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

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Geometrical Methods for Kinematics and Dynamics in Relativistic Theories of Gravity with Applications to Cosmology and Space Physics Geometrical methods for kinematics and dynamics in relativistic theories of gravity with applications to cosmology and space physics Dissertation zur Erlangung des Doktorgrades der Fakult¨at f¨ur Mathematik und Physik der Albert–Ludwigs–Universit¨at Freiburg im Breisgau vorgelegt von Matteo Carrera aus Tenero–Contra, Schweiz August 16, 2010 Dekan: Prof. Dr. Kay K¨onigsmann Betreuer der Dissertation: Prof. Dr. Domenico Giulini Referent: Prof. Dr. Domenico Giulini Korreferent: Prof. Dr. Hartmann R¨omer Datum der m¨undlichen Pr¨ufung: 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 ´epures, n’aboutissent, comme signes visibles, qu’`a la seule simplicit´e, comme s’il fallait l’exp´erience de plusieurs g´en´erations pour d´egager peu `apeu la courbe d’une colonne, d’une car`ene, ou d’un fuselage d’avion, jusqu’`aleur rendre la puret´e ´el´ementaire de la courbe d’un sein ou d’une ´epaule. Il semble que le tra- vail des ing´enieurs, des dessinateurs, des calculateurs du bureau d’´etudes ne soit ainsi en apparence, que de polir et d’effacer, d’all´eger ce raccord, d’´equilibrer cette aile, jusqu’`ace qu’on ne la remarque plus, jusqu’`ace qu’il n’y ait plus une aile accroch´ee `aun fuselage, mais une forme par- faitement ´epanouie, enfin d´egag´ee de sa gangue, une sorte d’ensemble spontan´e, myst´erieusement li´e, et de la mˆeme qualit´eque celle du po`eme. Il semble que la perfection soit atteinte non quand il n’y a plus rien `a ajouter, mais quand il n’y a plus rien `aretrancher. 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–Sch¨ucking 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 ten- sors (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
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