ µν,λ µ ,µ 3 EH1 = 2 h hµν,λ 2h hµλ,ν +2h h,µ h h,µ + (ǫ ) (C.17g) L κ0 4 − − O The second term in (C.17g) is equal, up to a divergence term, to 2hµh . This, − µ together with the substitution ǫ = κ0, shows that the ﬁrst part of the massive spin- 2 Lagrangian, (9.26), is indeed the leading order approximation of the Einstein– Hilbert Lagrangian of GR.

Appendix D

Observational data

In this appendix we collect the relevant cosmological data used in the text and the trajectory informations for the spacecrafts Pioneer 10 and 11.

D.1 Cosmological data

Here are summarized the deﬁnitions of the relevant cosmological parameters used in the text: Hubble parameter: H :=a/a ˙ • Hubble radius: R := c/H • H Reduced Hubble parameter (h): H = h 100 (km/s)/Mpc = h/(3.08 1017 s) • · · Deceleration parameter: q := (¨a/a)/(˙a/a)2 • − Critical density: ̺ := 3H2/8πG = h2 1.89 10−29 g/cm3 • c · · Ω parameters (subject to the relation Ω +Ω +Ω = 1): • m Λ k

Ωm := ̺m/ρc , (D.1a) 2 ΩΛ := ̺Λ/̺c =Λ/3H , (D.1b) Ω := kc2/(Ha)2 . (D.1c) k −

For our purpose the today values are approximately given by:

H 70 (km/s)/Mpc (D.2) 0 ≈ R 1.3 1023 km (D.3) H ≈ · h 0.7 (D.4) 0 ≈ q 1/2 (D.5) 0 ≈− ̺ 9.2 10−30 g/cm3 (D.6) c ≈ · Ω 1/3 (D.7) m ≈ Ω 2/3 (D.8) Λ ≈ Ω 0 . (D.9) k ≈

For the precise values and their errors we refer to [K+09].

199 200 Appendix D. Observational data

D.2 Pioneer 10 and 11 data

The following data on the trajectories of the spacecrafts Pioneer 10 and 11 are taken from the reference [ALL+02].

P10 P11 Launch 2 Mar 1972 5 Apr 1973 Planetary Jupiter: 4 Dec 1973 Jupiter: 2 Dec 1974 encounters Saturn: 1 Sep 1979 Tracking data 3 Jan 1987 – 22 Jul 1998 5 Jan 1987 – 1 Oct 1990 Distance from the Sun 40 AU – 70 AU 22.4 AU – 31.7 AU Light round-trip time 11h–19h 6h–9h Radial velocity 13.1 Km/s – 12.6 Km/s N.A.

Table D.1: Some orbital data of the Pioneer 10 and 11 spacecrafts.

Tracking system

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M. Carrera and D. Giulini, Generalization of McVittie’s model for an inho- • mogeneity in a cosmological spacetime, Physical Review D 81, 043521 (2010) [15 pages], Preprint 0908.3101 [gr-qc]

M. Carrera and D. Giulini, Inﬂuence of global cosmological expansion on local • dynamics and kinematics, Reviews of Modern Physics 82 (2010) 169–208, Preprint 0810.2712 [gr-qc]

M. Carrera and D. Giulini, On Doppler tracking in cosmological spacetimes, • Classical and Quantum Gravity 23 (2006) 7483–7492, Preprint gr-qc/0605078

M. Carrera and D. Giulini, On the inﬂuence of the global cosmological ex- • pansion on the local dynamics in the Solar System, European Space Agency, the Advanced Concepts Team, Ariadna Final Report 04-1302 (2005), Preprint gr-qc/0602098

M. Carrera and D. Giulini, Classical Analysis of the van Dam–Veltman Dis- • continuity, Preprint gr-qc/0107058

209

Index

anti-commutator, 169 derivative, 171 antiderivation, 173 projective, 19 areal radius, 56 exterior derivative, 173 exterior product, 172 boost, 20 extrinsic curvature, 181

Cartan’s formula, 173 Fermi derivative, 22 codifferential, 174 Fermi transport, 22 commutator, 169 Fermi observer-derivative, 45 comoving derivative, 27 FLRW cosmological models, 81 comoving transport, 27 Friedmann equations, 82 conformal related metrics, 182 Gaussian curvature, 179 transformation, 182 graded derivation, 172 corotating derivative, 29 gradient, 169 corotating transport, 29 cosmological time, 81 Hessian, 169 curvature, 168 Hodge star, 174 extrinsic, 181 spatial, 20 Gaussian, 179 Hubble parameter, 83 Ricci, 168 Hubble radius, 84 Riemann, 168 sectional, 179 indicator, 180 Weyl, 178 induced connection, 181 deceleration parameter, 85 metric, 180 derivation, 170 interior product, 173 graded, 172 irrotational, 30 derivative isometry group, 55 comoving, 27 isotropic spacetime, 81 corotating, 29 exterior, 173 junction conditions Fermi, 22 Darmois (DJC), 72 observer, 41 spherically symmetric (SSJC), 73 derivative extension for a derivation, 171 Killing equation, 33 for an endomorphism, 171 Killing field, 33 divergence, 169 kinematical operator, 41 Einstein tensor, 168 Kodama basis, 62 electric part, 19 Kodama observer field, 58 Euler equations, 15 Kodama current, 59 expansion, 30 Kodama vector field, 58 extension Kulkarni–Nomizu product, 177

211 212 Index

Laplace–de Rham operator, 175 shear tensor, 30 Laplacian, 169 shear-expansion tensor, 30 local rest reference system, 18 shearless, 30 local rest space, 18 skew-adjoint, 169 local temporal subspace, 18 part, 169 spacetime, 13 magnetic part, 19 spatial manifold Hodge star, 20 semi-Riemannian, 167 metric, 20 metricity defect, 168 projector, 18 Misner–Sharp (MS) energy, 57 tensor, 19 spatial Ricci-isotropy, 92 non-expanding, 30 spherically symmetric spacetime, 55 observer, 18 tensor field, 55 cosmological, 81 symmetric product, 176 field, 18 synchronizable, 36 geodesic, 33 locally, 36 inertial, 49 locally proper-time, 36 rigid, 36 proper-time, 36 static, 33 stationary, 33 temporal uniformly-accelerated, 33 metric, 20 observer derivative, 41 projector, 18 observer transport, 41 tensor, 19 outwardly-pointing, 58 tensor field, 167 tensor field along a map, 169 projector tidal operator, 31 spatial, 18 time function, 36 temporal, 18 torsion, 168 proper reference frame, 37 trace, 168 corotating, 37 transport non-rotating, 37 comoving, 27 observer, 42 corotating, 29 static, 37 Fermi, 22 stationary, 37 observer, 41 trapped surface, 96 rapidity, 40 relative acceleration, 43 warped product, 183 relative velocity, 40 warping function of, 183 Ricci part of the Riemann tensor, 178 wave vector, 51 Ricci scalar, 168 wedge product, 172 Ricci tensor, 168 Weyl tensor, 178 Ricci-isotropic, 61 worldline Riemann tensor, 168 geodesic, 33 rotation, 20 lightlike, 18 rotation tensor, 30 static, 33 stationary, 33 scalar curvature, 168 timelike, 18 sectional curvature, 179 uniformly-accelerated, 33 self-adjoint, 169 part, 169