GEM: the User Manual Understanding Spacetime Splittings and Their Relationships1

Home , Congruence (manifolds), Sign convention, Static spacetime

Robert T. Jantzen Paolo Carini and Donato Bini

June 15, 2021

1This manuscript is preliminary and incomplete and is made available with the understanding that it will not be cited or reproduced without the permission of the authors. Abstract

A comprehensive review of the various approaches to splitting spacetime into space-plus-time, establishing a common framework based on test observer families which is best referred to as gravitoelectromagnetism. Contents

Preface vi

1 Introduction 1 1.1 Motivation: Local Special Relativity plus Rotating Coordinates ...... 1 1.2 Why bother? ...... 4 1.3 Starting vocabulary ...... 6 1.4 Historical background ...... 8 1.5 Orthogonalization in the Lorentzian plane ...... 11 1.6 Notation and conventions ...... 14

2 The congruence point of view and the measurement process 17 2.1 Algebra ...... 17 2.1.1 Observer orthogonal decomposition ...... 18 2.1.2 Observer-adapted frames ...... 21 2.1.3 Relative kinematics: algebra ...... 23 2.1.4 Splitting along parametrized spacetime curves and test particle worldlines ...... 26 2.1.5 Addition of velocities and the aberration map ...... 28 2.2 Derivatives ...... 29 2.2.1 Natural derivatives ...... 29 2.2.2 Covariant derivatives ...... 30 2.2.3 Kinematical quantities ...... 31 2.2.4 Splitting the exterior derivative ...... 33 2.2.5 Splitting the differential form divergence operator ...... 35 2.2.6 Spatial vector analysis ...... 35 2.2.7 Ordinary and Co-rotating Fermi-Walker derivatives ...... 37 2.2.8 Relation between Lie and Fermi-Walker temporal derivatives ...... 39 2.2.9 Total spatial covariant derivatives ...... 43 2.2.10 Splitting the total covariant derivative ...... 46 2.3 Observer-adapted frame derivatives ...... 47 2.3.1 Natural frame derivatives ...... 47 2.3.2 Splitting the connection coefficients ...... 48 2.3.3 Observer-adapted connection components ...... 49 2.3.4 Splitting covariant derivatives ...... 50 2.3.5 Observer-adapted components of total spatial covariant derivatives ...... 52 2.4 Relative kinematics: applications ...... 55 2.4.1 Splitting the acceleration equation ...... 55 2.4.2 Analogy with electromagnetism: gravitoelectromagnetism ...... 57 2.4.3 Maxwell-like equations ...... 58 2.4.4 Splitting the spin transport equation ...... 60 2.4.5 Relative Fermi-Walker transport and gyro precession ...... 62 2.4.6 The Schiff Precession Formula ...... 65 2.4.7 The relative angular velocity as a boost derivative ...... 67

i 2.4.8 Relative kinematics: transformation of spatial gravitational fields ...... 68 2.5 Spatial curvature and torsion ...... 70 2.5.1 Definitions ...... 70 2.5.2 Algebraic symmetries ...... 70 2.5.3 Symmetry-obeying spatial curvature ...... 72 2.5.4 Spatial Ricci tensors and scalar curvatures ...... 73 2.5.5 Pair interchange symmetry ...... 73 2.5.6 Spatial covariant exterior derivative ...... 74 2.6 The symmetrized curl operator for symmetric spatial 2-tensors ...... 76 2.7 Splitting spacetime curvature ...... 77 2.7.1 Splitting definitions ...... 77 2.7.2 Spacetime duality and curvature ...... 78 2.7.3 Evaluation of splitting fields ...... 79 2.7.4 Maxwell-like equations ...... 81 2.8 Mixed commutation formulas ...... 82 2.8.1 Splitting the Ricci identities ...... 82 2.8.2 Commuting £(u)u and ∇(u) ...... 83 2.9 Splitting the Bianchi identities of the second kind ...... 84 2.9.1 Spacetime identities ...... 84 2.9.2 Spatial identities ...... 87 2.10 “Time without space defines space without time” and vice versa ...... 88

3 The slicing and threading points of view 89 3.1 Introduction ...... 89 3.2 Algebra ...... 89 3.2.1 The nonlinear reference frame ...... 89 3.2.2 Measurement and the lapse function ...... 90 3.2.3 The Shift ...... 92 3.2.4 Computational frames and the reference decomposition ...... 93 3.2.5 Decomposing the metric ...... 94 3.2.6 Relationship between the reference and observer decompositions ...... 96 3.2.7 The slicing, threading and reference representations ...... 97 3.2.8 Transformation between slicing and threading points of view ...... 98 3.2.9 So far: ...... 99 3.3 Derivatives ...... 101 3.3.1 Evolution ...... 101 3.3.2 Natural time derivatives ...... 101 3.3.3 Natural spatial derivatives ...... 102 3.3.4 Gauge transformations of the nonlinear reference frame ...... 103 3.3.5 Observer-adapted frame structure functions and kinematical quantities ...... 105 3.3.6 Spatial covariant derivative ...... 106 3.3.7 Spatial vector analysis ...... 107 3.3.8 Partially-observer-adapted frames: connection components ...... 108 3.3.9 Total spatial covariant derivatives ...... 109 3.3.10 Spatial gravitational forces ...... 111 3.3.11 Second-order acceleration equation ...... 112 3.3.12 The spin transport equation ...... 113 3.3.13 Transformation of spatial gravitational ﬁelds ...... 114 3.4 Spatial curvature ...... 115 3.5 Initial value problem? ...... 115 3.5.1 Hypersurface and slicing points of view ...... 115 3.5.2 Thin sandwich problem ...... 116 3.5.3 Congruence and threading points of view ...... 116

ii 3.5.4 Perfect ﬂuids ...... 117

4 Maxwell’s equations 118 4.1 Introduction ...... 118 4.2 Splitting the electromagnetic ﬁeld ...... 118 4.2.1 Congruence point of view ...... 118 4.2.2 Slicing and threading points of view ...... 119 4.2.3 Observer Boost ...... 120 4.2.4 Reference representation (Landau-Lifshitz-Hanni) ...... 120 4.3 Splitting the 4-current ...... 122 4.4 Splitting Maxwell’s equations ...... 123 4.4.1 Congruence point of view ...... 123 4.4.2 Slicing and threading points of view ...... 124 4.5 Vector potential ...... 126 4.6 Wave equations ...... 127 4.7 Computational 3-space representations ...... 128 4.8 Lines of force ...... 128

5 Stationary spacetimes 129 5.1 Stationary nonlinear reference frame ...... 129 5.2 Synchronization gap and Sagnac effect ...... 130 5.3 Rotating spatial Cartesian coordinates in flat spacetime ...... 133 5.4 Stationary axially-symmetric case: rotating Minkowski, Gödeland Kerr spacetimes ...... 135

6 Perturbation problems 139 6.1 Linearization about an orthogonal nonlinear reference frame ...... 139 6.2 Post-Newtonian approximation ...... 139 6.3 The Newtonian limit ...... 139 6.4 Friedmann-Robertson-Walker Perturbations ...... 139

A Formulas from differential geometry 1 A.1 Manifold ...... 1 A.2 Frame and dual frame ...... 1 A.3 Linear transformations ...... 2 A.4 Change of frame ...... 3 A.5 Metric ...... 3 A.6 Unit volume n-form ...... 4 A.7 Connection ...... 4 A.8 Metric connection ...... 6 A.9 Curvature ...... 6 A.10 Total covariant derivative ...... 7 A.11 Parallel transport and geodesics ...... 8 A.12 Generalized Kronecker deltas ...... 8 A.13 Symmetrization/antisymmetrization ...... 8 A.14 Exterior product ...... 9 A.15 Hodge star duality operation ...... 9 A.16 Complex Duality Operation ...... 10 A.17 Exterior derivative ...... 10 A.18 Differential form divergence operator ...... 11 A.19 De Rham Laplacian ...... 12 A.20 Covariant exterior derivative ...... 12 A.21 Ricci identities ...... 13 A.22 Bianchi identities of the first and second kind ...... 13 A.23 Ricci Tensor, Scalar Curvature and Einstein tensor ...... 14

iii A.24 Contracted Bianchi Identities of the Second Kind and the Weyl Tensor ...... 14 A.25 Conformal Transformations in 3 Dimensions ...... 15 A.26 n = 3 Structure Functions and Orthonormal Frame Connection Components . . . . . 16 A.27 Lie derivative ...... 17

iv List of Figures

1.1 Spacetime and Space plus Time. Although spacetime is the arena where calculations are simpler, we always interpret them through our space plus time worldview, which depends on the choice of inertial observer...... 2 1.2 General linear coordinates on E2 and their two orthogonalizations. The dashed lines indicate the unit coordinate rectangle and the dotted lines the projections of a typical vector along the coordinate axes...... 12 1.3 General linear coordinates on M 2 and their two orthogonalizations. The dashed lines indicate the unit coordinate rectangle and the dotted lines the projections of a typical vector along the coordinate axes...... 14

2.1 The relative observer plane, the relative velocities and the associated relative observer maps. 24

3.1 Computational 3-spaces ...... 90 3.2 Reference decomposition of tangent and cotangent spaces ...... 90 3.3 Shifting of points of the computational 3-space...... 92 3.4 Projected computational frame ...... 94 3.5 Decompositions of a vector ﬁeld ...... 96 3.6 Lorentz boost 2-plane ...... 99 3.7 A suggestive representation of the various total Lie spatial covariant derivatives for the hypersurface, slicing and threading points of view, normalized to correspond to the proper time of the appropriate observers. Each covariant derivative is the sum of a temporal Lie derivative and a spatial covariant derivative along the temporal and spatial parts of the relevant multiple of the 4-velocity U represented in the triangle, each vector decomposition given under the symbol for the corresponding covariant derivative...... 111

5.1 Lifts from the quotient space ...... 131 5.2 Sagnac tube geometry ...... 131

v Preface

I was a sophomore in John Wheeler’s modern physics course at Princeton University in 1972 when the proofs for some chapters of Gravitation began showing up. This book and its authors (Misner, Thorne and Wheeler) have helped to shape generations of relativists including myself and have done much to establish the use of more modern mathematical notation and style as commonplace in the field and in neighboring areas. In particular it is the one textbook which presents the 3 + 1 approach to general relativity to a wide audience. However, an alternative “1 + 3” approach to the splitting of spacetime into space plus time also existed and was most readily found in the text The Classical Theory of Fields by Landau and Lifshitz, but few versed in the 3 + 1 school took the time to understand this alternative in terms of the same beautiful language that made the 3 + 1 approach such a powerful tool in gravitational physics. The culprit is of course quantum gravity which was always lurking in the background as one of the prime motivations for using the 3 + 1 approach. In astrophysical applications the superiority of either approach is not obvious and it seems clear that both approaches together can reveal complementary features of a physical problem, shedding more insight than either one alone. Indeed the 1 + 3 approach underlies much of what is done in the post-Newtonian approximation of general relativity, although it is not generally recognized, and provides the variable decomposition for the stationary exact solution industry, while in the cosmological context it appears that gauge invariant perturbation theory for Friedmann-Robertson-Walker cosmological models may even be simpler in this approach. Unfortunately the 1 + 3 approach, which apparently flourished in the fifties before the 3 + 1 school took over, suffered a near fatal public relations blackout in the intervening decades preceding the nineties. This has left many of us who have entered the field since the seventies almost “completely unaware” of much of this formalism beyond recognition of it from limited exposure to it in The Classical Theory of Fields or in the closely related congruence approach systematized for applications in cosmology by Ehlers and more widely known through the work of Hawking and Ellis. When Paolo Carini as a student of Remo Ruffini in Rome in 1989 got me interested in trying to explain what was really underlying the peculiar coordinate decomposition of the electromagnetic field found in an exercise in the Landau-Lifshitz text and its relationship to the 3 + 1 and 1 + 3 approaches, we were drawn into an examination of the very foundations of the splitting formalisms independent of electromagnetism. We were also “completely unaware” of the work in the fifties by Møller, Zel’manov, and Cattaneo on the 1 + 3 point of view. Driven by our ignorance of these matters and the lack of a clear discussion of their mathematical structure in the literature, we embarked on a study in which we learned a great deal not only about the larger framework in which these questions fit together, but also about the work that had been done and effectively buried in the past. Later joined by Donato Bini, we went on to examine some new aspects of the application of splitting formalisms to questions in general relativity. It seems not only valuable to share with the larger community what we have understood in a notation that easily allows comparison of the different approaches but also appropriate that some of the historical roots of the subject be recalled and not forgotten. Meanwhile during the decade of the nineties the 3 + 1 approach using the more modern point of view of congruences has enjoyed increasing visibility through many articles by Ellis and collaborators, and the need for understanding the relationships among various approaches has only increased. In this spirit the following monograph has grown to its present form. Certainly it is not meant to be exhaustive, especially where the 3+1 approach is concerned given the enormous amount of existing literature which deals with that case alone. Rather it is meant to reveal more clearly the relationships between the

vi various approaches in an effort to break down the artificial barriers which divide them. If this attempt is only partially successful, our efforts will have been well spent. The International Center for Relativistic Astrophysics at the University of Rome and its director Remo Ruffini are especially thanked for suggesting and supporting this project.

bob jantzen

vii Chapter 1

Introduction

1.1 Motivation: Local Special Relativity plus Rotating Coordi- nates

Most of us know special relativity pretty well and are quite happy switching back and forth between the spacetime picture of 4-vector algebra and the space plus time picture of events occurring in space as time elapses, using 3-vector algebra. We have no difficulty using Lorentz transformations to transform 3-dimensional quantities from those measured by one inertial observer to another. We also have no problem extending our splitting algebra to spacetime derivative operators in inertial coordinates yielding the space plus time equivalent of time derivatives and spatial derivative operators. After all, these are the derivatives we began with before learning special relativity. Figure 1.1 suggestively compares these two pictures. Since old habits die hard, there is a strong incentive to push this habit into general relativity. This helps us interpret spacetime information in a curved spacetime using the same intuition that we have about classical 3-dimensional physics. The catch is that one no longer has a privileged class of “global inertial frames” as in special relativity which effectively allows a single inertial observer in flat spacetime to set up a preferred class of global inertial coordinate systems that may be used to interpret observations at all other points in spacetime. Essentially, one inertial observer in flat spacetime uniquely determines a family of inertial observers filling the spacetime with no relative velocities, and the “observations” of the original observer of an event not on his own worldline are understood to be those of the companion observer who is present. The only option allowing us to continue to make a spacetime splitting without relying on flatness is to give up the global splitting associated with a single preferred observer and settle for the local splitting of each member of a family of observers filling the spacetime and in arbitrary motion in the absence of any preference. Such a splitting takes place in the tangent space to each event in spacetime, describing the locally Minkowskian neighborhood of each observer in the family. If we agree to split each such tangent space based on the 4-velocity of the observer at a given event in the same way that we split flat spacetime globally based on the worldline of a single inertial observer, then all of our familiarity with special relativity can be transferred to general relativity in a rather straightforward way. The only difference is that what we did globally before, with the splitting at every spacetime point determined by the splitting at a single spacetime point, namely any point on the worldline of a chosen inertial observer, must be abandoned in favor of doing the same thing independently at each spacetime point, modulo continuity/differentiability conditions. There is one catch. Being creatures of habit, we like global splittings, so even though in general there is no preferred way of doing a global splitting, we can just do it arbitrarily, though clearly matters may be simplified if such a splitting can be adapted to any special structure that may exist in a particular spacetime. One then has to reconcile the local observer splittings with such a global splitting. This too is not anything particularly deep, and it involves using the linear algebra of nonorthogonal bases independently at each spacetime point (since such a global splitting will in general be nonorthogonal) to represent the orthogonal splitting of the local observers on spacetime. The new complication in general relativity is that in general one must deal with a family of so called

1 t ...... t ...... ~v ...... O ...... O ...... x ...... x

z z

......

...... ~v

...... y ...... y ...... x ...... x O O Figure 1.1: Spacetime and Space plus Time. Although spacetime is the arena where calculations are simpler, we always interpret them through our space plus time worldview, which depends on the choice of inertial observer.

2 “test observers” in arbitrary motion, and this introduces the well known effects that accompany noninertial (i.e., accelerated) observers even in nonrelativistic physics. However, in the classical example of a “rigidly rotating” family of noninertial observers in nonrelativistic physics, one still has a global correlation between the different members of the family of observers since one has a global (though not physical) Cartesian coordinate system which rotates with the passage of time. In the extension to general relativity one must consider such effects locally at each spacetime point. The effects of rigid rotation are well known and familiar. There is the centrifugal force that a body feels in the rotating frame even if it is not moving with respect to that frame due to the frame’s acceleration and the Coriolis force that a body feels if it is in motion with respect to that frame due to the frame’s rotation. Quantitatively in a system rotating with constant angular velocity Ω,~ there is a force on an otherwise free body ~x¨ = F~ /m = ~g + ~v × H,~ (1.1) where the “gravitoelectric” force (per unit mass)

~g = −A~ = −Ω~ × V~ = −Ω~ × (Ω~ × ~x) (1.2) is the negative of the acceleration field A~ of the rotating observers, whose velocity field is V = Ω~ ×~x. Similarly the “gravitomagnetic” force is the cross product of the body’s velocity ~v = ~x˙ in the rotating system with the “gravitomagnetic” vector field H~ = 2Ω~ = curl V~ , which is twice the angular velocity vector of the rotating frame and equals the local vorticity of the velocity field. The “gravitoelectromagnetic” terminology due to Thorne is in direct analogy with the Lorentz force of electromagnetism in a nonrotating system

(m/q)~x¨ = F~ /q = E~ + ~v × B.~ (1.3)

Note that the “gravitomagnetic” vector field H~ = curl V~ admits a vector potential V~ in the same way that the magnetic field B~ = curl A~ locally admits a vector potential. Similarly the “gravitoelectric” vector field

1 ~ ~ ~g = − grad[− 2 V · V ] (1.4)

admits a scalar potential in this time-independent case just like the conservative electric field E~ in electro- statics. The velocity field of the rotating observers and minus half its length squared (once multiplied by the mass m) serve as vector and scalar potentials for the noninertial forces we often call “fictitious” forces in classical mechanics, and they are directly interpretable in terms of kinematical properties of the velocity field of the family of noninertial observers used to describe the motion of the body being studied. These forces vanish as soon as we require the members of the family of observers to be inertial. In a curved spacetime such global frames are not immediately available, so one must analyse the situation in the local rest space of each observer in the family of test observers used to describe the physics in 3-dimensional form. The kinematical properties of the 4-velocity field of these observers in spacetime, with some extra complication, directly generalize the above problem, leading to the introduction of “gravitoelectromagnetic” forces which enter into the force equation when expressed in terms of a family of noninertial observers. Rather than doing a global comparison of their motion with respect to a global inertial frame which does not exist in curved spacetime, it must be a local comparison at each point of spacetime with a suitable inertial observer having the same 4-velocity and a set of nonrotating spatial axes. Of course there are new features in curved spacetime which have no analogy in electromagnetism and this has to do with the spatial metric which describes the relative distances between nearby test observers as well as the different proper times that different observers use at the same event in spacetime, both of which depend on the gravitational field. These features, as well as the potentials for the gravitoelectromagnetic vector forces, are contained in the spacetime metric. Taking into account the local proper time complicates slightly the above analogy which relies on a global proper time function on flat spacetime. Thus we need relativistic definitions of acceleration, gradient, curl, time derivative, etc. This together with the details of the way in which an observer in spacetime measures quantities at a given event will enable us to push our present knowledge about special relativity and noninertial motion to the case of

3 general relativity. It helps explain the so called ADM or three-plus-one approach to general relativity as well as the slightly diﬀerent Landau-Lifshitz approach and shows their similarities and diﬀerences, both of which involve the structure of a family of test observers together with a global time function on spacetime. It also shows how both relate to the Ehlers-Hawking-Ellis splitting approach, which is based only on a family of test observers with no global time function assumed to be available.

1.2 Why bother?

You might say, why invest a lot of time into understanding the details of this way of looking at all the different splitting approaches used in general relativity? After all, relativity physics liberated us from the prison of 3- dimensional language into the arena of spacetime where the true nature of kinematics and dynamics became much simpler to understand. Why try to climb back into the cage of 3-dimensional physics? Wasn’t centuries of solitary confinement enough? Well, many spacetimes and idealized problems we use in understanding gravitational theory practically beg us to do this. A rotating black hole spacetime, for example, has two very different privileged families of test observers, one of which is suited to the ADM picture and the other to the Landau-Lifshitz picture. The one we use depends on the question we want to study. Both turn out to be useful. The Ehlers-Hawking- Ellis picture provides the means to relate these two different pictures to each other, which is important if they both turn out to be needed, as they in fact do. The failure to investigate this “relativity of spacetime splitting formalisms” has kept most of us from having better intuition not only about black holes, or even the relativistic picture of the nonrelativistic problem of rotating coordinate systems (which continues to confuse people even now), but other interesting rotating spacetimes like the Gödeluniverse. Many of us are somewhat familiar with (or at least aware of) the initial value problem for the ADM splitting of spacetime based on a family of spacelike hypersurfaces, but the corresponding problem for the splitting of Landau and Lifshitz based on a timelike congruence is almost unknown. In fact the exact solution industry for stationary spacetimes studies precisely this problem without the difficulties that breaking the stationarity symmetry introduces, since only the initial value problem remains of the Einstein equations in that case. The choice of metric variables made in studying this problem is adapted to the Landau-Lifshitz splitting, though few stop to think about its geometric significance or realize it is a decomposition parallel to the ADM splitting. Even if none of these special spacetimes interests you, if you are interested in any post-Newtonian calculations of more realistic, say isolated self-gravitating systems, then understanding the splitting game helps to make a little more sense out of what is universally done in that field. In recent years many references to “the gravitomagnetic field” have sprung up, but since no one took the time to unambiguously define just what this field was, controversy has blossomed between different schools of thought about just what a “real gravitomagnetic field” is. People can agree on a naive definition of the field in this stationary weak-field limit but for strong fields many different definitions are possible depending on the choices one makes for the way in which observers make measurements. All of these different choices can be fit into a general framework and related to each other. Of course the big difficulty in general relativity has always been the limited analytical computations that can be done exactly. An entire industry has grown up around the approximation schemes for general relativity which in practice involves rather complicated details. Certainly the naive things one can do for relatively simple exact solutions in terms of interpreting them in 3-dimensional form cannot be extended easily to the more realistic calculations done in post-Newtonian approximations to general relativity. However, one can interpret features of the approximate analysis in terms of the exact spacetimes which it is meant to approximate. Unfortunately many aspects of this exact geometry being approximated seem to be lost in the details of the approximation scheme itself. The language of the present discussion can help put these details into some perspective. The splitting of the gravitational field is closely related to the splitting of the electromagnetic field, though historically reversed in direction, since the individual electric and magnetic fields of classical Newtonian physics were unified into a single spacetime field by special relativity, while the spacetime metric of general relativity only later gave birth to the electric- and magnetic-like gravitational fields accompanied by the spatial metric. The four-dimensional form of Maxwell’s equations in a curved spacetime is very elegant

4 and powerful, perhaps because it is independent of any particular observers or local coordinates. However, in many practical applications, spacetime is endowed at least locally with either a preferred congruence of integral curves of a timelike vector field or a preferred slicing by a family of spacelike hypersurfaces or both, and it is convenient to decompose the electromagnetic field in some way using this additional structure. Of course the electric and magnetic fields measured by an observer in spacetime are well defined and easily expressed in an orthonormal frame adapted to the observer’s local rest space, but often coordinate systems and nonorthonormal frames prove more convenient for studying the field equations. In this context one must reinterpret the computational quantities which are naturally introduced in terms of some family of test observers. When the present work began, a clear discussion of the various approaches to formulating Maxwell’s equations in terms of three-dimensional quantities and their relationship to each other did not exist. More- over, the analogous discussion for the gravitational field itself was even more conspicuously absent in the literature. It therefore seemed useful to carefully develop the mathematics of the splitting formalisms in general relativity which provide the foundation for the subsequent splittings of the electromagnetic field as well as other matter fields on spacetime. In so doing one obtains a precise description of spatial gravitational force fields in the different points of view and of their interrelationships, as well as a clear exposition of the similarities between those nonlinear fields and the linear electric and magnetic fields. In view of the evolution of terminology which has taken place, it seems natural to refer to this analogy by the name of gravitoelectromagnetism. The analogy between the linear theory of electromagnetism and the linearized theory of general relativity was noted by Einstein even before arriving at his final formulation of the Einstein equations [Verbin and Nielsen, 2004], who compared the geodesic equations with the Lorentz force law. This analogy for the final theory was then soon spelled out explicitly by Thirring [1918]. Although the analogy in practice is usually considered for the linearized gravitational field, the implications and limitations of this analogy are best seen in the fully nonlinear context of general relativity. Of course as previously noted, an obvious question to ask is “Why bother to split spacetime at all?” Certainly the idea of a four-dimensional spacetime and its local Lorentzian geometry has been an important advance of this century. However, our intuition and experience are decidedly three-dimensional in character, and splittings of spacetime into space plus time allow us to interface better with the four-dimensional information, even when a splitting does not occur naturally. When it does, it can considerably simplify the presentation and interpretation of both the gravitational field and whatever matter fields are present. This is not to say that space-plus-time splittings are always useful. Sometimes 2 + 2-splittings are important, examples of which occur in spherical symmetry or in the null initial value problem. In many other instances splittings are actually unproductive, obscuring the spacetime structure of a problem. However, when splittings are useful, they are worth doing carefully. Many different points of view may be taken in splitting spacetime into space plus time but their interrelationships are rarely considered. This is a useful thing to do since our intuition is based on the standard splitting of flat spacetime, but the three-dimensional quantities which serve as a foundation for this intuition often find themselves associated with distinct points of view in the context of a general splitting of an arbitrary spacetime. The result is that no single point of view captures all aspects of our three-dimensional intuition, and the particular application really should determine the choice which is most appropriate. How- ever, no common mathematical framework and no common notation yet exist to enable one to easily switch point of view or compare different points of view. Most relativists are consequently prisoners of the language of one particular choice. The goal of the present work is to establish such a common mathematical framework to help break down the barriers which exist between different schools of relativists who have settled upon a single choice of point of view. Unfortunately no text on general relativity can spare the space to do justice to this idea of a relativity of splitting formalisms, so we learn general relativity from one school or another but rarely appreciate more than one approach in our working lives even if we are relativists. The present text attempts to provide both a universal language and the detailed formulas which describe the relatively straightforward but systematic analysis of these various approaches.

5 1.3 Starting vocabulary

An overriding necessity in this enterprise is a careful definition of the vocabulary to be employed since the standard labels which occur in these discussions do not have universally accepted interpretations. All of the various splitting points of view can be nicely classified and will be assigned labels according to a neutral scheme independent of any particular surnames, thus sidestepping the issue of who “owns” which ideas. Each splitting point of view is based on two fundamental concepts: measurement and evolution, the realization of which differ for each of the possible choices. Before sketching the history of this topic, it is useful to establish some of the basic terminology to be used in what follows. Of course in order to encompass all of the approaches one finds in the literature in a simple scheme, one must be a bit loose about exactly what mathematical details characterize each of the basic categories into which these approaches will be divided. The notions of time and space are complementary since a “time line” represents “time elapsing at a point fixed in space” while a “time hypersurface” represents “space at a moment of time”. These two different notions of time, the first which focuses on measuring time at a single point of space and the second which is associated with some kind of synchronization of times at different points of space, will be assigned the labels “time” and “space” respectively. These divide the splitting points of view into two categories, those in which a local time direction is fundamental, and those in which a nonlocal correlation of local times, i.e., space is fundamental. In the time category, the “time lines” must be timelike in order to represent a local time direction at each event in spacetime, while in the space category, the “time hypersurfaces” or “spaces” must be spacelike in order to be associated with a moment of time in the usual sense of a Riemannian space (alternatively, in order that orthogonality define a local time direction). Given this division, one may consider a partial splitting or a full splitting depending on whether any additional structure is assumed. Table 1.1 establishes this general classification of points of view and the terminology that will be used to describe it. Given no additional structure, one has only a partial splitting of spacetime, splitting off either the time or the space alone. In the first case, to be called the “congruence point of view,” one has only a timelike congruence at one’s disposal, with a unit timelike tangent vector field u. Spacetime will be assumed to be time-oriented as well as oriented, so one may assume that u is future-pointing. It may then be interpreted as the 4-velocity of a family of test observers whose worldlines are the curves of the congruence, and it determines the local time direction at each point of spacetime. The orthogonal complement of this local time direction in the tangent space is the local rest space LRSu of the test observer at that event. It is exactly this structure that one needs for the measurement process which will be the same for the full and partial splittings in a given category in Table 1. The orthogonal decomposition of the tensor algebra induced by this decomposition of the tangent space at each event will define the measurement process, modulo a final step in which projection along the local time direction is replaced by contraction with u, yielding a collection of “spatial tensor fields” of different rank for each spacetime tensor field that is split. In general the splitting of the tangent spaces does not extend to the spacetime manifold. Only in the special case that the rotation of u vanishes does such an extension exist and one has a family of orthogonal spacelike hypersurfaces which slice the spacetime, leading to a full splitting in this category to be discussed below. The second case in the category of partial splittings of spacetime, that of a spacelike slicing of spacetime with no additional structure, is essentially equivalent to the special case of a nonrotating congruence since every spacelike slicing admits a family of timelike orthogonal trajectories. These are the integral curves of the (rotation-free) unit normal vector field n to the slicing, which may be assumed to be future-pointing. The accompanying point of view, for the sake of completeness, might be called the “hypersurface point of view”. Its measurement process is associated with the normal congruence, taking u = n as the 4-velocity of the family of test observers who do the measuring. The local rest spaces of this family are integrable and coincide with the subspaces of the tangent space which are tangent to the slicing. A full splitting of spacetime at the manifold level requires both a slicing of the spacetime and a congruence, to be referred to as a “threading” of the spacetime, together with a compatibility condition that the two families be everywhere transversal. Such a structure will be called a “nonlinear reference frame” in order to distinguish it first from the terms “reference frame,” “frame of reference,” “reference system” and “system of reference” that one finds in the literature, second from the connotation of “frame” in the context of a linear frame of vector fields, and third from related terminology which occurs in the discussion of globally

6 time: 1 space: 3

(“single-observer” time) (“moment of time”)

time: 1 space: 3 ...... PARTIAL SPLITTING ...... u or LRSu ...... GE, GM ﬁelds ...... (3) congruence p.o.v. (4) hypersurface p.o.v.

time + space: 1 + 3 space + time: 3 + 1

...... FULL SPLITTING ...... parametrized nonlinear ...... a ...... reference frame: {t, x } ...... GE, GM ﬁelds ...... and potentials ...... (2) threading p.o.v. (1) slicing p.o.v.

spacelike local rest timelike observers spaces (timelike normal TIME gauge ↔ threading observers) ↔ slicing

arbitrary synchronization arbitrary identiﬁcation SPACE gauge of observer times of “points of space” ↔ slicing ↔ threading

Table 1.1: A characterization of the diﬀerent points of view (p.o.v.) that may be adopted in splitting spacetime. Solid lines in diagrams imply the use of the appropriate causality condition while dashed lines indicate that no causality condition is assumed. The hypersurface p.o.v. is essentially equivalent to the vorticity-free congruence p.o.v. The reference p.o.v. corresponds to a full splitting in which no causality assumptions are made.

7 constant frames in flat spacetime. A “parametrized nonlinear reference frame” will consist of a nonlinear reference frame together with a choice of parametrization of the family of slices. Such a parametrization defines a specific “time function” t on the spacetime which in turn provides an obvious parametrization for each curve in the threading congruence. In the category of full splittings, the distinguishing criterion is the causality condition imposed on the nonlinear reference frame. In the “slicing point of view” the slicing is assumed to be spacelike, but no assumption is made about the causality properties of the threading, which serves only as a way of identifying the points on different slices. In the “threading point of view,” the threading is assumed to be timelike, but no assumption is made about the causality properties of the slicing, which serves only to synchronize in some arbitrary fashion points on different curves in the congruence. If both causality conditions hold, then both points of view hold and one can transform between them. On the other hand it can also be useful in the case that at least one of the two causality conditions holds to not take advantage of that condition and exploit only the structure of the nonlinear reference frame that does not depend on it. This leads to the “reference point of view,” whose measurement process is associated with the nonorthogonal decomposition of the tangent space into the direct sum of a 1-dimensional subspace tangent to the threading and a three- dimensional subspace tangent to the slicing. One can always relate either the slicing or threading points of view to this acausal approach, which is the way in which they are usually represented in a local coordinate system adapted to the nonlinear reference frame. The partial splittings may be related to the full splittings in different ways. In the threading point of view one may define a (future-pointing) unit timelike tangent vector field m along the threading congruence, while in the slicing point of view one has the (future-pointing) timelike unit normal vector field n. By making the respective choices o = m and o = n (“o” for “observer”) of the 4-velocity of a privileged family of test observers in these two points of view, the identification u = o relates each of them to a corresponding congruence point of view described above, defining for each a measurement process. When both the slicing and threading points of view hold, then a unique boost in each tangent space relates the two timelike unit vectors m and n and this may be extended to a transformation of the measurement process. In the special case of an orthogonal nonlinear reference frame (one for which both the causality conditions hold and the slicing and threading are everywhere orthogonal), then m = n and the two points of view coincide. Evolution is defined first by a choice of a 1-parameter group of diffeomorphisms of the spacetime into itself which in some sense advances into the future (either its orbits are timelike or it pushes certain spacelike hypersurfaces into their future), and second by a choice of transport along its orbits for the spatial fields of the given point of view. For a partial splitting only one congruence is available and it is timelike. In the absense of additional structure one can take u or n respectively in the congruence or hypersurface points of view as the generator of such a group, and choose either spatially-projected (“spatial”) Lie transport (“noncovariant” but integrable) or spatially-projected parallel transport (“covariant” but in general nonintegrable) along this congruence. The latter transport of spatial fields coincides with Fermi-Walker transport which defines locally nonrotating axes along a worldline. Each of these choices may be extended to the full splitting in its category but it is the spatial Lie transport along the threading congruence which defines the evolution relative to the nonlinear reference frame, since fields which are “rigidly” attached to this frame do not evolve with this choice. However, unlike Fermi-Walker transport, spatial Lie transport is in general incompatible with orthonormal frames. A compromise between the two kinds of transport leads to co-rotating Fermi-Walker transport, which is the closest one can get to attaching an orthonormal frame to the nonlinear reference frame.

1.4 Historical background

Armed with this initial vocabulary, the historical background may be sketched in a way that puts the diﬀerent formalisms into some perspective. The slicing and threading points of view today are introduced to most of us through two leading textbooks, respectively Gravitation by Misner, Thorne, and Wheeler [1973] and The Classical Theory of Fields by Landau and Lifshitz [1975], each of which carefully avoids mention of the “competing” point of view. Both points of view can be traced back to the early forties when the ﬁrst edition of the Landau-Lifshitz text [1941] introduced the threading point of view splitting of the spacetime metric and, in the stationary case, of the spacetime connection to yield spatial gravitational

8 forces, as still described in their last edition. Soon after, Lichnerowicz [1944] introduced the beginnings of the slicing point of view with an article discussing the initial value problem in an orthogonal nonlinear reference frame. His later book on general relativity and electromagnetism [Lichnerowicz 1955] curiously enough makes use of the threading split, but in actual applications uses an orthogonal nonlinear reference frame in which the two points of view agree. The threading point of view apparently dominated during the fifties when much interest was focused on the equations of motion for test particles. Møller discussed a parametrization-dependent definition of spatial gravitational forces for a general spacetime in the first edition of his text The Theory of Relativity [Møller 1952] at the beginning of the decade. This was then refined to a parametrization-independent splitting later in the decade by Zel’manov [1956, 1959] in the Soviet Union and then independently by Cattaneo [1958,1959a,b,c] in Italy. Unfortunately most of the few references to these works that do appear in the literature cite papers written in Russian, Italian or French, so one must dig to find English versions. The most accessible discussion of much of this material is the second edition of Møller’s text [1972] which describes it in detail and also contrasts it with his original splitting. However, for some reason this text itself is not very prominent among the relativity texts one usually encounters for those of us who have entered the field since the sixties, perhaps because of its old-fashioned viewpoint. Of these authors, only Zel’manov [1956] discussed the splitting of Einstein’s equations in the general case. Meanwhile the slicing point of view was further developed during the fifties by Choquet-Bruhat [1956] and Dirac [1959]. Choquet-Bruhat extended Lichnerowicz’s initial value discussion to the general case, identifying but not naming the lapse and shift variables of the slicing point of view using orthonormal frame techniques. Dirac recognized the significance of the slicing point of view metric decomposition for the Hamiltonian dynamics of general relativity and its relation to his theory of constrained Hamiltonian systems. This was then refined in a series of papers at the turn of the decade by Arnowit, Deser and Misner who used the Hamiltonian formulation permitted by the slicing point of view to study the true degrees of freedom of the gravitational field, culminating in an often cited review article [Arnowit, Deser and Misner 1962]. This ushered in the new era of domination of the splitting scene by the slicing point of view, pushed by the problem of quantum gravity where Hamiltonian techniques have played a rather important role in an endless quest that has not yet met success. The notation of Arnowit, Deser and Misner, soon labeled by Wheeler’s lapse and shift terminology [Wheeler 1964] and later effectively propagated by the text of Misner, Thorne and Wheeler [1973], has found widespread acceptance. The slicing point of view is also commonly referred to as the “3+1” or ADM formalism. A number of useful reviews of various aspects of this formalism exist, among them being articles by York [1979], Isenberg and Nester [1980], Fischer and Marsden [1978, 1979] and Gotay et al [1991]. The term “1 + 3” formalism with its obvious change in emphasis has been suggested as an alternative label for the threading point of view. Although Cattaneo stopped his analysis of the threading point of view at the connection level, Cattaneo- Gasperini [1961, 1963] and Ferrarese [1963, 1965] continued it to the curvature level, studying the splitting of the spacetime curvature and of Einstein’s equations, and the various definitions of spatial curvature that are possible. This previously unexplored area of differential geometry dealing with a degenerate but nonintegrable connection, namely the spatial covariant derivative or transverse covariant derivative which occurs in the threading and congruence points of view, has never been fully understood in the context of the initial value problem for those points of view. This problem, which has been discussed by Stachel [1980] and Ferrarese [1987,1988, 1989], is of a completely different character than in the slicing point of view where it is rather well understood, and its resolution is still an open problem. The approach of Cattaneo and Ferrarese to the threading point of view was reformulated by Massa [1974a,b,c] and used to discuss gyroscope precession [Massa and Zordan, 1975]. This latter problem, more than any other, has focused attention on the effects of spatial gravitational fields. More recently Perjes [1988] and Abramowicz [1988, 1990] have considered variations of Møller’s parametrization-dependent threading approach. In the slicing point of view the natural extension of the splitting of the metric to the splitting of the connection and the discussion of spatial gravitational forces has only recently been considered, perhaps forgotten in the emphasis on the initial value problem and Hamiltonian dynamics. The roots of this discussion can be traced back to the original threading point of view work, although this link is not made apparent in citations. At the crossover point between the popularity of the threading and slicing points of view, Forward [1961] described the analogy between electromagnetism and linearized general relativity using the reference point of view, in a slight variation of Møller’s formalism. In the late seventies this article then

9 inspired a reference point of view discussion of the PPN formalism by Braginsky, Caves and Thorne [1977] who introduced an “electric-type” gravitational field and a “magnetic-type” gravitational field, the latter of which became the “gravitomagnetic” field of the eighties in a discussion of linearized general relativity by Braginsky, Polnarev and Thorne [1984]. In linearized gravity, in the usual weak field slow motion discussions, the corresponding spatial gravitational fields defined in the threading, slicing and reference points of view are very closely related and agree to the lowest order, although not to full post-Newtonian order. Curiously enough, it is always the threading fields which are used in post-Newtonian discussions even when the slicing point of view is advocated before linearization. The introduction of the terminology “gravitoelectric” field and the first discussion of slicing spatial gravitational forces finally appeared in the text Black Holes: The Membrane Paradigm by Thorne et al [1986], but only in the case of a stationary gravitational field in their treatment of black hole spacetimes. This may be extended in an obvious way to general spacetimes in a notation which shows the close relationship to the threading point of view as will be described below. The congruence point of view is briefly introduced in an article by Hawking [1966] and at great length in a pair of articles by Ellis [1971, 1973] at the beginning of the seventies (later updated by Ellis and van Elst [1998]), all based on earlier unifying work of Ehlers [1961] and of Kundt and Trümper [1962] unavailable in English until the Ehlers article alone finally appeared in tranlation over three decades later [1994]. Completing the congruence vector field u to an orthonormal frame leads to the explicit orthonormal frame approach of Estabrook and Wahlquist [1964], who in a note added in proof in their article thank Pirani for calling their attention to Cattaneo’s work and cite articles in French; they were not the first or the last to have been as they say “completely unaware of this work.” (Cattaneo himself appears to have been completely unaware of Zel’manov’s work, while the present authors were completely unaware of both when this project was begun.) The hypersurface point of view is described in the article by Zel’manov [1973] in the early seventies in terms of a nonlinear reference frame, but some work is required to decipher his notation. Ehlers [1961] and Ellis [1971] also treat the hypersurface point of view as a special case of the congruence point of view. These various splittings of spacetime are particularly interesting in the case of electromagnetism, where all of our intuition is tied to individual electric and magnetic fields, and astrophysical applications can be aided by allowing this intuition to find expression in the context of a splitting. Early work by Ruffini and collaborators [Hanni and Ruffini 1973, Hanni and Ruffini 1975, Ruffini and Wilson 1975, Damour and Ruffini 1975, Hanni 1977, Damour et al 1978, Ruffini 1978] followed up in more detail by Damour [1978, 1982] revealed the utility of introducing the concept of electric and magnetic fields in studying black hole systems. This was discussed later in great detail from the slicing point of view by Thorne and Macdonald [1982], who summarize the history of the different splittings in general and as applied to Maxwell’s equations, and by Thorne et al [1986] for application to black hole systems. Maxwell’s equations may be expressed in the congruence point of view as done by Ellis [1973], in the threading point of view as done by Benvenuti [1960] in an application of Cattaneo’s formalism unfortunately appearing only in Italian, and in the slicing point of view as described by Misner, Thorne and Wheeler [1973] (for the correct vector potential splitting, see Isenberg and Nester [1980], for example). The reference point of view splitting is much older, dating back to the beginnings of general relativity in an article by Tamm [1924], as noted by Skrotsky [1957] and Plebanski [1960]. This appears in a peculiar mix of the reference and threading points of view in an exercise in the Landau and Lifshitz text [1975]. The mystery of this latter approach has caused its share of confusion about how one should define electric and magnetic fields in applications in general relativity. Hanni [1977] has given the complementary version which mixes the reference and slicing points of view. It is rather interesting to compare each of these numerous splittings of Maxwell’s equations with the others. Maxwell’s equations from the slicing point of view, first considered by Misner and Wheeler [1957] using the language of differential forms, are discussed in great detail by Thorne and Macdonald [1982] and by Thorne et al [1986]. An enormous language barrier exists at present between the slicing and threading points of view, prevent- ing those versed in the formalism and notation of one from easily penetrating the other or understanding how the two are related. This is exactly the problem the present exposition hopes to address, namely the lack of a common mathematical framework to discuss both approaches on an equal footing. Both of these splitting formalisms can be developed in a completely parallel way as complementary aspects of a single geometrical structure imposed on spacetime (the nonlinear reference frame), aspects which in a close way are related by

10 the same duality that links contravariant and covariant fields on the spacetime manifold. Furthermore, each of these approaches has important ties with the congruence point of view which invariantly describes the geometry of the observer congruence and with the reference point of view which links these discussions to adapted coordinate systems in practice. The style of the threading approach as usually presented is somewhat more cumbersome than the slicing one, so it will be recast in the slicing style, generalizing from adapted coordinate systems to adapted local frames. In the special case of an orthogonal slicing and threading of a spacetime, the two descriptions will then coincide. This standardization of ideas and notation can also prove useful in approximation techniques, including the recent axiomatization of the idea of a Newtonian limit of general relativity based on a family of spacetime splittings as discussed by Ehlers [1989], Lottermoser [1989], and Schmidt [Ehlers, Schmidt and Lottermoser 1990]. The relatively unknown geometry of the spatial connection of the congruence and threading points of view is also currently of interest in the reformulation of gauge-invariant perturbation theory for Friedmann- Robertson-Walker spacetimes by Ellis and coworkers [Ellis and Bruni 1989, Ellis, Hwang, and Bruni 1989]. Each of these limits concern special cases of the problem of perturbation theory for a general spacetime from the somewhat unfamiliar alternative points of view to be discussed in this monograph. Chapter 2 describes the congruence point of view, which is later used to define the measurement process for the slicing and threading points of view. Chapter 3 studies the splitting geometry associated with a nonlinear reference frame and discusses both the slicing and threading points of view and their relationship to the congruence point of view and the reference point of view. Chapter 4 discusses electromagnetism in detail from each of the points of view while Chapter 5 considers the very important case of stationary spacetimes and the Sagnac effect and synchronization questions. The special case of flat spacetime in rotating coordinates clarifies the relationship of the spatial gravitational forces to the centrifugal and Coriolis forces. Chapter 6 discusses the weak field, slow motion limit and almost Friedmann-Robertson-Walker spacetimes. Although this text contains many formulas, only a few simple ideas applied in a methodical way are be- hind much of the detail. Orthogonal decomposition, by itself or represented in the context of a nonorthogonal decomposition, is at the heart of the algebra of respectively partial and full splittings of spacetime. Orthog- onal projection of differential operators in the same spirit then provides the differential tools necessary to extend the splitting algebra to include derivatives.

1.5 Orthogonalization in the Lorentzian plane

The heart of linking spacetime splittings to coordinate systems is a simple idea, but one which receives little attention in our academic preparation: the use of nonorthogonal coordinate systems. It is very useful to look at the Euclidean and Lorentzian 2-planes E2 and M 2 to recall ﬁrst a case for which our geometric intuition holds and then see how it diﬀers in the spacetime arena.

The Euclidean example: general linear coordinates Given any two nonzero linearly independent vectors (X,Y ) in R2 with its usual Euclidean structure, then expressing the position vector r = xX + yY deﬁnes its coordinates (x, y), which may be thought of as functions on R2. Then the self-dot product of the position vector deﬁnes the quadratic distance formula

~r · ~r = Ax2 + Bxy + Cy2 , (1.5) where A = X · X,B = X · Y,C = Y · Y. (1.6) Assume this is positive-deﬁnite: B2 − 4AC < 0, A > 0, C > 0. Figure 1.5 illustrates the geometry. There are two choices for completing the square on the quadratic form:

B B2 B2 B r · r = A(x + y)2 + (C − )y2 = (A − )x2 + C(y + x)2 , 2A 4A 2C 2C (1.7) B2 B2 = A(x)2 + (C − )(y)2 = (A − )(x=)2 + C(y=)2 , 4A 4C

11 ...... y ...... Y ...... ~r ...... x ... X ...... = .... y ...... B y ...... − X .. . 2A ...... = ...... Y . ... Y . Y = Y ...... ~r . . . ~r ...... x ...... X .. . X = X ...... B ...... − Y ...... 2C = ...... X ...... x=

Figure 1.2: General linear coordinates on E2 and their two orthogonalizations. The dashed lines indicate the unit coordinate rectangle and the dotted lines the projections of a typical vector along the coordinate axes. where new adapted coordinates are deﬁned by B B x = x + y , x = x − y , x= = x , x = x= , 2A 2A (1.8) B B y = y , y = y , y= = y + x , y = y= − x= , 2C 2C The new coordinate systems correspond to the new bases

= = r = xX + yY = xX + yY = x=X + y=Y (1.9) which transform in a complimentary way

= B = B = X = X,X = X, X = X − Y,X = X + Y, 2C 2C (1.10) B B = = Y = Y − X,Y = Y + X, Y = Y,Y = Y, 2A 2A In each case one basis vector is retained and the additional new basis vector is obtained by orthogonal projection of the other, a process complimentary to completing the square on the corresponding coordinates. Figure 1.5 illustrates these projections and the new unit coordinate rectangles.

The Lorentzian case: general linear coordinates The Lorentzian case of two-dimensional Minkowski space M 2 is similar but our geometric intuition no longer holds.

12 Given a pair of vectors (X,T ) in M 2, one spacelike, the other timelike, then expressing the position vector r = xX + tT deﬁnes its coordinates (x, t), which may be thought of as functions on M 2. Then the self-dot product of the position vector deﬁnes the quadratic distance formula

~r · ~r = Ax2 + Bxt + Ct2 , (1.11) where A = X · X,B = X · T,C = T · T. (1.12) Assume this is a Lorentzian inner product: B2 −4AC > 0, A > 0, C < 0. Figure 1.5 illustrates the geometry. Again there are two choices for completing the square on the quadratic form:

B B2 B2 B r · r = A(x + t)2 + (C − )t2 = (A − )x2 + C(t + x)2 , 2A 4A 2C 2C (1.13) B2 B2 = = A(x)2 + (C − )(t)2 = (A − )(x=)2 + C(t )2 , 4A 4C where new adapted coordinates are deﬁned by B B x = x + t , x = x − t , x= = x , x = x= , 2A 2A (1.14) = B = B t = t , t = t , t = t + x , t = t − x= , 2C 2C The new coordinate systems correspond to the new bases

= == r = xX + tT = xX + tT = x=X + t T (1.15) which transform in a complimentary way

= B = B = X = X,X = X, X = X − T,X = X + T, 2C 2C (1.16) B B = = T = T − X,T = T + X, T = T,T = T, 2A 2A In each case one basis vector is retained and the additional new basis vector is obtained by orthogonal projection of the other, a process complimentary to completing the square on the corresponding coordinates. Figure 1.5 illustrates these projections and the new unit coordinate rectangles. The first orthogonalization remains the same, but the second orthogonalization changes due to the change in sign of C. This whole discussion may be transferred to a tangent space where the starting basis vectors are two frame vectors and the coordinate functions correspond exactly to the dual 1-forms. If one pictures a 1-form σ geometrically in the tangent space by associating with it the two subspaces σ(~r) = 0, σ(~r) = 1, then the parallel sides of the unit coordinate rectangle serve this purpose for the two dual 1-forms. Note that the dual 1-form parallel sides for one basis vector are parallel to the other basis vector so that the natural evaluation gives 0. If we assume that the original basis of the tangent space is a coordinate frame X = ∂x,T = ∂t, then the dual 1-forms are dx and dt. The figure shows that the first orthogonalization does not change dt, while the second one does not change dx. The quadratic form is then the line element

2 2 2 ds = gxx dx + 2gtx dt dx + gtt dt , (1.17) and the space orthogonalization is

2 2 gtx 2 gtx 2 ds = gxx (dx + dt) + (gtt − ) dt gxx gxx x 2 2 2 = gxx (dx + N dt) − N dt , (1.18)

13 ...... t ...... ~r ...... T ...... x ...... X ...... = ...... t ...... t ...... B ...... − X ...... 2A ...... = ...... x ...... ~r . . ... ~r ...... = ...... T . . T .. . . T = T ...... = . B ...... − T ...... 2C ...... X ...... x .. X . X = X ...... Figure 1.3: General linear coordinates on M 2 and their two orthogonalizations. The dashed lines indicate the unit coordinate rectangle and the dotted lines the projections of a typical vector along the coordinate axes. while the time orthogonalization is

2 2 2 gtx 2 gtx 2 ds = (gxx − ) dx + gtt (dt + dx) gtt gtt 2 2 2 = γxx dx − M (dt + Mx dx) . (1.19)

These two distinct completions of the square lead immediately to a suggestive lapse function and shift vector notation for each case which describes the geometry of the associated projections in the context of the original coordinate frame X = ∂x , X · X = gxx , x 2 T = ∂t + N ∂x , T · T = −N , (1.20) = = = X = ∂x + Mx∂t , X · X = γxx , = = = 2 T = ∂t , T · T = −M ,

1.6 Notation and conventions

The conventions of Misner, Thorne and Wheeler [1973] will be followed unless stated otherwise or unless an ambiguity arises. Lower case Greek indices will take the values 0,1,2,3 and lower case Latin indices the values 1,2,3. The signature of the spacetime metric will be (-+++). A convenient mix of index and index- free notation will be used in our discussion in order to bridge the gap between those who feel comfortable without indices and those who do not. This leads to a problem when in the index notation the kernel symbol of an object is used to denote its trace, determinant, norm or other scalar property or if the same kernel symbol is used for two diﬀerent objects. In these cases obvious distinguishing marks must be added

14 to the index-free symbols which in our applications will turn out not to be too cumbersome. For example, “index shifting” with the spacetime metric associates a single kernel symbol with many different objects. The “index lowering” and “index raising” maps associated with the spacetime metric will be denoted by [ and ] respectively, and the symbols S[ and S] will refer to the fully covariant and fully contravariant forms respectively of a given tensor field S. This gives a unique symbol for one-index objects if we identify the symbol alone with the index position which best characterizes the properties of the object or for two-index objects if we agree to use the kernel symbol alone for the mixed object. (4) (4) The spacetime metric tensor itself gαβ will be denoted by g in an index-free notation (and the inverse (4) αβ (4) −1 (4) ] (4) (4) or contravariant metric tensor g by g = g ) so that the symbol g ≡ | det( gαβ)| can denote (the absolute value of) its determinant as is customary in the index notation. The prefix “ (4) ” will distinguish certain four-dimensional objects from related three-dimensional objects for which it is convenient to use the same kernel symbol. Similarly the lapse function N and the shift vector field N α of the slicing formalism lead to the need for using N~ to indicate the shift in an index-free notation. The same problem will exist with the lapse function M and shift 1-form Mα to be introduced below for the threading point of view, so the 1-form = will be distinguished by the index-free symbol M. The two parallel lines over the kernel symbol recall the geometrical interpretation of a covector in terms of a family of parallel planes in its related vector space in the same way that the arrow oversymbol recalls the geometrical interpretation of a vector. These symbols = = interact with those for index-shifting in an obvious way, with N ≡ N~ [ and M~ ≡ M ]. One also needs an index-free notation for contraction of tensors. A generalized vector contraction notation will prove useful. Let the left contraction S T denote the tensor product of the two tensors S and T with a contraction between the rightmost contravariant index of S with the leftmost covariant index of T (i.e., ...α ... S ... T α...), and the right contraction S T the tensor product with a contraction between the leftmost ... α... contravariant index of T with the rightmost covariant index of S (i.e., S ...αT ... ), assuming in each case that such indices exist. Each of these contractions may themselves be generalized to an ordered contraction of sets of p adjacent indices, indicated by p and p respectively, reducing to the previous contractions when p = 1. Finally the spacetime or portion of spacetime under discussion will always be assumed to be orientable and time-orientable and local coordinates and frames will be assumed to be compatible with these orientations when appropriate. The unit oriented volume 4-form (4)η is defined by the component expression in an (4) 1/2 oriented frame ηαβδγ = g αβδγ with 0123 = 1, so that in an (oriented) orthonormal frame one has 0123 η0123 = 1 = −η . The dual of a p-form will then be represented in index form as a left contraction with (4)η as in Misner, Thorne and Wheeler. For a 2-form F the order does not matter and the dual is

∗ 1 (4) γδ Fαβ = 2 Fγδ η αβ . (1.21)

One must be careful when comparing formulas involving (4)η with the convention which uses the ordered indices 1,2,3,4 as in Hawking and Ellis [1973] rather than 0,1,2,3. This problem (among others) aﬀects formulas for the vorticity of a timelike congruence. The one exception to the conventions of Misner, Thorne and Wheeler made in the present text is a reversal of the order of the covariant indices on the symbol for the components of the connection in a spacetime frame {eα} following Hawking and Ellis [1977]

(4) (4) γ ∇eα eβ = Γ αβeγ = eβ;α . (1.22) This convention might be called the “del” convention in contrast with the “semicolon” convention in which the order of the covariant indices instead follows the semicolon notation. Both the comma and subscripted partial notation will be used to indicate the frame derivatives of functions

eαf = ∂αf = f,α . (1.23)

One notational conﬂict between the old fashioned component notation and modern frame notation is the convention which doesn’t distinguish between the derivatives of components and the components of α α derivatives. For example, £XY = [X,Y ] conventionally denotes the components of the Lie derivative of

15 the vector ﬁeld Y and not the Lie derivative of its components, which are instead represented by XY α = α β Y ,βX . On the other hand in the frame notation it is natural to write for the product rule

α α α α £X(Y eα) = (£XY )eα + Y (£Xeα) = [£XY ] eα , (1.24)

α α and here the symbol £XY = XY instead really does stand for the Lie derivative of the components. The context should always make clear which meaning is intended, and if it doesn’t, an explicit comment will. The notation introduced here may be found to be somewhat cumbersome in a particular application by someone already familiar with that particular application and accustomed to a more streamlined choice of symbols. However, the point is not to develop a notation that is the simplest to use in a specific application, but one which is capable of describing unambiguously all possible applications and their relationships. One may always later adopt an abbreviated notation for specific calculations by elimination of some of the qualifying marks on the kernel symbols. Given this introduction to the notational philosophy to be employed in the text, one can find in Appendix A a list of actual formulas from differential geometry that will be required. All discussion will be local in character. The region of four-dimensional spacetime where the discussion is valid will be designated by (4)M, and this region will be referred to simply as spacetime. A good example to keep in mind is the exterior of the event horizon in a black hole spacetime, for example, where the Boyer-Lindquist coordinates are valid. The threading point of view is valid outside the ergosphere (the time coordinate lines are are timelike), while the slicing point of view is valid outside the event horizon (the time coordinate hypersurfaces are spacelike).

16 Chapter 2

The congruence point of view and the measurement process

Although special relativity may be studied with all of the tools of general relativity, the existence of a special class of “inertial observers” in Minkowski spacetime enables one to simplify considerably the analysis of its spacetime geometry. With each such observer in Minkowski spacetime one can associate an entire family of such observers of zero relative velocity filling the spacetime and one may synchronize the proper times of all of them so that they describe a global time function for the family. This family constitutes the global reference frame of the given observer. The 4-velocity field of this family of observers is a covariant constant timelike unit vector field on Minkowski space. Two inertial observers in relative motion lead to two distinct such global reference systems. Although the worldlines of two particular inertial observers in relative motion may be transformed into each other by an active Poincarétransformation of Minkowski space, their associated reference frames may be transformed into each other with a pure Lorentz tranformation or “boost” of Minkowski spacetime into itself about some arbitrary point. This in turn may be identified with the boost which maps the 4-velocity associated with the first reference frame onto that of the second in the tangent space at each point of Minkowski spacetime. It is this globally constant boost (the mixed tensor field which represents it is covariant constant) which allows one transform the measurements of physical quantities at some spacetime point from one family of observers to the other. However, in a general spacetime in general relativity, no such preferred families of inertial observers exist, and one must rely on arbitrary families of observers to establish a reference frame with which to measure physical quantities. The global boost of special relativity relating two families of inertial observers gives way to a point by point boost between the 4-velocities of the observers in the tangent space at each spacetime point associated with two families of observers covering the spacetime. Thus all of the algebra associated with a “single” boost of Minkowski space between two inertial reference frames must be repeated at each point independently in general relativity in order to relate the two reference frames associated with two families of observers. Furthermore, the differential operators associated with a single family of observers necessarily reflect the state of motion of those observers, leading to the description of noninertial effects due to their acceleration and relative rotation. The analysis of these questions is the natural generalization of special relativity kinematics to general relativity.

2.1 Algebra

Suppose u is a future-pointing timelike unit vector ﬁeld on our spacetime (4)M

(4) α β (4) αβ α −1 = gαβu u = g uαuβ = uαu , (2.1) −1 = (4)g(u, u) = (4)g−1(u[, u[) = u[(u) .

17 It may be interpreted as the 4-velocity field of a family of test observers whose worldlines are the integral curves of u, each of which may be parametrized by the proper time τu which is defined up to an additive constant on each curve. This defines an observer congruence on the spacetime which may be used to induce a pointwise orthogonal decomposition of the spacetime tensor algebra. (Sachs and Wu [1977] call u itself a “reference frame” and its proper time parametrized integral curves the “observers”.)

2.1.1 Observer orthogonal decomposition The orthogonal decomposition of the spacetime tensor algebra begins with the orthogonal direct sum of each tangent space into the “local time axis” along u and the “local rest space” LRSu orthogonal to u. This may be accomplished using the orthogonal projection operators T (u) for the temporal projection and P (u) for 1 the spatial projection. Each of them can be identified with a 1 -tensor field, defined by α α [ T (u) β = −u uβ ,T (u) = −u ⊗ u , (2.2) α α α (4) [ P (u) β = δ β + u uβ ,P (u) = id + u ⊗ u , which are related to the identity tensor, the metric tensor and its inverse tensor in the following way

α α α (4) δ β = T (u) β + P (u) β , id = T (u) + P (u) , (4) (4) [ [ gαβ = T (u)αβ + P (u)αβ , g = T (u) + P (u) , (2.3) (4)gαβ = T (u)αβ + P (u)αβ , (4)g−1 = T (u)] + P (u)] . Here T suggests “time” or “tangential” (to the congruence), while P suggests “perpendicular” or “projection”. The latter makes sense since often attention is focused on the spatial projection tensor P (u) which is conventionally denoted by the kernel symbol h, while the temporal projection T (u) is usually suppressed in favor of contraction with the timelike unit vector ﬁeld or 1-form. 1 1 -tensor ﬁelds may be interpreted as linear transformations of each tangent space into itself α α β X → A βX ,X → A X (2.4) and the right contraction corresponds to matrix multiplication of the component matrices in the index- notation, both in this action and in their successive action by composition

α γ α A γ B β = [A B] β . (2.5) It is convenient to extend the “square” notation for the self-composition of a linear map with the corresponding self-right-contraction of the tensor ﬁeld

2 α α γ 2 [A ] β = A γ A β ,A = A A. (2.6) With these notational conventions, the orthogonal projection tensor ﬁelds then satisfy the usual relations for an orthogonal decomposition α γ α 2 P (u) γ P (u) β = P (u) β ,P (u) = P (u) , α γ α 2 T (u) γ T (u) β = T (u) β ,T (u) = T (u) , (2.7) α γ α γ P (u) γ T (u) β = T (u) γ P (u) β = 0 ,P (u) T (u) = T (u) P (u) = 0 , which follow from the unit character of u and its orthogonality to the spatial projection tensor

α β α P (u) βu = 0 = uαP (u) β , (2.8) P (u) u = 0 = u[ P (u) . It is convenient to extend the projection operators to act on a tensor S by understanding P (u)S to mean the projection of S on each index by the projection P (u), i.e., contraction of S on each index by the appropriate index of P (u) α... α δ γ... [P (u)S] = P (u) γ ··· P (u) β ··· S , β... δ... (2.9) α... α δ γ... [T (u)S] β... = T (u) γ ··· T (u) β ··· S δ... .

18 These give the purely spatial and purely temporal parts of S. The orthogonal decomposition of S is accomplished by extending the orthogonal decomposition to each of its indices, and expanding out the result

α... α δ γ... S β... = δ γ ··· δ β ··· S δ... α α δ δ γ... = [T (u) γ + P (u) γ ] ··· [T (u) β + P (u) β] ··· S δ... (2.10) α... α... = [T (u)S] β... + (mixed terms) + [P (u)S] β... . The coeﬃcent tensors of those factors of u with a free index in this formula deﬁne a family of tensors of all ranks between zero and the rank of S. These are the tensors which result from replacing the projection δ along u with contraction by −u or −uγ as appropriate. They represent the spatial projections of all possible contractions of S with any number of factors of −u or −u[ as appropriate, and can be easily used to reconstruct S by factoring in the appropriate missing factors of u[ and u respectively. The reduction of a tensor S to such a family of tensors will be called the measurement of S by the observer congruence. A tensor which has been spatially projected on all of its free indices is called a spatial tensor and gives zero upon contraction of any index with u. All of the members of the family of tensors which result from the 1 measurement of a single spacetime tensor are spatial tensors. For example, the measurement of a 1 -tensor S leads to the following result in the index notation

α S β ↔ γ δ α γ δ γ δ α γ δ {S δu uγ , −P (u) γ S δu , −uγ S δP (u) β,P (u) γ S δP (u) β} . (2.11) | {z } | {z } | {z } | {z } scalar vector 1−form 1 (1)−tensor

P (u) and its index-shifted forms are all spatial tensors which result from the purely spatial part of the measurement of the identity tensor and the metric and its inverse. The remaining parts of these measurements are trivial (4)id ↔ {−1, 0, 0,P (u)} , (4)g ↔ {−1, 0,P (u)[} , (2.12) (4)g−1 ↔ {−1, 0,P (u)]} . For the symmetric metric tensor and its inverse the two rank-one mixed parts are equal and vanish, while for the mixed identity tensor, the two distinct rank one parts are the zero vector field and the zero 1-form. α In each case the rank zero or spatial scalar part is the constant −1 = uαu . When tensors have symmetries among their contravariant indices or among their covariant indices, the measurement process may be collapsed to list only the distinct spatial fields which result from the measurement, as done above for the metric and its inverse. The simplest case occurs for p-forms, which give only two distinct fields since at most one contraction with u is possible before spatially projecting the result. No contraction yields the purely spatial part or “magnetic part” p-form and one contraction yields the single mixed part or “electric part” (p − 1)-form. This splitting may be expressed in the form

(E) α (E) [S (u)]α ...α = −u Sαα ...α ,S (u) = −u S, 1 p−1 1 p−1 (2.13) (M) (M) [S (u)]α1...αp = [P (u)S]α1...αp ,S (u) = P (u)S, while the orthogonal decomposition of S has the form

(E) (M) Sα ...α = p!u[α S (u)α ...α ] + S (u)α ...α , 1 p 1 2 p 1 p (2.14) S = u[ ∧ S(E)(u) + S(M)(u) .

This is equivalent to the following decomposition of the identity operator ALT (the antisymmetrizer) on diﬀerential forms ALT = u[ ∧ (−u ) + ALT ◦P (u) , (2.15)

19 or in index notation, of the generalized Kronecker delta tensor δ(p)

α ...α δ 1 p = p!δα1 ··· δαp β1...βp [β1 βp] α ...α (2.16) = −p!u[α1 u δα2 ··· δαp] + δ(u) 1 p , [β1 β2 βp] β1...βp where the spatial generalized Kronecker delta tensor δ(u)(p) = P (u)δ(p) is deﬁned by

α ...α δ(u) 1 p = p!P (u)[α1 ··· P (u)αp] . (2.17) β1...βp [β1 βp]

Notice that the contraction u S is automatically spatial due to the antisymmetry of S (since u (u S) = 0 α β or equivalently u u Sβαγ3...γp = 0) and so requires no subsequent spatial projection. The (unit oriented) volume 4-form (4)η may be split in this way but only its electric part survives since it is easily shown that the magnetic part P (u)(4)η vanishes. The electric part reversed in sign

(4) δ (4) η(u) = u η , η(u)αβγ = u ηδαβγ (2.18)

deﬁnes the (unit oriented) volume 3-form for the local rest space LRSu. In an (oriented, time-oriented) 0123 123 orthonormal frame adapted to u and LRSu, then η0123 = 1 = −η and η(u)123 = 1 = η(u) . This spatial volume 3-form deﬁnes the spatial duality operation ∗(u) for antisymmetric spatial tensors and the usual cross product of two vectors X and Y in the local rest space

∗(u) α α β γ X ×u Y = [X ∧ Y ] , [X ×u Y ] = η(u) βγ X Y . (2.19)

One can also introduce the spatial “dot product” between two spatial vectors in order to mimic Euclidean vector analysis [ α β X ·u Y = P (u) (X,Y ) = P (u)αβX Y , (2.20) This just gives the spacetime inner product of the two spatial vectors. Note that one can replace the spacetime metric tensor in any contraction with spatial tensors by the appropriate valence spatial projection which acts as the metric on the local rest space and its tensor algebra. The spatial volume 3-form η(u) is used to define the spatial duality operation “ ∗(u) ” for spatial differential forms 1 ∗(u)S = S η(u)β1···βp , (2.21) α1···α3−p p! β1···βp α1···α3−p It satisfies ∗(u) ∗(u)S = S. Using this spatial operation, one can split the spacetime duality operation “ ∗ ” on differential forms. Assuming the right dual convention of Misner, Thorne and Wheeler [1973] this is given by 1 ∗S = S ηβ1···βp , (2.22) α1···α4−p p! β1···βp α1···α4−p This operation instead satisfies ∗ ∗S = (−1)p−1S for a p-form S. Using the definition of the electric and magnetic parts of a differential form, one finds for a p-form S

[ ∗S](E)(u) = (−1)p−1 ∗(u)S(M)(u) , (2.23) [ ∗S](M)(u) = ∗(u)S(E)(u) .

α In order to have unique index-free symbols for the three valence forms of a 2-form, let F (F β) be the [ ] αβ mixed valence form of a 2-form F (Fαβ) and let F (F ) be the 2-vector or contravariant form of the ﬁeld. Then the above relationship with S = F [ is

[ ∗F [](E)(u) = −∗(u)F [(M)(u) , [ ∗F [](M)(u) = ∗(u)F [(E)(u) . (2.24)

20 2.1.2 Observer-adapted frames Components with respect to a frame adapted to the observer orthogonal decomposition can be quite useful α in the splitting game. An observer-adapted frame {eα} with dual frame {ω } will be any frame for which e0 is along u and the “spatial frame” {ea} spans the local rest space LRSu

−1 [ u = L e0 ≡ e> , u (ea) = 0 , (2.25) u[ = −Lω0 ≡ −ω> , ωa(u) = 0 .

The orientability and time-orientability assumptions imply L > 0 and η(u)123 > 0. Thus for a time-oriented oriented orthonormal observer-adapted frame, one has L = 1 or e0 = u. A frame which only has L = 1 will be called partially normalized. The splitting of a tensor field S amounts to a partitioning of the components in an observer-adapted frame according to whether or not individual indices are zero or not. The purely spatial part corresponds to those components which have only “spatial indices” 1,2,3, while the purely temporal part corresponds to the single component with every index equal to the “temporal index” 0. The remaining components parametrize the mixed parts of the tensor. Spatial tensors have only the spatially-indexed components nonzero. Thus any tensor equation in index-form involving only spatial tensor fields, when expressed in an observer-adapted frame, reduces to an equation whose only nonzero components correspond to replacing Greek indices with Latin indices. 1 For a 1 -tensor S one has 0 a 0 a S ↔ {S 0,S 0,S a,S b} (2.26) > a > a ↔ {S >,S >,S a,S b} , where the index “>” (pronounced “tan”) suggests “tangential” to the congruence (or even “time” in this context) and corresponds to the orthonormal temporal component obtained by scaling the zero-indexed frame component by the normalization factor L (evaluation on contravariant arguments on e> and covariant arguments on ω>). This family of spatial fields differs by a sign from the contraction/projection algorithm [ > [ described above which picks off the coefficients of u and u rather than of e> = u and ω = −u in the expansion of a tensor. One can represent this decomposition in index-free notation by using an index “⊥” (pronounced “perp”) standing for “perpendicular” to the congruence in place of the spatial index

> ⊥ > ⊥ S ↔ {S >,S >,S ⊥,S ⊥} . (2.27)

The symbol > was selected for being the reﬂection of the symbol ⊥ to a dual position. Similarly a p-form S splits in component form in the following way

{Sα1...αp } ↔ {S0a2...ap ,Sa1...ap }

↔ {S>a2...ap ,Sa1...ap } (2.28) (E) (M) ↔ {S>⊥...⊥,S⊥...⊥} = {−S (u),S (u)} .

The components of the spacetime metric and its inverse in such a frame may be represented in the conventional form (4) 2 (4) (4) g00 = −L , g0a = 0 , gab = P (u)ab = hab , (2.29) (4)g00 = −L−2 , (4)g0a = 0 , (4)gab = P (u)ab = hab , or equivalently (4) 2 0 0 a b g = −L ω ⊗ ω + habω ⊗ ω > > a b = −ω ⊗ ω + habω ⊗ ω , (2.30) (4) −1 −2 ab g = −L e0 ⊗ e0 + h ea ⊗ eb ab = −e> ⊗ e> + h ea ⊗ eb . The kernel symbol P (u) is here being replaced by h as often used in speciﬁc formalisms where the frame is understood to be adapted to a given congruence. The spacetime metric determinant factor has the expression

21 (4) 1/2 1/2 (4) 1/2 g = Lh , while the oriented spatial volume 3-form has components η(u)abc = η>abc = h abc. The three valence forms of the spatial projection are

[ a b ] ab a b a P (u) = habω ⊗ ω ,P (u) = h ea ⊗ eb ,P (u) = h bea ⊗ ω = ea ⊗ ω . (2.31)

Note that the cross product and dot product have the explicit expressions

a a b c 1/2 ad b c [X ×u Y ] = η(u) bcX Y = h h dbcX Y , (2.32) a b X ·u Y = habX Y .

It is also quite useful to consider frames which are only adapted to one of the two subspaces of the orthogonal decomposition of the tangent spaces associated with a given family of test observers. An “observer- partially-adapted frame” {eα} will be a frame for which either 1) e0 is along u or 2) {ea} spans LRSu (equivalently ω0 is along u[), i.e., that spacetime frame contains either a subframe for the distribution of local time directions or for the distribution of local rest spaces. Both conditions together characterize the observer-adapted frames. Those observer-partially-adapted frames which are not observer-adapted will be considered below in the context of a nonlinear reference frame. For an electromagnetic 2-form F [, the electric and magnetic parts defined above correspond directly to α the electric field and the spatial dual of the magnetic field. It is convenient to let F (in index notation F β) be the mixed form of the 2-form and E(u) = F u and B(u) be the vector fields in the index-free notation, so one has E(u)[ = F [(E)(u) = −u F [ = F [ u , (2.33) B(u)[ = ∗(u)F [(M)(u) ,F [(M)(u) = ∗(u)B(u)[ , or equivalently β E(u)α = Fαβu , (2.34) α 1 αβγ 1 αβγδ B(u) = 2 η(u) Fβγ = − 2 η Fβγ uδ . In an observer-adapted frame, these definitions take the index-form

E(u)a = −F>a = Fa> , (2.35) 1 bc c B(u)a = 2 η(u)a Fbc ,Fab = η(u)ab B(u)c . One may re-express the 2-form and its spacetime dual in terms of the electric and magnetic 1-forms in the following way

[ [ [ ∗(u) [ γδ F = u ∧ E(u) + B(u) ,Fαβ = 2u[αE(u)β] + ηαβ uγ B(u)δ , (2.36) ∗ [ [ [ ∗(u) [ ∗ γδ F = −u ∧ B(u) + E(u) , Fαβ = −2u[αB(u)β] + ηαβ uγ E(u)δ .

22 2.1.3 Relative kinematics: algebra Suppose U is another unit timelike vector field representing a different family of test observers. One may then examine how the measurements of the two families of observers are related to each other. This involves as well the measurement of the various projection and boost maps between their local rest spaces which are necessary to interpret the “transformation” from the spatial quantities and operators of one set to those of the other. The relative velocities of one set of observers with respect to the other are defined by measuring the other’s 4-velocity U = γ(U, u)[u + ν(U, u)] , (2.37) u = γ(u, U)[U + ν(u, U)] . The relative velocity ν(U, u) of U with respect to u is spatial with respect to u and vice versa. Both have α 1/2 the same magnitude ||ν(U, u)|| = [ν(U, u)αν(U, u) ] , while the common gamma factor is related to that magnitude by −(4)g(U, u) = γ(U, u) = γ(u, U) = [1 − ||ν(U, u)||2]−1/2 . (2.38) Letν ˆ(U, u) = ||ν(U, u)||−1ν(U, u) be the unit vector giving the direction of the relative velocity ν(U, u), when nonzero. Introduce also the energy and spatial momentum per unit mass relative to u

E˜(U, u) = γ(U, u) , p˜(U, u) = γ(U, u)ν(U, u) . (2.39)

Multiplication of these quantities by the nonzero rest mass m of a test particle whose worldline coincides with one of the curves of the congruence of U yields its energy E(U, u) and spatial momentum p(U, u) as seen by the observers with 4-velocity u. These in turn are the result of the measurement by u of the 4-momentum P = mU of the test particle. In addition to the natural parametrization of the worldlines of U by the proper time τU , one may introduce a new parametrization τ(U,u) by dτ(U,u)/dτU = γ(U, u) . (2.40) This corresponds to the sequence of proper times of the family of observers from the u congruence which cross paths with a given worldline of the U congruence. It is convenient to abbreviate γ(U, u) by γ when its meaning is clear from the context in order to simplify the appearance of equations which involve this factor. When the relative velocity is nonzero, U and u deﬁne a 2-dimensional subspace of each tangent space called the relative observer plane. Its orthogonal complement is the subspace LRSu ∩LRSU common to both local rest spaces representing in each the directions orthogonal to the direction of relative motion. Equations (2.37) describe a unique active “relative observer boost” B(U, u) which acts as the identity on LRSu ∩LRSU such that in the relative observer plane

B(U, u)u = U,B(U, u)ν(U, u) = −ν(u, U) . (2.41)

The inverse boost B(u, U) “brings U to rest” relative to u. It will be convenient to use the same symbol for 1 a linear map of the tangent space into itself and the corresponding 1 -tensor acting by contraction. The right contraction between two such maps will represent their composition. When the contraction symbol is suppressed, the linear map will be implied. The boost B(U, u) restricts to an invertible map B(lrs)(U, u) ≡ P (U) ◦ B(U, u) ◦ P (u): LRSu → LRSU between the local rest spaces which acts as the identity on their common subspace. Similarly the projection P (U) restricts to an invertible relative projection map P (U, u) = P (U) ◦ P (u): LRSu → LRSU with −1 inverse P (U, u) : LRSU → LRSu and vice versa, and these maps also act as the identity on the common subspace of the local rest spaces. The boosts and projections between the local rest spaces diﬀer only by a gamma factor along the direction of motion. It is exactly the inverse projection map which describes Lorentz contraction of lengths along the direction of motion. Figure 2.1.3 illustrates these maps and the relative velocities on the relative observer plane. α If Y ∈ LRSu, then the orthogonality condition 0 = uαY implies that Y has the form

Y = [ν(u, U) ·U P (U, u)Y ]U + P (U, u)Y. (2.42)

23 ...... ν(u, U) ...... ν(U, u) ...... X u... U ...... −νˆ(u, U) ...... νˆ(U, u) ...... −1 −1 . . . P (U, u) X = γ B(u, U)X ...... −1 .. . . B(u, U)X = γ P (u, U)X ...... P (u, U)X ......

Figure 2.1: The relative observer plane, the relative velocities and the associated relative observer maps.

−1 If X = P (U, u)Y ∈ LRSU is the ﬁeld seen by U, then Y = P (U, u) X and

−1 P (U, u) X = [ν(u, U) ·U X]U + X (2.43) = [P (U) + U ⊗ ν(u, U)[] X, which gives a useful expression for the inverse projection in terms of the tensor which represents it. Any map between the local rest spaces may be “measured” by one of the observers, i.e., expressed entirely in terms of quantities which are spatial with respect to that observer. For example, the mixed tensor representing the relative observer projection P (U, u) = P (U) P (u) (2.44) = P (U) P (U, u) = P (U, u) P (u)

(which expands to P (u)+γU ⊗ν(U, u)), corresponding to the linear map P (U, u): LRSu → LRSU , is spatial with respect to u in its covariant index and with respect to U in its contravariant index, i.e., is a “connecting tensor” in the terminology of Schouten [1954]. It has associated with it two tensors

P (U) = P (U, u) P (U, u)−1 , (2.45) P (u) = P (U, u)−1 P (U, u) , which are spatial with respect to U and u respectively and correspond to identity transformations of each local rest space into itself. In the same way any linear map M(U, u): LRSu → LRSU is represented by such a connecting tensor and has associated with it two tensors MU (U, u) and Mu(U, u) which are spatial with respect to U and u respectively and act as linear transformations of the respective local rest spaces into themselves M(U, u) = MU (U, u) P (U, u) = P (U, u) Mu(U, u) , −1 MU (U, u) = M(U, u) P (U, u) , (2.46) −1 Mu(U, u) = P (U, u) M(U, u) . These latter tensors enable one to express the map in terms of the spatial projections of just one of the observers.

24 The relative boosts and projections all involve the further decomposition of each local rest space into the subspaces parallel and perpendicular to the direction of relative motion. Introducing the notation (||) (⊥) LRSu = LRSu ⊕ LRSu (2.47) (⊥) (⊥) for these subspaces, then LRSu = LRSU = LRSu ∩ LRSU is the common subspace of the two local rest spaces and and each of the above maps decompose into maps M (||)(U, u): LRS(||) → LRS(||) , u u (2.48) (⊥) (⊥) (⊥) M (U, u): LRSu → LRSu , where the latter map is the identity map for both the boosts and projections. The individual projections parallel and perpendicular to the direction of relative motion between the local rest spaces and within each local rest space have the representations P (||)(U, u) = −γνˆ(u, U) ⊗ νˆ(U, u)[ ,P (⊥)(U, u) = P (U, u) − P (||)(U, u) , (||) [ (⊥) (||) PU (U, u) =ν ˆ(u, U) ⊗ νˆ(u, U) ,PU (U, u) = P (U) − PU (U, u) , (2.49) (||) [ (⊥) (||) Pu (U, u) =ν ˆ(U, u) ⊗ νˆ(U, u) ,Pu (U, u) = P (u) − Pu (U, u) , where P (U, u)ˆν(u, U) = −γνˆ(U, u) explains the γ factor in the ﬁrst relation. The vector identity

A ×u [B ×u C] = (A ·u C)B − (A ·u B)C (2.50) may be used to express this decomposition of a spatial vector X in the following way (||) (⊥) Pu (U, u)X =ν ˆ(U, u)[ˆν(U, u) ·u X] ,Pu (U, u)X =ν ˆ(U, u) ×u [ˆν(U, u) ×u X] . (2.51) These parallel and perpendicular projections in turn may be used to similarly decompose the boost −1 B(lrs)(U, u) and the inverse projection P (u, U) , for which one has the obvious relations (see Figure 2.1.3)

(||) −1 −1 (||) −2 (||) P (u, U) = γ B(lrs)(U, u) = γ P (U, u) , (2.52) (⊥) −1 (⊥) (⊥) P (u, U) = B(lrs)(U, u) = P (U, u) which may be used to reconstruct the spatial tensors associated with the boost and inverse projection. For example, for the inverse boost B(lrs)(u, U) one has −1 [ B(lrs)u(u, U) = P (u) − γ(γ + 1) ν(U, u) ⊗ ν(U, u) , (2.53) −1 [ B(lrs)U (u, U) = P (U) − γ(γ + 1) ν(u, U) ⊗ ν(u, U) , which follows from the expansion of

(||) (⊥) B(lrs)u(u, U) = B(lrs) u(u, U) + B (u, U)u (lrs) (2.54) (||) −1 (⊥) = Pu (u, U) + γ Pu (u, U) .

Thus if S ∈ LRSU , then its inverse boost is −1 [ B(lrs)(u, U)S = [P (u) − γ(γ + 1) ν(U, u) ⊗ ν(U, u) ] P (u, U)S. (2.55) The map P (u, U, u) = P (u, U)P (U, u) = P (u)P (U)P (u) (2.56) is an isomorphism of LRSu into itself which turns up in manipulations with these maps. It and its inverse (in the sense P (u, U, u)−1P (u, U, u) = P (u)) can be expressed in the following equivalent ways

(⊥) 2 (||) P (u, U, u) = Pu (u) + γ Pu (u, U) , = P (u) + γ2ν(U, u) ⊗ ν(U, u)[ , −1 −1 −1 −1 P (u, U, u) = P (U, u) P (u, U) = Pu(U, u) (2.57) (⊥) −2 (||) = Pu (U, u) + γ Pu (U, u) = P (u) − ν(U, u) ⊗ ν(U, u)[ .

25 The inverse map represents a “double Lorentz contraction” along the direction of relative motion. The map P (U, u)−1 appears in the transformation law for the electric and magnetic ﬁelds. Using the fact that (4) [P (U) F ] ν(u, U) = ν(u, U) ×U B(U) , (2.58) one ﬁnds P (U, u)E(u) = γP (U){(4)F [U + ν(u, U)]} (2.59) = γ[E(U) + ν(u, U) ×U B(U)] , and similarly P (U, u)B(u) = γ[B(U) − ν(u, U) ×U E(U)] . (2.60) Equivalently one may write

−1 E(u) = γP (U, u) [E(U) + ν(u, U) ×U B(U)] , −1 (2.61) B(u) = γP (U, u) [B(U) − ν(u, U) ×U E(U)] .

The transformation of the electric and magnetic ﬁelds takes a more familiar form if one re-expresses it in terms of the parallel/perpendicular decomposition of the boost using equation (2.52)

(||) (||) (||) E (u) = B(lrs)(u, U)E (U) , (2.62) (⊥) (⊥) (⊥) (⊥) E (u) = γB(lrs)(u, U)[E (U) − ν(u, U) ×U B (U)], with analogous expressions for the magnetic ﬁeld. When expressed in a pair of orthonormal frames adapted to the two local rest spaces and related by the boost, these reduce to the familiar component expressions in a direct way. The map γP (u, U)−1 appears rather than the projection P (U, u) in these transformations due to the spatial duality operation. Suppose F (u) is a spatial 2-vector with respect to u. Then F (U) = P (U, u)F (u) is the purely spatial part of this 2-vector as seen by U. Its spatial dual with respect to U is

∗ ∗ F~ (U) = (U )F (U) = [ (U )P (U, u)∗(u)]F~ (u) (2.63) = γP (u, U)−1F~ (u) .

This is easily veriﬁed by an index calculation

α δα βγ σ F (U) = Uδη βγ u η σF (u) δα σ = −Uδu δ σF (u) δ α α σ (2.64) = −Uδu F (u) + u UσF (u) α α σ −1 α = γ[F (u) + u ν(U, u)σF (u) ] = [γP (u, U) F~ (u)] .

2.1.4 Splitting along parametrized spacetime curves and test particle worldlines The gravitational field is felt through its action on test particles, whose paths in spacetime (world lines) are either timelike (nonzero rest mass) or null (zero rest mass) curves. However, other gravitational effects manifest themselves along spacelike curves—synchonization of clocks, for example. It is therefore necessary to consider curves of all three causality types when viewing them in terms of a given observer family. Given any parametrized curve c in spacetime, with parameter λ, its tangent vector V (λ) = c0(λ) is defined in local coordinates by α α 0 α V (τU ) = dx (c (λ)) = d[x ◦ c(λ)]/dλ . (2.65) This tangent vector can be decomposed into its parts parallel and perpendicular to the observer 4-velocity

V (λ)α = V (λ)(||u) uα + [P (u)V (λ)]α , (2.66)

26 where its temporal and spatial parts are respectively

(||u) γ α α γ V (λ) = −uγ V (λ) , [P (u)V (λ)] = P (u) γ V (λ) . (2.67)

Dividing the spatial part by the temporal part defines the relative velocity of the path in spacetime repesented by the curve as seen by the observer family, regardless of its causality type. This relative velocity and (when nonzero) the unit vector defining its direction are defined by

ν(V (λ), u)α = [P (u)V (λ)]α/V (λ)(||u) , (2.68)

and νˆ(V (λ), u)α = sgn V (λ)(||u)[P (u)V (λ)]α/||P (u)V (λ)|| , (2.69) while the relative speed is the magnitude of the relative velocity

||ν(V (λ), u)|| = ||P (u)V (λ)|| . (2.70)

For a null curve, the relative velocity is itself a unit vector and the relative speed is 1. For a curve whose tangent vector is spatial with respect to the observer family, this speed goes to inﬁnity. The relative velocity, direction and speed are invariant under a change of parametrization which scales the tangent vector by the rate of change of one parameter with respect to the other by the chain rule d d dλ d λ → λ(λ0) , → = . (2.71) dλ dλ0 dλ0 dλ

Suppose one has a timelike worldline c parametrized by its proper time λ = τU with unit timelike tangent vector U = c0. In local coordinates one has

α α 0 α U (τU ) = dx (c (τU )) = d[x ◦ c(τU )]/dτU . (2.72)

This worldine may represent the path of a test particle of mass m and charge q in spacetime. Measurements made by a test observer following the same worldline can be related to the sequence of measurements made by the family of diﬀerent test observers of the observer congruence whose worldlines intersect the given worldline. All of the algebra developed above for two families of test observers may be restricted to the given worldline to relate the test particle quantities to the single observer congruence. In particular multiplying the energy and spatial momentum per unit mass by the rest mass yields the the energy and spatial momentum of the test particle E(P, u) = mE˜(U, u) = mγ(U, u) , (2.73) p(P, u) = mp˜(U, u) = mγ(U, u)ν(U, u) , which themselves result from the measurement of the 4-momentum P = mU

P = E(P, u)u + p(P, u) = E(P, u)[u + ν(P, u)] . (2.74)

One may also extend to a null worldline those quantities which do not rely on the timelike nature of P in order to study a test particle with zero rest mass m = 0. In this case one must work directly with the 4-momentum P and the energy and spatial momentum.

2 α 2 2 m = −PαP = E(P, u) − ||p(P, u)|| (2.75)

In the null case m = 0, the relative velocity is a unit vector ν(P, u) =ν ˆ(P, u), corresponding to the unit speed of light. To handle simultaneously the cases of zero and nonzero rest mass, introduce the “rest mass per unit mass” parameterm ˜ which has the value 0 for the ﬁrst case and 1 for the second.

27 2.1.5 Addition of velocities and the aberration map Another map of physical interest between the local rest spaces of two observers is the nonlinear aberration map between their unit spheres which describes the relationship between the apparent directions of a light ray as seen by the two observers. This in turn is a special case of the addition of velocity formula which relates the relative velocities of a given test particle with 4-momentum P as seen by two diﬀerent observers with 4-velocities u and U

P = E(P, u)[u + ν(P, u)] = E(P,U)[U + ν(P,U)] . (2.76)

One easily ﬁnds

E(P,U) = γ(U, u)[1 − ν(U, u) ·u ν(P, u)]E(P, u) , −1 −1 (2.77) ν(P,U) = γ(U, u) [1 − ν(U, u) ·u ν(P, u)] P (U, u)[ν(P, u) − ν(U, u)] .

Re-expressing the relative projection in terms of the boost leads to the formulas

(⊥) −1 −1 (⊥) ν(P,U) = γ(U, u) [1 − ν(U, u) ·u ν(P, u)] ν(P, u) , (2.78) (||) −1 (||) ν(P,U) = [1 − ν(U, u) ·u ν(P, u)] B(lrs)(u, U)[ν(P, u) − ν(U, u)] .

The null case describes the 4-momentum of a photon or the tangent vector to the path of a light ray, depending on the choice of language. The relative velocity is then a unit vector giving the direction of the photon or light ray. Dividing the energy and spatial momentum by Planck’s constant yields the frequency ω(P, u) = E(P, u)/¯h and wave vector k(P, u) = p(P, u)/¯h = ω(P, u)ν(P, u) of the photon. The relationship between the observed frequencies for the two diﬀerent observer congruences determines the relativistic Doppler eﬀect ω(P, u) = γ[1 − ν(U, u) ·u ν(P, u)]ω(P, u) . (2.79) The above relationship between the unit relative velocites is the aberration map. For the special case

ν(P, u)(||) = 0 , ||ν(P, u)(⊥)|| = 1 (2.80) of a light ray orthogonal to the relative motion as seen by u, one has the familiar result

||ν(P,U)(||)|| tan θ = = γ||ν(U, u)|| (2.81) U ||ν(P,U)(⊥)|| for the angle away from the perpendicular direction as seen by U, using the fact that boosts preserve lengths.

28 2.2 Derivatives

The orthogonal decomposition associated with a choice of observer congruence has been used to split each spacetime tensor field into a family of spatial fields which represent its measurement by the observers. The action of a spacetime differential operator on a spacetime tensor field leads to a new tensor field which may be measured. This leads to a representation of the differential operator itself in terms of two independent derivative operators which act on the collection of spatial fields which represent the spacetime tensor field being differentiated. The spatial projection of the derivative along u (when appropriate) leads to a temporal differential operator, while the spatial projection of the differential operator itself yields a spatial version of the differential operator. This splitting process may be applied to the natural derivatives, namely Lie and exterior derivatives, and to the covariant derivative and variations of it like the Fermi-Walker derivative. While the operators so defined preserve the spatial character of tensor fields they differentiate, they may also be applied to nonspatial tensor fields and this is actually useful in certain cases. However, because of the spatial projection, any terms containing undifferented factors of the observer 4-velocity will vanish, breaking the appropriate product rule for tensor (or wedge) products of nonspatial fields, while differentiated factors lead to the appearance of kinematical factors. When a second observer congruence is present, one must also introduce relative derivative operators which preserve the spatial character of fields which are spatial with respect to the original observer congruence. This is necessary in order to measure derivatives along the second observer congruence of the quantities which result from the measurement by the original observer congruence of fields associated with the second congruence. These operators may be restricted to the case of a single timelike worldline of a test particle of nonzero rest mass and suitably extended to the case of zero rest mass in order to study test particles of either type. This leads to the subject of apparent spatial gravitational forces acting on test particles due to the motion of the family of test observers, as well as their consequences for the behavior of test gyroscopes or spinning test particles.

2.2.1 Natural derivatives Two natural derivatives are important for various considerations, the Lie derivative, which for vector fields becomes the Lie bracket, and the exterior derivative. Each of these may be split as described above. The spatial Lie bracket is defined by for arbitrary vector fields X and Y by

[X,Y ](u) = P (u)[X,Y ] = £(u)XY, (2.82) and in general the spatial Lie derivative of an arbitrary tensor S is deﬁned in the obvious way

£(u)X S = P (u)[£X S] . (2.83)

Similarly the spatial exterior derivative of an arbitrary diﬀerential form S is deﬁned by

d(u)S = P (u)dS . (2.84)

Each of these spatial derivative operators can be applied to nonspatial fields S but are perhaps most useful when applied to spatial fields. However, when applied to nonspatial fields both operators violate the various product rules that usually hold since the spatial projection eliminates terms with temporal factors. For example, when S is spatial and the vector field X = fu is a temporal field, the spatial Lie derivative along X has the special property of being linear in the temporal component X> = f due to the orthogonality properties £(u)fuS = f£(u)uS,S spatial . (2.85) This property also holds for the spatial Lie derivative of the metric itself

(4) (4) £(u)fu g = f£(u)u g (2.86) since it diﬀers from the spatial ﬁeld P (u)[ only by the term u[ ⊗ u[ which has zero spatial Lie derivative along u.

29 For the same reason, the spatial exterior derivative of a temporal 1-form X[ = fu[ is linear in the temporal component X> = −f

d(u)[fu[] = P (u)[df ∧ u[ + fdu[] = fd(u)u[ . (2.87)

The derivative £(u)u provides a natural time derivative which will be referred to both as the temporal Lie derivative and as the Lie temporal derivative, while the derivative £(u)X along a spatial vector field X provides a natural spatial derivative. These two operators are necessary to describe the splitting of the Lie derivative. On the other hand, the temporal Lie derivative and the spatial exterior derivative together are necessary to describe the splitting of the exterior derivative. It is convenient to introduce the alternate notation ∇(lie)(u) = £(u)u for the Lie temporal derivative associated with u. This will be needed for efficient handling of three different preferred temporal derivatives.

2.2.2 Covariant derivatives Each spacetime tensor field S can be split into its representative family of spatial fields by the measurement process. Its covariant derivative (4)∇S can be split in exactly the same way and the result can be represented in terms of two projected differential operators acting either on members of the family of spatial fields representing S or on u itself, generating the kinematical quantities associated with the observer congruence. The two projected differential operators are the spatial covariant derivative, for differentiating along spatial directions, and the spatial Fermi derivative, or spatial Fermi-Walker derivative, for differentiating along the local time direction of the observer congruence. They are defined for an arbitrary tensor field S by ∇(u)S = P (u)[(4)∇S] , (2.88) (4) ∇(fw)(u)S = P (u)[ ∇uS] . In index notation it is convenient to introduce the double vertical bar symbol in place of the semicolon for the alternative notation for the spatial covariant derivative

(4) α... (4) α... α... [ ∇S] β...γ = ∇γ S β... = S β...;γ , α... α... α... (2.89) [∇(u)S] β...γ = ∇(u)γ S β... = S β...||γ . For a spatial tensor S, the spatial Fermi-Walker derivative coincides with the Fermi-Walker derivative [Walker 1932] which in turn reduces to the Fermi derivative [Fermi 1922] defined only for spatial tensors. For a nonspatial tensor, the spatial Fermi-Walker derivative and the Fermi-Walker derivative yield distinct results. One might also refer to the spatial Fermi-Walker derivative as the Fermi-Walker temporal derivative in analogy with the Lie temporal derivative, a term which is more suggestive of the direction of differentiation. Similarly the spatial covariant derivative may be applied to a nonspatial tensor, although it is most useful when applied to spatial tensors. One can speak of its spatial part ∇(u) ◦ P (u) and its temporal part ∇(u) ◦ T (u). In the literature the spatial covariant derivative is consistently defined to act only on spatial tensor fields, so only the geometry of the spatial part of spatial covariant derivative ∇(u) as defined here has been explored. This restricted derivative was mentioned by Møller [1952] and studied by Cattaneo [1958,1959] who called it the “transverse covariant derivative”, while its associated curvature was mentioned by Zel’manov [1956] and more deeply explored by Cattaneo-Gasperini [1961] and Ferrarese [1963,1965]. Ellis has recently used the terminology “orthogonal covariant derivative” [Ellis, Hwang and Bruni, 1990]. The spacetime metric and the volume 4-form are obviously covariant constant with respect to the spatial covariant derivative. It is straightforward to show that the spatial projection and each of its index-shifted forms are all covariant constant with respect to the spatial covariant derivative, as is the spatial volume 3-form ∇(u)(4)g = ∇(u)P (u) = ∇(u)P (u)[ = 0 , etc. (2.90) Thus index shifting of spatial fields and the spatial duality operation commute with the spatial covariant derivative. Fermi-Walker transport of a spatial tensor along the observer congruence is defined by requiring its spatial Fermi-Walker derivative to vanish. Similarly spatial parallel transport of a spatial tensor along a

30 curve everywhere orthogonal to the observer congruence is deﬁned by requiring that its spatial covariant derivative along the curve vanish. Under both transports spatial tensors remain spatial by undergoing a unique 1-parameter family of boosts relative to parallel transport in order to preserve the orthogonality with u. Since parallel transport itself preserves inner products, so do these boosted transports, which means that the spatial projections and volume 3-form are also covariant constant with respect to the spatial Fermi-Walker derivative [ ∇(fw)(u)P (u) = ∇(fw)(u)P (u) = ∇(fw)(u)η(u) = 0 , etc. (2.91) This also easily follows from the spatial projection of the vanishing of the covariant derivatives of the spacetime metric and 4-volume form. Thus index-shifting and the spatial duality operations commute with the spatial Fermi-Walker derivative, exactly as for the spatial covariant derivative. Fermi-Walker transport of a spatial vector along the observer congruence has the physical interpretation of describing the behavior of the spin vector of a test torque-free gyroscope (gyro) carried by an observer.

2.2.3 Kinematical quantities (4) α The first nontrivial covariant derivative to split is ∇u, or u ;β in index-notation, namely the covariant derivative of the observer congruence 4-velocity vector field. This mixed tensor may be identified with a linear transformation of the tangent space, which is connected with its physical interpretation. The time projection on the contravariant index vanishes due to the unit nature of u

α 1 α uαu ;β = 2 (uαu );β = 0 , (2.92) so only two of the four terms in the splitting survive. In index-free form one has

(4)∇u = −a(u) ⊗ u[ + ∇(u)u , (4) a(u) = ∇uu = ∇(fw)(u)u , (2.93) ∇(u)u = −k(u) = θ(u) − ω(u) ,

while the index-lowered form is usually given when expressed in the index form

(4) [ (4) [ ∇u ]αβ = ∇βuα = uα;β = θ(u)αβ − ω(u)αβ − a(u)αuβ , γ δ k(u)αβ = −P (u) αP (u) βuγ;δ = −θ(u)αβ + ω(u)αβ , γ δ θ(u)αβ = P (u) αP (u) βu(γ;δ) , (2.94) γ δ ω(u)αβ = −P (u) αP (u) βu[γ;δ] , β a(u)α = uα;βu ,

The expansion tensor θ(u)[ and the rotation or vorticity tensor ω(u)[ are just the symmetric and sign-reversed antisymmetric parts of the spatial covariant derivative of u[. The expansion tensor is then decomposed into its pure-trace and tracefree parts to deﬁne the expansion scalar

α Θ(u) = Tr θ(u) = θ(u) α (2.95)

and the shear tensor

1 1 γ σ(u) = θ(u) − 3 Θ(u)P (u) , σ(u)αβ = θ(u)αβ − 3 θ(u) γ P (u)αβ . (2.96) The expansion scalar, shear tensor and rotation tensor, together with the acceleration vector, are referred to as the kinematical quantities of the vector ﬁeld u or of its associated congruence of integral curves. The acceleration vector ﬁeld is spatial due to the unit character of u

α α β 1 α β uαa(u) = uαu ;βu = 2 [uαu ];βu = 0 . (2.97) If it is identically zero the observer worldlines are geodesics and the observers are inertial observers. However, in contrast with the ﬂat spacetime case, families of inertial observers are often less convenient to use than

31 noninertial observers unless they happen to admit additional special properties shared by the usual translation invariant families of inertial observers in the flat case. This is due to the attractive nature of gravitation which causes the inertial observers to collapse together unless other conditions oppose this behavior. The remaining kinematical properties reside in the spatial tensor k(u). This mixed tensor may be thought of as a linear transformation of the local rest space into itself, and the decomposition of this linear transformation into its irreducible parts has a very physical interpretation. This interpretation involves a comparison with Lie dragging along u, for which the operator £(u)u is relevant. One may express the kinematical quantities of u entirely in terms of natural derivatives acting on u[ or its associated covariant or contravariant spatial projections du[ = −u[ ∧ a(u)[ + 2ω(u)[ , a(u)[ = −[du[](E) = u du[ = £uu[ = £(u)uu[ , [ 1 [ (M) 1 [ ω(u) = 2 [du ] = 2 d(u)u , (2.98) £uP (u)[ = £(u)uP (u)[ = £(u)u(4)g = 2θ(u)[ , £uP (u)] = £(u)uP (u)] = £(u)u(4)g−1 = −2θ(u)] , £uP (u) = u ⊗ a(u)[ , a(u)[ = −u [£uP (u)][ , £(u)uP (u) = 0 . The negative sign in the antisymmetrized covariant derivative used to define the rotation tensor arises because of the convention of adding the additional exterior derivative index to the left while the additional covariant derivative index adds to the right. For a 1-form, the two differ by a sign. The factor of two arises since the antisymmetrized covariant derivative (for a symmetric connection) is half the exterior derivative, modulo the sign difference. The definition of the rotation tensor by Ehlers [1961, 1993] and Ellis [1971] has the opposite sign, while Misner, Thorne and Wheeler [1973] use different signs in the two sections in which kinematical quantities are discussed. Since the spatial Lie derivative by u of the spatial metric yields (twice) the expansion tensor, this derivative operator does not commute with index-shifting on spatial fields when the expansion tensor is nonzero. Similarly the result

£uη(u) = £(u)uη(u) = u £u(4)η = Θ(u)η(u) (2.99) shows that it does not commute with the spatial duality operation either as long as the expansion scalar is nonzero. For example, for a spatial diﬀerential form S one has the relation

[∗(u)£(u)u∗(u)S]] = [£(u)u + Θ(u)]S] . (2.100)

However, the spatial projection tensor P (u) does have zero spatial Lie derivative along u, so the spatial projection operator and the spatial Lie derivative along u do commute

[£(u)u,P (u)] = 0 , (2.101) which means £(u)uP (u)S = P (u)£uS = £(u)uS. (2.102) Thus if a tensor is Lie dragged along u, its Lie derivative by u is zero which implies that the spatial Lie derivative of its spatial projection is also zero. It is natural to define spatial Lie transport of a spatial field along u by requiring that its spatial Lie derivative along u be zero. Then this result means that the spatial projection of Lie transport along u of a tensor yields the spatial Lie transport along u of the spatial projection of that tensor. The rotation tensor characterizes the failure of the Lie bracket of two spatial vector fields to again be spatial. The following short calculation for a pair of spatial vector fields X and Y using the invariant definition of the exterior derivative of a 1-form 2ω[(u)(X,Y ) = du[(X,Y ) = Xu[(Y ) − Y u[(X) − u[([X,Y ]) = −u[([X,Y ]) , (2.103) T (u)[X,Y ] = −u[([X,Y ])u = 2ω(u)[(X,Y )u ,

32 shows that the nonspatial part of this Lie bracket vanishes only if the rotation vanishes. The spatial part defines the spatial Lie bracket of the two spatial vector fields. When the rotation vanishes this corresponds to the Lie bracket defined within the hypersurfaces orthogonal to the congruence. The covariant rotation tensor is just half the spatial exterior derivative of u[, thus agreeing with its classical 3-dimensional definition in the Euclidean case [Synge and Schild 1949]. Its index-raised dual defines the spatial rotation or vorticity vector

~ω(u) = ∗(u)ω(u) = 1 [ ∗(u[ ∧ du[)]] , 2 (2.104) α 1 (4) βαγδ(4) 1 (4) αβγδ ω(u) = 2 uβ η ∇γ uδ = 2 η uβuγ;δ . The sign of this formula is affected both by the choice of index range (0,1,2,3 or 1,2,3,4) and the signature, each of which varies in the literature. For the index range 1,2,3,4, the spatial volume 3-form is defined by the right contraction (4)η u rather than the left contraction, while for the opposite signature (+ − −− or − − −+) the covariant spatial components of a vector field change sign relative to the contravariant spatial components, forcing another sign change to correspond to the classical spatial definition. For example, Ellis (4) [1973] in one of his review articles uses the convention η0123 = −1 in an orthonormal frame with signature − + ++ and defines the covariant rotation tensor as minus the spatial exterior derivative of u[ for a total of two sign changes, leading to the same formula as given here. A stationary spacetime is characterized by the existence of a timelike Killing vector field ξ which Lie derives the metric (4) £ξ gαβ = 2ξ(α;β) = 0 . (2.105) The 4-velocity vector field of a congruence of “Killing observers” can be defined by normalizing the Killing vector field u = `−1ξ , ` = |(4)g(ξ, ξ)|1/2 . (2.106) The normalization factor is also stationary, i.e., invariant under this symmetry, since

(4) α β (4) α β £ξ[ gαβξ ξ ] = [£ξ gαβ]ξ ξ = 0 (2.107) implies £ξ` = 0. For such an observer congruence, a “stationary observer congruence”, the expansion tensor vanishes

1 (4) 1 −1 (4) 1 −1 (4) θ(u) = 2 £(u)u g = 2 ` £(u)ξ g = 2 ` P (u)£ξ g = 0 . (2.108) Similarly the acceleration admits a spatial potential which is the natural logarithm of the normalization factor ` [ [ −1 [ a(u) = £(u)uu = P (u)£ −1 [` ξ ] = ... ` ξ (2.109) = d(u) ln ` = d ln ` , where the last equality follows from the stationarity of `. Only in the event that u itself is a Killing vector ﬁeld must the normalization factor be constant, leading to vanishing acceleration.

2.2.4 Splitting the exterior derivative The rotation and acceleration appear in the splitting of the spacetime exterior derivative, which may be expressed in terms of the spatial exterior derivative and the temporal Lie derivative in the same way that the spacetime covariant derivative can be expressed in terms of the spatial covariant derivative and the spatial Fermi-Walker derivative, together with the kinematical quantities. To carry out the splitting of the exterior derivative, one needs several identities. Wedging the splitting identity (2.14) by u[ yields the useful preliminary identity u[ ∧ S = u[ ∧ P (u)S. (2.110) The Lie derivative of a diﬀerential form is related to the exterior derivative by the well known identity

£uS = d(u S) + u dS . (2.111)

33 For a spatial diﬀerential form S, the contraction of S by u vanishes (and u dS is automatically spatial) so

u dS = £uS = £(u)uS , u S = 0 , (2.112) and hence applying the splitting identity (2.14) one obtains

dS = −u[ ∧ £(u)uS + d(u)S , u S = 0 . (2.113)

Using these identities it is straightforward to evaluate the electric and magnetic parts of the exterior derivative of an arbitrary diﬀerential form

[dS](E) = −[d(u) + a(u)[∧]S(E) − £(u)uS(M) , (2.114) [dS](M) = d(u)S(M) + 2ω(u)[ ∧ S(E) .

The measurement of the identity 0 = d2u[ leads to identities for the spatial exterior derivatives of the acceleration and rotation forms. Just substituting S = du[ = −u[ ∧ a(u)[ + 2ω(u)[ into the splitting identity for the exterior derivative leads to

0 = (d2u[)(E) = [d(u) + a(u)[∧]a(u)[ − 2£(u)uω(u)[ , (2.115) 0 = (d2u[)(M) = 2[d(u)ω(u)[ − ω(u)[ ∧ a(u)[] ,

leading to the relations d(u)a(u)[ = 2£(u)uω(u)[ , (2.116) d(u)ω(u)[ = ω(u)[ ∧ a(u)[ . For an arbitrary diﬀerential form S, the splitting of d2S is obtained by iterating the splitting of the exterior derivative, with the result

0 = (d2S)(E) = {d(u)2 − 2ω(u)[ ∧ £(u)u}S(E) + {d(u)a(u)[ − 2£(u)uω(u)[} ∧ S(E) + {−[£(u)u, d(u)] + a(u)[ ∧ £(u)u}S(M) , (2.117) 0 = (d2S)(M) = {d(u)2 − 2ω(u)[ ∧ £(u)u}S(M) + 2{d(u)ω(u)[ − ω(u)[ ∧ a(u)[} ∧ S(E) .

Since S(E) and S(M) are independent, the terms involving each must separately vanish. Two of these terms vanish identically due to the identity for u[ itself (S(E) = 1 and S(M) = 0), while a third gives an identity for the commutator of the two natural derivatives acting on a spatial diﬀerential form

[£(u)u, d(u)] = a(u)[ ∧ £(u)uS , u S = 0 . (2.118)

This leaves the result that the measurement of d2 acting on a differential form leads to the operator d(u)2 − 2ω(u)[ ∧ £(u)u acting on each of the two splitting fields which represent that differential form. The vanishing of this operator then gives an identity for d(u)2 acting on any spatial differential form

d(u)2S = 2ω(u)[ ∧ £(u)uS , u S = 0 . (2.119)

Using this result, one can evaluate d(u)2 acting on an arbitrary diﬀerential form, leading to the result

d(u)2S = 2ω(u)[ ∧ [£(u)uS − d(u)(u S)] . (2.120)

34 2.2.5 Splitting the differential form divergence operator One may merge the splitting of the exterior derivative d and of the duality operation ∗ to obtain the splitting of the divergence operator δ, defined for a differential form S on spacetime by

δS = ∗d ∗S. (2.121)

This takes a more familiar form in index notation using the covariant derivative (assuming zero torsion)

(4) β [δS]α1...αp−1 = − ∇βS α1...αp−1 . (2.122)

The spatial divergence operator has an additional sign; for a spatial p-form S one has

δ(u)S = (−1)p∗(u)d(u)∗(u)S, β (2.123) [δ(u)S]α1...αp−1 = −∇(u)βS α1...αp−1 .

By manipulating the deﬁnitions and identities one ﬁnds the result

[δS](E) = −δ(u)S(E) − 2ω(u)] 2 S(M) , (2.124) [δS](M) = [δ(u)−??a(u) ]S(M) − ∗(u)£(u)u∗(u)S(E) .

In applying the last formula, the identity (2.100) is useful in re-expressing the term ∗(u)£(u)u∗(u)S, especially for a spatial 1-form S.

2.2.6 Spatial vector analysis For functions and spatial vector fields, one can introduce the covariant analogs of the gradient, curl and divergence operations in order to make contact with traditional vector analysis which takes place in Euclidean space. These spatial operations are defined in an obvious way in terms of the spatial covariant derivative. Let ∇~ (u) denote the covariant derivative operator with its differentiating index raised, i.e., [∇~ (u)f]α = ∇(u)αf. These operators are related to the spatial exterior derivative in the usual manner

~ ] gradu f = ∇(u)f = [d(u)f] ,

∗(u) [ ] curlu X = ∇~ (u) ×u X = [ (d(u)X )] , (2.125)

∗(u) ∗(u) [ divu X = ∇~ (u) ·u X = [d(u) X ] ,

where X is assumed to be a spatial vector field. However, nothing prevents their application to nonspatial fields. The spatial curl of an arbitrary vector field can be written in index form as

α αβγ (4) [curlu X] = η(u) ∇βXγ . (2.126)

Since many second rank tensors also enter into vector analysis in the form of linear transformations of vectors, one can follow the leftright arrow notation of Thorne et al [1986] for contravariant second rank spatial tensors ↔ α αβ γ [ S ·u X~ ] = S P (u)βγ X . (2.127) In the old-fashioned language such tensors are called “dyads”. Thus the spatial vorticity vector of u is just the spatial dual of half the contravariant form of the spatial exterior derivative of u, which deﬁnes the vorticity dyad ↔ω (u) = ω(u)] , (2.128) ∗(u)↔ 1 ~ω(u) = ω (u) = 2 curlu u . Allowing the spatial curl operator to act on the nonspatial vector u, one sees that the vorticity vector is half its curl. Expressing this in an orthonormal frame in terms of which u has a nonrelativistic relative spatial

35 velocity leads to the classical expression of nonrelativistic mechanics [Synge and Schild 1949]. With these conventions the linear transformation corresponding to the rotation tensor becomes

α β α β γ ω(u) X = −~ω(u) ×u X , ω(u) βX = −η βγ ω(u) X (2.129) when expressed in terms of the rotation vector. The spatial exterior derivative of the rotation 2-form itself as given above may also be rewritten in the 3-vector notation α α divu ~ω(u) = a(u) ·u ~ω(u) , ∇(u)αω(u) = a(u)αω(u) . (2.130) The identity for the exterior derivative of the acceleration 1-form becomes the following when rewritten

1 ∗(u) [ 2 curlu a(u) = [£(u)uω(u) ] = £(u)u~ω(u) + Θ(u)~ω(u) = ∇(fw)(u)~ω(u) − [θ(u) − P (u)Θ(u)] ~ω(u) , (2.131) 1 α 2 α α β 2 [curlu a(u)] = [∇(fw)(u) + 3 Θ(u)]~ω(u) − σ(u) βω(u) , where the expansion scalar appears from the commutation of the spatial Lie derivative and the spatial duality operation.

36 2.2.7 Ordinary and Co-rotating Fermi-Walker derivatives Suppose one refers to a tensor as being of “of type σ”, where σ is the gl(4,R)-representation corresponding to the GL(4,R)-representation under which the tensor transforms under a change of frame. Then the representation function σ is deﬁned by

α... α γ... γ α... [σ(A)S] β... = A γ S β... + · · · − A βS γ... − · · · . (2.132) With this notation, the Fermi-Walker derivative along the observer congruence may be deﬁned by

(4) (4) [ ∇(fw)(u) = ∇u − σ(u ∧ a(u) ) , (2.133) where the argument of σ is the antisymmetric tensor (mixed form)

[ α α α [u ∧ a(u) ] β = u a(u)β − a(u) uβ . (2.134)

Such an operator satisfies the obvious product laws with respect to tensor products and contractions thereof. It is easily verified the (4)g, (4)η, u, P (u), T (u), and all index-shifted variations of these tensor fields have an identically vanishing Fermi-Walker derivative. This means that the Fermi-Walker derivative commutes with the measurement process: the measurement of the Fermi-Walker derivative of a spacetime tensor yields the Fermi-Walker derivative acting on each member of the family of spatial tensors which represent that tensor. A tensor field is said to undergo Fermi-Walker transport along u if its Fermi-Walker derivative is zero, in analogy with parallel transport along u, to which it reduces for a geodesic congruence with vanishing acceleration. Thus (4)g, (4)η, u, P (u), and T (u) all undergo such Fermi-Walker transport, while only (4)g and (4)η are parallel transported along u. The effect of the difference term in the two derivatives is to generate an active boost in the velocity-acceleration plane which maps the parallel transport of u along u onto u itself. This same boost maps the parallel transport of any tensor field along u onto the Fermi-Walker transported tensor field along u. Both transports preserve inner products and commute with index-shifting and duality operations, but the Fermi-Walker derivative preserves the orthogonal projection and measurement operations associated with u as well. The Fermi-Walker derivative of a spatial tensor field is also spatial with respect to u. 1 In fact, the addition to the covariant derivative of a term −σ(A), where A is a 1 -tensor field, will generate a continuous Lorentz transformation relative to parallel transport along the curve or congruence as long as A is antisymmetric with respect to the metric, i.e., is the mixed form of a 2-form. Such a derivative will automatically commute with index shifting and its corresponding transport will preserve inner products, as does the Fermi-Walker derivative. If A is also spatial with respect to u then it will generate a continuous rotation of the local rest space of u while leaving u itself uneffected, and both u and u[ will still have vanishing derivative with respect to the new derivative (as will the metric and the spatial and temporal projection tensors). For a congruence of timelike curves, one may define a co-rotating Fermi-Walker derivative which further adapts the covariant derivative to the congruence by adding a term to the Fermi-Walker derivative which leads to a co-rotation of the local rest spaces relative to the congruence under the corresponding transport

(4) (4) ∇(cfw)(u) = ∇(fw)(u) + σ(ω(u)) (2.135) = (4)∇u − σ(u ∧ a(u)[ − ω(u)) .

A vector which is transported by the co-rotating Fermi-Walker transport has the same angular velocity as the relative velocity vector to be deﬁned below, but does not exhibit the eﬀects of the expansion and shear that the latter does. The spatial Fermi-Walker derivative has been introduced by spatially projecting the covariant derivative (4) (4) ∇u and not the Fermi-Walker derivative ∇(fw)(u), in order to have a temporal derivative one can apply (4) to u itself (since ∇(fw)(u)u = 0 while ∇(fw)(u)u = a(u))

(4) [ P (u) ∇(fw)(u) = ∇(fw)(u) − P (u)σ(u ∧ a(u) ) . (2.136)

37 The difference term only contributes to the derivative of nonspatial fields. In the same spirit one can define the spatial co-rotating Fermi-Walker derivative along u or equivalently the co-rotating Fermi-Walker temporal derivative along u ∇(cfw)(u) = ∇(fw)(u) + σ(ω(u)) . (2.137) This spatial derivative is related to its corresponding spacetime derivative in the same way as the ordinary Fermi-Walker derivatives are related

(4) [ P (u) ∇(cfw)(u) = ∇(cfw)(u) − P (u)σ(u ∧ a(u) ) . (2.138)

On spatial fields the spacetime and spatial ordinary and co-rotating Fermi-Walker derivatives agree, and so the measurement of the spacetime derivative leads to the action of the corresponding spatial derivative on each member of the family of spatial fields which represent a given spacetime field. With the present definitions, the derivatives of a multiple of u itself are

∇(cfw)(u)[fu] = ∇(fw)(u)[fu] = fa(u) , (2.139) which determines their action on the nonspatial parts of a tensor field. The alternative would be to define all the spatial operators by projection of the corresponding spacetime operators and introduce a new temporal derivative to handle nonspatial tensor fields (identical to the present definition of ∇(fw)(u)).

38 2.2.8 Relation between Lie and Fermi-Walker temporal derivatives The interpretation of the kinematical quanitites depends on the relationship between the Lie temporal derivative and the Fermi-Walker temporal derivatives. This in turn depends on the more general relationship between the Lie and covariant derivatives themselves. The formulas for the components of the covariant and Lie derivatives in a coordinate frame {eα} charac- α α terized by C βγ = ω ([eβ, eγ ]) = 0 are

(4) α... α... γ (4) α... [ ∇uS] β... = S β...,γ u + [σ( Γ(u))S] β... , α... α... γ α... [£uS] β... = S β...,γ u − [σ(∂u)S] β... , (2.140) α... γ (4) α... = S β...;γ u − [σ( ∇u)S] β... ,

(4) α α γ α α where the matrix arguments of σ are the suggestive abbreviations Γ(u) β = Γ γβu ,(∂u) β = u ,β, and (4) α α ( ∇u) β = u ;β. The last expression for the Lie derivative is true for an arbitrary frame and is the result of the “comma to semicolon rule” valid for any symmetric connection in a coordinate frame

£uS = (4)∇uS − σ((4)∇u)S. (2.141) Taking the spatial projection of this equation yields

(4) £(u)uS = ∇(fw)(u)S − P (u)σ( ∇u)S (2.142) for an arbitrary tensor but £(u)uS = ∇(fw)(u)S + σ(k(u))S (2.143) for a spatial tensor. The kinematical tensor k(u) thus acts as the linear transformation of the local rest space which describes the difference between spatial Lie transport and Fermi-Walker transport of spatial tensors along u. These two transports and their corresponding differential operators embody two possible choices for describing “evolution”, which might be called “Lie” and “Fermi-Walker” evolution respectively from their origins in derivative operators which correspond to those labels. In the absence of further structure, evolution can only be defined independently along each observer worldline and then only relative to a transport of the local rest space along each such worldline which defines “no evolution”. Note that for k(u) itself the difference between the spatial Fermi-Walker derivative and the spatial Lie derivative along u vanishes so

£(u)uk(u) = ∇(fw)(u)k(u) . (2.144) The Lie derivative along u has the following representation in terms of the ordinary and co-rotating Fermi-Walker derivatives

£u = (4)∇u − σ((4)∇u) = (4)∇u + σ(ω(u) − θ(u) + a(u) ⊗ u[) (4) [ = ∇(fw)(u) + σ(ω(u) − θ(u) + u ⊗ a(u) ) (2.145) (4) [ = ∇(cfw)(u) + σ(−θ(u) + u ⊗ a(u) ) . In the diﬀerence term relative to the co-rotating Fermi-Walker derivative, the expansion term in the Lie derivative generates a deformation of the local rest space of u while the acceleration term tilts the local rest space relative to u along the acceleration direction, breaking the orthogonality. Only in the case that u is a Killing vector ﬁeld do both of these terms vanish

£u(4)g = 0 → θ(u) = 0 = a(u) , (2.146) in which case the Lie and co-rotating Fermi-Walker derivatives coincide. However, a unit timelike Killing vector ﬁeld is perhaps only interesting in ﬂat spacetime. Spatially projecting the representation of the Lie derivative along u in terms ∇(cfw)(u) leads to the following relation between that operator and the Lie temporal derivative

[ ∇(cfw)(u) = £(u)u + σ(θ(u)) − P (u)σ(a(u) ⊗ u ) . (2.147)

39 The last term only contributes for nonspatial fields. Thus when acting on spatial fields the three temporal derivatives are related by ∇ (u) = ∇ (u) + σ(ω(u)) (cfw) (fw) (2.148) = £(u)u + σ(θ(u)) . The co-rotating Fermi-Walker temporal derivative is therefore an obvious compromise between the Fermi- Walker temporal derivative (retaining the latter’s compatibility with spatial orthonormality) and the Lie temporal derivative (retaining the latter’s co-rotation feature). If one wants to use an orthonormal spatial frame, the best one can do to adapt it to the congruence is to transport it along the congruence by co-rotating Fermi-Walker transport, since under Lie transport, the frame will not remain orthonormal or spatial. Only when u is a Killing vector field is Lie transport compatible with orthonormality, since in this case it coincides with the co-rotating Fermi-Walker transport. An immediate consequence of the “comma to semicolon” formula for the Lie derivative is the following “product rule” £fuS = f£uS − σ(u ⊗ df)S, (2.149) with spatial projection £(u)fuS = f£(u)uS − P (u)σ(u ⊗ df)S. (2.150) For a stationary spacetime with a timelike Killing vector field ξ and its corresponding normalized 4-velocity u = `−1ξ, invariance under the symmetry group is associated with the Lie derivative by ξ and its spatial projection, not the corresponding operators for u. The previous identities together with d ln ` = a(u)[ then imply −1 £uS = ` £ξS + σ(u ⊗ a(u))S, −1 (2.151) £(u)uS = ` £(u)ξS + P (u)σ(u ⊗ a(u))S. These lead to the following relationships between the co-rotating Fermi-Walker derivative and co-rotating Fermi-Walker temporal derivative along u and the rescalings of the respective Killing vector field operators

(4) −1 ∇(cfw)(u) = ` £ξ , [ ∇(cfw)(u) = £(u)u − P (u)σ(a(u) ⊗ u ) (2.152) −1 [ = ` £(u)ξ − P (u)σ(a(u) ∧ u ) . Without stationarity, the co-rotating Fermi-Walker operators are the best that one can do to extend the corresponding Lie operators in the stationary case. Both the ordinary and co-rotating Fermi-Walker derivatives along u of a spatial tensor field reduce to the respective spatial derivatives of that tensor field. Thus the spacetime transport of a spatial tensor field along u for either of these two types of derivatives reduces to the corresponding spatial transport. The spatial Lie transport of spatial fields is rather simple. Lie transport a spatial tensor along the observer congruence using the 1-parameter group of diffeomorphisms generated by u and then spatially project it. If eα is any frame with e0 along u and {ea} Lie dragged along u, then {e0,P (u)ea} is an observer-adapted frame and {P (u)ea} is a spatial frame which undergoes spatial Lie transport. Any spacetime tensor which does not evolve in the Lie sense is represented by a family of spatial tensors which have components in such a spatial frame which are constant along the observer worldlines. Such tensors are “anchored” in the observer congruence. The spatial Lie derivative along u measures the rate at which a field is evolving in this sense. The Fermi-Walker evolution instead describes how a field changes with respect to what the observers see using an orthonormal spatial frame which is locally nonrotating, as determined by a set of torque-free gyros carried by the observer, rather than by comparing the field to the nearby observers seen by that observer. Fermi-Walker transport of the spatial frame along the observer congruence defines mathematically what locally nonrotating means. A spatial field which has constant components with respect to such a frame exhibits no evolution in the covariant sense. A spacetime tensor field which is Fermi-Walker transported along the observer congruence is represented by a family of spatial tensors which undergo spatial Fermi-Walker transport.

40 The interpretation of the kinematical quantities is associated with the comparison of the two kinds of evolution. These quantities describe the (limiting) relative motion of the nearby observers as seen by each observer moving through spacetime and as compared to a locally nonrotating orthonormal spatial frame. Physically an observer can assign a relative position vector in his local rest space to each neighboring observer at each moment of his proper time by using light signals, obtaining the relative distance from half the light travel time and the direction as an average direction. This relative position vector in the flat local rest space will undergo a time-dependent linear transformation compared to the locally nonrotating orthonormal spatial frame. The rate of change of this transformation is described by the kinematical tensor −k(u). Suppose X is a tangent vector which is Lie dragged along a given observer worldline. If X is sufficiently small, it has the interpretation of being a “connecting vector” whose tip lies on the worldline of some other fixed nearby observer worldline, identifying nearby points in the spacetime manifold with points in the tangent space. If it is initially spatial, it will not remain spatial since its tip moves at unit speed relative to the nearby observer’s proper time, while its initial point moves at unit speed relative to the given observer. Its spatial projection P (u)X after an interval ∆τu of the given observer’s proper time represents the observed position vector (“relative position vector”) of the nearby observer at the same value of the elapsed proper α time measured by that nearby observer, not by the given observer. The component along u, namely −uαX , represents the additional lapse of the given observer’s proper time necessary to reach the event at which P (u)X is the relative position vector. Neglecting this additional synchronization question helps define the spacetime neighborhood in which this discussion [Ehlers 1961, 1993, Ellis 1971] makes sense. Let Y = P (u)X be this relative position vector, which undergoes spatial Lie transport. Its spatial Fermi-Walker derivative (“relative velocity vector”) is therefore

∇(fw)(u)Y = −k(u) Y = [θ(u) − ω(u)] Y = ~ω(u) ×u Y + θ(u) Y. (2.153)

This shows that nearby observers appear to be rotating with angular velocity ~ω(u) and shearing and expanding by σ(u) and Θ(u) relative to locally nonrotating orthonormal axes. The co-rotating Fermi-Walker temporal derivative describes the deviation from pure rotation

∇(cfw)(u)Y = θ(u) Y. (2.154)

Thus the relative position vector fails to undergo spatial co-rotating Fermi-Walker transport along u only because of the eﬀects of the expansion of the congruence. One can further decompose the relative position vector into a relative distance ||Y || and a relative direction vector Yˆ Y = ||Y ||Y.ˆ (2.155) This leads to a corresponding decomposition of the spatial Fermi-Walker derivative ˆ ˆ ˆ ˆ ˆ ∇(fw)(u)Y = ~ω(u) ×u Y + [σ(u) − σ(Y, Y )P (u)] Y, 1 (2.156) ∇ (u) ln ||Y || = Θ(u) + σ(Y,ˆ Yˆ ) . (fw) 3

The corresponding “relative acceleration” is obtained by taking one more spatial Fermi-Walker derivative

∇ (u)2Y = −[∇ (u)k(u)] Y − k(u) ∇ (u)Y (fw) (fw) (fw) (2.157) = −[∇(fw)(u)k(u) − k(u) k(u)] Y,

but since k(u) = −∇(u)u and ∇(fw)(u)u = a(u) this becomes

2 ∇(fw)(u) Y = {[∇(fw)(u), ∇(u)]u + k(u) k(u) + ∇(u)a(u)} Y. (2.158)

To ﬁnish this evaluation, one needs to consider the commutator of the spatial covariant derivative and the spatial Fermi-Walker derivative, which involves the curvature tensor through the Ricci identity. This will be postponed until the splitting of the curvature tensor is discussed.

41 On the other hand, a spatial vector Z which is not evolving relative to the locally nonrotating orthonormal spatial frame does evolve with respect to the observer congruence at a rate described by the kinematical tensor itself. If ∇(fw)(u)Z = 0 then

£(u)uZ = k(u) Z = −~ω(u) ×u Z − θ(u) Z. (2.159)

This would describe the evolution of the spin vector of a gyro carried by the observers relative to the observer congruence itself. Equivalently, the rotation tensor alone describes the evolution with respect to a co-rotating Fermi-Walker spatial frame ∇(cfw)(u)uZ = ω(u) Z = −~ω(u) ×u Z. (2.160)

42 2.2.9 Total spatial covariant derivatives Suppose one has a parametrized timelike or null curve c with parameter λ and tangent vector c0 in spacetime, thought of as the worldline of a test particle of nonzero rest mass m 6= 0 (m ˜ = 1) or of zero rest mass m = 0 (m ˜ = 0) respectively. It is then of interest to split the absolute or intrinsic derivative, here called the total (4) (4) covariant derivative operator D/dλ = ∇c0(λ) along this curve. This operator is discussed in section A.10 of the appendix. Before splitting this operator, it is useful to explore ﬁrst its spatial projection and several related operators which are useful in representing the splitting. To identify this operator more explicitly, introduce the expanded notation (4)D(c0(λ))/dλ. In the case that c is a proper-time-parametrized worldline belonging to another observer congruence 0 (4) with 4-velocity U, with c (λ) = U ◦ c(λ), then this is just the splitting of the operator ∇U with respect −1 0 to u. However, if one uses the parameter λ = m τU , then the tangent vector c = P = mU equals the 4-momentum of the test particle, and one can handle the case of zero rest mass in a parallel fashion. Recall the notation of section (2.1.4). In both of these cases a new parametrization of the curve c can be deﬁned which corresponds to the continuous sequence of proper times of the observers whose worldlines intersect this curve

dτ(P,u)/dλ = E(P, u) . (2.161)

Here the subscript notation on τ indicates the proper time with respect to u along P . For the casem ˜ = 1 of a timelike worldline, substituting m = 1 replaces P everywhere by the 4-velocity U, E(P, u) by the gamma factor γ(U, u), and the affine parameter λ by the proper time τU . This particular relation then connects the proper times dτ(U,u)/dτU = γ(U, u) , (2.162) corresponding to the Lorentz dilation of the test particle’s proper time in relative motion with respect to the observer congruence. It is convenient to follow the custom of not distinguishing the operator (4)D(c0(λ))/dλ which acts on (4) tensors defined only along the curve c from the operator ∇c0(λ) which acts on tensor fields defined in a neighborhood of c. When necessary they will be understood to apply to any smooth extension of a tensor along c to a tensor field, since only the values along the curve enter into these derivatives. From the total covariant derivative one may extract a derivative operator along the curve (equivalently, along P ) which preserves the spatial character of a tensor and can therefore be used in the representation of the splitting of the total covariant derivative of a spacetime tensor in terms of its representative spatial fields. The obvious way to do this is simply by spatial projection of the total covariant derivative

(4) (4) D(fw)(P, u)/dλ = P (u) D(P )/dλ = P (u) ∇P (2.163) = E(P, u)[∇(fw)(u) + ∇ν(P, u)] ,

and hence changing the parametrization to observer proper time

−1 D(fw)(P, u)/dτ(P,u) = E(P, u) D(fw)(P, u)/dλ (2.164) = ∇(fw)(u) + ∇ν(P, u) .

The qualifying notation (P, u) indicates the derivative along P as seen by u. However, two other useful possibilities exist. P itself can be split into spatial and temporal parts. Clearly one can use the spatial covariant derivative along the spatial part of P but one has the choice of the ordinary or co-rotating spatial Fermi-Walker derivative or the spatial Lie derivative along the temporal part. This leads to three different possibilities. One then has the option to rescale the derivative to correspond to the rate of change with respect to the observer proper time. Suppose S is a spatial tensor defined only along the worldline with tangent U. In order to discuss derivative operators along this worldline in terms of the three temporal derivatives {∇(tem)(u)}tem = fw,cfw,lie and the spatial covariant derivative ∇(u) already introduced for tensor fields, assume that S may be smoothly extended to a spatial field on (4)M. The three possible total spatial covariant derivatives that one can define

43 for such spatial tensors along the worldline can then be represented in terms of the following operators on the corresponding extended spatial tensor ﬁelds

D(tem)(P, u)/dλ = E(P, u)[∇(tem)(u) + ∇(u)ν(P, u)] (2.165) = E(P, u)∇(tem)(u) + ∇(u)p(P, u) , tem = fw,cfw,lie , which may then be rescaled to correspond to a derivative with respect to the observer proper time

−1 D(tem)(P, u)/dτ(P,u) = E(P, u) D(tem)(P, u)/dλ (2.166) = ∇(tem)(u) + ∇(u)ν(P, u) tem = fw,cfw,lie . These three operators will be called respectively the Lie, Fermi-Walker, and co-rotating Fermi-Walker total spatial covariant derivatives along P with respect to u. All of them, like the Lie and Fermi-Walker temporal derivatives and the spatial covariant derivative, may be applied to nonspatial tensor fields but are most useful for spatial tensor fields. When applied to functions, all of these operators reduce to the ordinary parameter derivative along the worldline if the function is only defined on the worldline or the ordinary derivative by P or E(P, u)−1P if the function is defined in a neighborhood of the worldline

D(tem)(P, u)f/dτ(P,u) ≡ df/dτ(P,u) = P f . (2.167) Since the spatial metric is covariant constant with respect to the spatial covariant derivative and has zero ordinary and co-rotating spatial Fermi-Walker derivative along u, the ordinary and co-rotating Fermi-Walker total spatial covariant derivatives of the spatial metric vanish, so that each of these derivatives commute with index-shifting on spatial ﬁelds. However, the Lie total spatial covariant derivative of the spatial metric, which coincides with its spatial Lie derivative, is not zero as long as the expansion tensor is nonvanishing

[ [ [ D(lie)(P, u)P (u) /dτ(P,u) = £(u)uP (u) = 2θ(u) . (2.168) Thus the Lie total spatial covariant derivative does not commute with index-shifting on spatial tensor ﬁelds and extra expansion terms must be taken into account when shifting indices. The co-rotating Fermi-Walker total spatial covariant derivative is a compromise between the Fermi- Walker and Lie total spatial covariant derivatives. It has the Fermi-Walker property of commuting with the shifting of indices but its associated transport has the same rotational properties as the Lie total spatial covariant derivative. One simply absorbs the expansion tensor into the latter derivative or equivalently the rotation tensor into the former to obtain this result

D(fw)(P, u)/dτ(P,u) + σ(ω(u)) = D(cfw)(P, u)/dτ(P,u) = D(lie)(P, u)/dτ(P,u) + σ(θ(u)) . (2.169) This new total covariant derivative was introduced by Massa [1974b, 1990]. The diﬀerence between the rotation and expansion tensors parametrizes the diﬀerence between the Lie and Fermi-Walker total spatial covariant derivatives

D(lie)(P, u)/dτ(P,u) − D(fw)(P, u)/dτ(P,u) = σ(k(u)) . (2.170) The Lie total spatial covariant derivative was used by Zel’manov [1956] and Cattaneo [1958]. Every timelike worldline with 4-velocity U has one spatial field defined along it, namely its relative velocity vector ν(U, u) as seen by u. The temporal derivative of this relative velocity defines the relative acceleration of U. Introduce therefore the ordinary and co-rotating and the Lie relative accelerations of U with respect to u by a (U, u) = D (U, u)ν(U, u)/dτ , (tem) (tem) (U,u) (2.171) tem = fw,cfw,lie . These are related to each other in the same way as the corresponding derivative operators

a(cfw)(U, u) = a(lie)(U, u) + θ(u) ν(U, u)

= a(fw)(U, u) + ω(u) ν(U, u) (2.172)

= a(fw)(U, u) − ~ω(u) ×u ν(U, u) .

44 The relative acceleration may in turn be related to the temporal derivative of the spatial momentum (per unit mass)p ˜(U, u) = γν(U, u) using the product rule and explicitly diﬀerentiating the expression for the gamma factor

γ = [1 − ||ν(U, u)||2]−1/2 , 2 1 [ (2.173) d ln γ/dτ(U,u) = γ ν(U, u) ·u [a(tem)(U, u) + 2 D(tem)(U, u)P (u) /dτ(U,u) ν(U, u)] , tem = fw,cfw,lie to obtain

D(tem)(U, u)˜p(U, u)/dτ(U,u) = γ[D(tem)(U, u)˜ν(U, u)/dτ(U,u) + d ln γ/dτ(U,u) ν(U, u)]

= γ[P (u, U) P (U, u) a(tem)(U, u) (2.174) 1 2 [ + 2 γ D(tem)(U, u)P (u) /dτ(U,u)(ν(U, u), ν(U, u)) ν(U, u)] .

Only in the Lie case is the derivative of the spatial metric nonzero, producing the expansion tensor θ(u)[ = 1 [ 2 D(lie)(U, u)P (u) /dτ(U,u). In all cases this relationship may be inverted to give the relative acceleration as a function of the temporal derivative of the spatial momentum either from the previous equation using equation (2.57) or by directly diﬀerentiating γ−1p˜(U, u). The result is

−1 −1 a(tem)(U, u) = γ Pu(U, u) D(tem)(U, u)˜p(U, u)/dτ(U,u) (2.175) 1 2 [ − 2 γ D(tem)(U, u)P (u) /dτ(U,u)(ν(U, u), ν(U, u)) ν(U, u) .

45 2.2.10 Splitting the total covariant derivative Consider the splitting of the total covariant derivative along an arbitrary timelike or null parametrized curve in spacetime representing the worldline of a test particle of nonzero or zero rest mass m respectively with 4-momentum P and associated parameter λ. Although one could derive formulas for the splitting of this operator acting on arbitrary tensors deﬁned along the curve, the main application will be for a vector deﬁned along the curve. Let Y = Y >(Y, u) + Y~ (Y, u) , Y~ (Y, u) ≡ P (u)Y (2.176) be such a vector. Recalling the splitting of P P = E(P, u)[u + ν(P, u)] , (2.177) and the Fermi-Walker total spatial covariant derivative of u

D(fw)(P, u)u/dτ(P,u) = [∇(fw)(u) + ∇(u)ν(P, u)]u = a(u) − k(u) ν(P, u) , (2.178) one ﬁnds

(4) −1 (4) P (u) D(P )Y/dτ(P,u) = E(P, u) P (u) D(P )Y/dτ(P,U) = D(fw)(P, u)Y/dτ(P,U) (2.179) > ~ = Y (Y,U)D(fw)(P, u)u/dτ(P,U) + D(fw)(P, u)Y (Y, u)/dτ(P,U) for the spatial projection. For the temporal projection recall that

(4) α α β α β α β ∇P u = u ;βP = −a(u) uβP − k(u) βP α α β = E(P, u)[a(u) − k(u) βν(P, u) ] (2.180)

= E(P, u)D(fw)(P, u)u/dτ(P,u)

and (4) > α(4) (4) α (4) α [ ∇P Y ] = −u ∇P Yα = ∇P [−u Yα] + Yα ∇P u > ~ [ = PY (Y, u) + E(P, u)Y (Y, u) D(fw)(P, u)u/dτ(P,u) (2.181) > ~ [ = E(P, u)[dY (Y,U)/dτ(P,u) + Y (Y, u) D(fw)(P, u)u/dτ(P,u)] so (4) > −1 (4) > [ D(P )Y/dτ(P,u)] = E(P, u) [ D(P )Y/dλ] (2.182) > ~ = dY (Y,U)/dτ(P,u) + Y (Y, u) ·u D(fw)(P, u)u/dτ(P,u) . In each case the splitting of the total covariant derivative leads to the Fermi-Walker total spatial covariant derivative of the splitting ﬁelds plus a term involving the Fermi-Walker total spatial covariant derivative of the observer 4-velocity, which introduces kinematical terms. The same result applies to the covector Y [ since index shifting commutes with these operators. This allows one to generate a formula for the splitting of the total covariant derivative of any tensor.

46 2.3 Observer-adapted frame derivatives

The previous section does not rely on the introduction of an explicit spatial frame to perform calculations. However, spatial frames can be quite useful in making calculations much more concrete and automatic. This is especially true in splitting spacetime ﬁelds involving derivatives, where the use of an observer-adapted spacetime frame makes calculations almost routine.

2.3.1 Natural frame derivatives It is convenient to have several notations for the frame derivatives of functions (including component functions) eαf = ∂αf = f,α . (2.183) α The commutators of these derivatives deﬁne the structure functions C βγ of the frame {eα}

γ α 1 α β γ [eα, eβ] = C αβeγ , dω = − 2 C βγ ω ∧ ω , (2.184) or γ β β γ £eα eβ = C αβeγ , £eα ω = −C αγ ω . (2.185) These functions satisfy the Jacobi identity

0 = [eα, [eβ, eγ ]] + [eβ, [eγ , eα]] + [eγ , [eα, eβ]] δ δ (2.186) = 2(∂[αC βγ] − C [αC βγ])eδ , equivalent to d2ωα = 0. The structure functions for an observer-adapted frame

c 0 a 1 a b c a 0 b [ea, eb] = C abec + C abe0 , dω = − C bcω ∧ ω − C 0bω ∧ ω , 2 (2.187) b 0 0 1 0 a b 0 0 a [e0, ea] = C 0aeb + C 0ae0 , dω = − 2 C abω ∧ ω − C 0aω ∧ ω , may be interpreted in terms of the geometry of the orthogonal decomposition, using only noncovariant derivatives. The purely spatial structure functions govern the spatial Lie bracket and spatial exterior derivative for purely spatial ﬁelds c [ea, eb](u) = £(u)e eb = C abec , a (2.188) c 1 c a b d(u)ω = − 2 C abω ∧ ω , leading to the expressions

b a b a c a b [X,Y ](u) = (X ∂bY − Y ∂bX + C abX Y )ea , (2.189) 1 c a b d(u)σ = 2 (∂aσb − ∂bσa − σcC ab)ω ∧ ω , for a spatial vector ﬁelds and 1-forms. 0 The structure functions C ab are proportional to the components of the rotation 2-form

[ 1 [ 1 0 1 0 a b ω(u) = d(u)u = − Ld(u)ω = LC abω ∧ ω , 2 2 4 (2.190) 1 0 ω(u)ab = 2 LC ab . These structure functions, or equivalently the rotation, directly measure the failure of the spatial frame vector ﬁelds to be hypersurface-forming since in the latter case their Lie brackets would again lie in the distribution determining the hypersurfaces, i.e., would again be spatial vector ﬁelds

[X,Y ] = [X,Y ](u) + 2ω(u)[(X,Y )u (2.191)

Similarly −1 −1 b £(u)uea = L £(u)e ea = L C 0aeb , 0 (2.192) a a −1 a b £(u)uω = u dω = −L C 0bω ,

47 b show that the structure functions C 0a directly describe the spatial Lie derivatives along u of the spatial frame and spatial dual frame and which appear in the component expression for the expansion tensor

1 −1 θ(u)ab = [£uP (u)]ab = (2L) [hab,0 − 2C(a|0|b)] , 2 (2.193) c 1/2 −1 c Θ(u) = θ(u) c = − Tr k(u) = £(u)u ln h − L C 0c

as follows from its Lie derivative deﬁnition. For an orthonormal spatial frame, as used by Durrer and Straumann [1988a,b, 1989] in the slicing point of view, for example, then hab = δab and hab,0 = 0, so that the symmetric combination of these structure functions alone determine the expansion tensor. 0 The structure functions C 0a enter into the expression for the acceleration of the congruence

[ [ 0 a a(u) = £(u)uu = d(u) ln L + C 0aω , (2.194)

and directly give the spatial acceleration components in a partially-normalized observer-adapted frame (for which L = 1). For a given observer congruence, one may impose conditions on the structure functions which do not directly represent kinematical quantities, provided that they are compatible with the Jacobi identities. The a structure functions C 0b determine how the spatial frame is transported along the observer congruence. Introducing the quantities b ∇(tem)(u)ea = C(tem)(u) aeb , (2.195) one finds b −1 b C(lie)(u) a = L C 0a , b −1 b b C(cfw)(u) a = L C 0a + θ(u) a , (2.196) b −1 b b b C(fw)(u) a = L C 0a + θ(u) a − ω(u) a . a Setting the matrix C(tem)(u) b to zero for each of the three choices in turn respectively defines the spatial frame’s spatial Lie transport, its co-rotating Fermi-Walker transport, and its Fermi-Walker transport along u. An observer-adapted frame for which one of these three conditions hold will be referred to respectively as a spatially-comoving observer-adapted frame, a spatially co-rotating observer-adapted frame or a locally nonrotating observer-adapted frame. For the spatially-comoving observer-adapted frame, letting (α, β, γ, δ) = (0, b, c, d) in 2.186 reduces it to d a C bc,0 = 0 under the condition C 0b = 0, showing that the spatial structure functions are constant along the congruence for this class of frame. That such a frame always exists is clear since one can always take any partially-observer-adapted frame for which e0 is along u and the spatial frame is Lie dragged along e0 (a comoving partially-observer-adapted frame) and simply spatially project that spatial frame. Similarly one can always choose a spatially-holonomic observer-adapted frame by taking comoving coordinates for the congruence and spatially projecting the coordinate frame vector fields for the three coordinates which a parametrize the congruence. Such a frame is then also spatially comoving and has C bc = 0, two conditions which make it the simplest choice of frame in which to perform certain calculations. In a spatially-comoving observer-adapted frame, the spatial Lie derivative of a spatial tensor with respect to either e0 or u just reduces to the ordinary derivative by these vector fields of the components. Introduce the spatial tensor C(u) a a C(u) bc = C bc (2.197) whose spatial components in the observer-adapted frame equal the corresponding structure functions. One may then freely shift the indices on C(u) unambiguously with the spatial metric. This tensor represents the freedom left in the choice of the spatial distribution of the spatial frame, once the evolution along the observer congruence has been fixed.

2.3.2 Splitting the connection coeﬃcients So far, a temporal and a spatial covariant derivative operator have been extracted from the spacetime covariant derivative, together with the kinematical quantities of the observer congruence. Each of the derivative operators, namely the spatial Fermi-Walker derivative and the spatial covariant derivative, may

48 act on any spacetime tensor fields, but preserve the spatial property of spatial fields. These operators, together with the spatial kinematical fields, are enough to express the measurement of the spacetime covariant derivative of a spacetime tensor in terms of spatial operators acting on the spatial fields which represent that tensor. The simplest way to obtain such measurement formulas is to use a spatially-comoving spatially-holonomic observer-adapted frame and simply evaluate the standard formulas (see Appendix A) for the components of the connection and of the covariant derivative of those spacetime tensors which are of interest. Such a frame is used as a fundamental tool in the approaches of both Zel’manov [1956] and Cattaneo [1958,1960] to the threading point of view. The components of the spacetime metric connection with respect to a frame {eα} will here be defined with the ordering of the covariant components following the “del convention” of Hawking and Ellis [1977]

(4) α α (4) Γ βγ = ω ( ∇eβ eγ ) (2.198)

(4) α α δ rather than the “semi-colon convention” Γ βγ = ω (eβ;δeγ ) of Misner, Thorne and Wheeler [1973]. Using the notation

A{δβγ}− = Aδβγ − Aβγδ + Aγδβ . (2.199) and shifting indices on the structure functions with the frame component matrix of the spacetime metric, the connection components can be written

(4) α 1 (4) αδ (4) Γ βγ = 2 g ( g{δβ,γ}− + C{δβγ}− ) . (2.200) 1 Consider the 2 -tensor (4) α β γ Γ βγ eα ⊗ ω ⊗ ω (2.201) which represents the connection coefficients in a particular choice of frame and of course changes to a different tensor under a change of frame, according to the inhomogeneous transformation law for the connection coefficients. This tensor can be split as any other, but the results will be frame dependent like the tensor itself. However, if the frame is at least a partially-observer-adapted frame, then some of the splitting information is frame independent and one can interpret some of the fields which result from the splitting in a geometric way, similar to the interpretation of the structure functions of an observer-adapted frame. The more general partially-observer-adapted frames will be considered below in the discussion of a nonlinear reference frame. Here attention will be restricted to the class of observer-adapted frames.

2.3.3 Observer-adapted connection components Measurement of the connection coeﬃcient “tensor” in the observer-adapted frame leads to the splitting

(4) α (4) 0 (4) 0 (4) 0 (4) a (4) 0 (4) a (4) a (4) a Γ βγ ↔ Γ 00; Γ a0, Γ 0a, Γ 00; Γ ab, Γ 0b, Γ b0; Γ bc (2.202) (4) > (4) > (4) > (4) a (4) > (4) a (4) a (4) a ↔ Γ >>; Γ a>, Γ >a, Γ >>; Γ ab, Γ >b, Γ b>; Γ bc

> where the 0 indices have been rescaled by L to correspond to components along e> and ω . These ﬁelds are also connected closely with the geometry of the congruence

(4) > Γ >> = £(u)u(ln L) , (4) a a Γ >> = a(u) , (4) > Γ a> = [d(u)(ln L)]a , (4) > Γ >a = a(u)a , (2.203) (4) > Γ ab = −k(u)ba , (4) a a a Γ >b = −k(u) b + C(lie)(u) b , (4) a a Γ b> = −k(u) b , (4) a a Γ bc = Γ(u) bc .

49 The spatial components of the spacetime connection equal the components of the spatial part of the spatial connection ∇(u) ◦ P (u)

c ∇(u)e eb = Γ(u) ab , a (2.204) a 1 ad Γ(u) bc = 2 h (h{db,c}− + C(u){dbc}− ) , For the hypersurface point of view, one makes the replacements u → n, L → N, > → ⊥ for “perpendicular” to n, ω(u) → 0, and k(u) → K(n), the extrinsic curvature tensor. The spatial part of the spatial connection is then equivalent to the connection of the induced metric on each slice. Only when L = 1 do the above formulas give the components of the connection in the partially-normalized (4) > (4) > frame. This distinction only matters for the explicitly L-dependent fields Γ >> and Γ a>, which are (4) associated with the threading parametrization. These two fields are not components of ∇e> as conventional −1(4) notation would imply, but rather of L ∇e0, unless L = 1 when e0 = e>, in which case they vanish. Otherwise they serve the purpose of converting temporal and spatial derivatives of 0-indexed components to derivatives of the corresponding >-indexed components. The purely spatial components determine the spatial part of the spatial connection while the remaining connection components dependent only on the kinematical a quantities, which determine the temporal part of the spatial connection. The contribution C(lie)(u) b to the (4) a field Γ >b comes from the spatial Lie derivatives of the spatial frame vectors and depends on their choice. Using a partially-normalized spatially-comoving spatially-holonomic observer-adapted frame

a a L = 1 ,C 0b = 0 = C bc , (2.205) eliminates the “gauge terms” which depend on the choice of e0 and the spatial frame, making it the most convenient frame to use in splitting the covariant derivative of a tensor field, leading to a family of spatial fields which arise from the original family representing the split of the tensor itself by a linear operation involving the spatial covariant derivative, the spatial Lie derivative along u and the kinematical quantities of the congruence. This was already done for the geodesic operator without the aid of such a frame. Furthermore, the spatial Lie derivative may be replaced by the spatial Fermi-Walker derivative plus a kinematical term to yield alternative expressions for the measured fields. The resulting formulas, in either case, are valid in any observer-adapted frame, the above “gauge terms” entering into the expressions for the observer-adapted components of the spatial Lie and covariant derivatives.

2.3.4 Splitting covariant derivatives 1 It is quite useful to explicitly consider splitting the covariant derivatives of vector fields, 1-forms and 1 - α > a α tensor fields. A vector field X splits into the family {X ,X } and its covariant derivative X ;β splits into the family > > d X ;> = £(u)uX + a(u)dX , a a a d a > X ;> = £(u)uX − k(u) dX + a(u) X , (2.206) > > d X ;a = ∇(u)aX − k(u)daX , a a a > X ;b = ∇(u)bX − k(u) bX , with trace α (4) > c X ;α = div X = {£(u)u + Θ(u)}X + {∇(u)c + a(u)c}X . (2.207) In these expressions and in similar ones below, the symbols £(u)uX> and £(u)uXa indicate respectively the spatial Lie derivative of a scalar X> and the spatial components of the spatial Lie derivative of the spatial vector Xa. The same applies to the spatial covariant derivative. In other words scalar projections along e> are intended before the action of spatial derivative operators. If X and Y are spatial vector fields, then these formulas imply the result

(4) ∇X Y − ∇(u)X Y = −k(u)(Y,X) e> = [θ(u)(Y,X) − ω(u)(Y,X)]e> . (2.208) Thus k(u) describes the diﬀerence between spatial and spacetime diﬀerentiation in the local rest space of the observer. When the vorticity ω(u) is zero, then k(u) reduces to the extrinsic curvature tensor −θ(u)

50 of the hypersurfaces orthogonal to the observer family, namely the sign-reversed second fundamental form of those hypersurfaces. As one parallel transports Y along X in the local rest space using the spatial connection, −k(u)(Y,X) describes the spacetime rate of change of the component of the transported ﬁeld along the direction of the observer 4-velocity necessary to keep Y spatial compared to the spacetime parallel transported ﬁeld. α... It is worth noting that for any tensor S β... one has

−1/2 1/2 −1 c {£(u)u + Θ(u)}S = h £(u)u(h S) − L C 0cS, (2.209) which gives an alternative expression for the temporal part of the spacetime divergence, the second term of which vanishes in a spatially-comoving frame. The spatial part on the other hand is the spacetime divergence of the spatial part of the vector ﬁeld α c [P (u)X] ;α = {∇(u)c + a(u)c}X , (2.210) leading to the addition of the acceleration term to the spatial divergence. Such formulas can also be obtained without using an adapted frame by relying only on the properties of the spatial projection operator, as described by Ehlers [1961, 1993] and Ellis [1971]. For example

(4) α (4) α β α (4) β β(4) α ∇αX = ∇α(P (u) βX ) = P (u) β ∇αX + X ∇αP (u) β (2.211) β γ β (4) α β = ∇(u)βX + X P (u) γ ∇α(u uβ) = [∇(u)β + a(u)β]X illustrates this technique for a spatial vector ﬁeld X. For a 1-form Xα represented by the spatial ﬁelds {X>,Xc}, the covariant derivative Xα;β has the following splitting d X>;> = £(u)uX> − Xda(u) , d Xa;> = £(u)uXa + Xdk(u) a − X>a(u)a , (2.212) d X>;a = ∇(u)aX> + Xdk(u) a ,

Xa;b = ∇(u)bXa + X>k(u)ba . Twice the symmetrized covariant derivative of the 1-form equals the Lie derivative of the spacetime metric with respect to its corresponding vector ﬁeld

(4) d [£X g]>> = 2£(u)uX> − 2Xda(u) , (4) b [£X g]>a = {∇(u)a − a(u)a}X> + £(u)uXa + 2Xbk(u) a , b b (2.213) = {∇(u)a − a(u)a}X> + hab£(u)uX + 2Xbω(u) a , (4) [£X g]ab = £(u)P (u)XP (u)ab − 2X>θ(u)ab = £(u)XP (u)ab , where £(u)P (u)XP (u)ab = 2∇(u)(aXb) , (2.214) and b b £(u)uXa − 2Xbθ(u) a = hab£(u)uX . (2.215) 1 α Finally consider splitting the covariant derivative of the 1 -tensor S β > > d > d S >;> = £(u)uS > + a(u)dS > − S da(u) , a a a d a d S >;> = £(u)uS > − k(u) dS > − S da(u) , > > > d > d S a;> = £(u)uS a + S dk(u) a − a(u)aS > + a(u)dS a , > > d > d S >;a = ∇(u)aS > − k(u)adS > + S dk(u) a , (2.216) a a a d a d a > a S b;> = £(u)uS b − k(u) dS b + S dk(u) b + a(u) S b − S >a(u)b , a a a d > a S >;b = ∇(u)bS > + S dk(u) b − S >k(u) b , > > > d S a;b = ∇(u)bS a + S >k(u)ba − k(u)bdS a , a a a > a S b;c = ∇(u)cS b − k(u) cS b + S >k(u)bc ,

51 with trace α > c c d c > S >;α = [£(u)u + Θ(u)]S > + [∇(u)c + a(u)c]S > + S dk(u) c − a(u) S c , α > > c c > S a;α = [£(u)u + Θ(u)]S a + S ck(u) a + S >k(u)ca − a(u)aS > (2.217) c + [∇(u)c + a(u)c]S a . Notice that the temporal derivative which occurs in the divergence is the Θ(u)-augmented Lie derivative, while the spatial divergence occurs in each case accompanied by an acceleration term to form the spacetime a divergence of the spatial ﬁeld S > and the spatial projection of the spacetime divergence of the spatial ﬁeld a S b. These formulas have interesting special cases for symmetric and antisymmetric tensors. For a symmetric tensor one has α > c c d S >;α = [£(u)u + Θ(u)]S > + [∇(u)c + 2a(u)c]S > − S dθ(u) c , (2.218) α > > c S a;α = [£(u)u + Θ(u)]S a − a(u)aS > + [∇(u)c + a(u)c]S a . while for an antisymmetric tensor one has

α c cab (E)c ∗(u) (M) ] S >;α = ∇(u)cS > − η(u) ω(u)cSab = ∇(u)cS − 2~ω(u) ·u [ S ] , α (E)c (M)c S a;α = hac[£(u)u + Θ(u)]S + [∇(u)c + a(u)c]S a (2.219)

(E)] b ~ ∗(u) (M) ] = hab[£(u)u + Θ(u)]S + {[∇(u) + a(u)] ×u [ S ] }a .

Note that analogous to the relation for a spatial vector ﬁeld, the spatial projection of the spacetime 1 divergence and the spatial divergence of a 1 -tensor diﬀer by an acceleration term γ c γ X ;γ = [∇(u)c + a(u)c]X = [∇(u)γ + a(u)γ ]X , γ c δ γ γ (2.220) Xa ;γ = [∇(u)c + a(u)c]Xa , or P (u) αXδ ;γ = [∇(u)γ + a(u)γ ]Xα , while for a spatial 2-form one has

γ ∗(u) ] Xa ;γ = {[∇~ (u) + a(u)] ×u X }a , (2.221) leading to the acceleration-augmented curl of its spatial dual which represents the spatial projection of the spacetime divergence. In the case of a stationary spacetime, where one may deﬁne u by normalizing a timelike Killing vector −1 (4) ﬁeld u = ` ξ, then the Killing equation £ξ g = 0 may be expressed in terms of u using the splitting of (4) > a £X g given above with X = ξ, X = ` 6= 0 and X = 0. This immediately yields three conditions on u (the spatial scalar, spatial vector and spatial tensor equations which represent the splitting of the Killing equation) £(u)u` = 0 , a(u)a = ∇(u)a ln ` , θ(u)ab = 0 . (2.222) The normalization factor must be a stationary function, the logarithm of which serves as an acceleration potential (modulo sign conventions), and the expansion tensor must vanish.

2.3.5 Observer-adapted components of total spatial covariant derivatives Given the deﬁnitions (2.195) and (2.204), the total spatial covariant derivative of the spatial frame along the worldline of a test particle with 4-momentum P is given by

D(tem)(U, u)ea/dτ(P,u) = [∇(tem)(u) + ∇(u)ν]ea b b c = [C (u) a + Γ(u) caν(P, u) ]eb (tem) (2.223) b b = [C(tem)(u) a + ωωω(u)(ν(P, u)) a]eb , tem = fw,cfw,lie .

b where (ωωω(u)(ν(P, u)) a) is the matrix of spatial connection 1-forms

b b c ωωω(u) a = Γ(u) caω (2.224)

52 evaluated on the relative velocity vector. The spatial connection 1-form is discussed below in section (2.5.6). From the appropriate product rule, one then has for a spatial vector ﬁeld the expression

a a a b a b c D(tem)(P, u)X /dτ(P,u) = dX /dτ(P,u) + C(tem)(u) bX + Γ(u) bcν(P, u) X , (2.225) or in general for a spatial tensor ﬁeld

a... D(tem)(P, u)S b.../dτ(P,u) a... a... (2.226) = dS b.../dτ(P,u) + [σ([C(tem)(u) + ωωω(u)(ν(P, u)] X)S] b... . When the spatial frame undergoes the corresponding transport along the observer congruence, the term a involving C(tem)(u) b = 0 vanishes and the total spatial covariant derivative of that type reduces to a simpler expression. One may find the Lie total spatial covariant derivative in this form mentioned by Møller [1952] and in subsequent work by Zel’manov [1956] and Cattaneo [1958], all of whom use spatially-comoving spatially-holonomic spatial frames. In general only the ordinary and co-rotating Fermi-Walker transport conditions for the spatial frame along u are compatible with the condition of orthonormality. An orthonormal spatial frame of this latter type has an arbitrary distribution of orientation on a given hypersurface transversal to the observer congruence, but its subsequent behavior away from that hypersurface along the observer congruence is completely fixed by the transport condition. For a given choice of such a frame, one may examine the relative rotation of a spatial frame which is transported along an arbitrary timelike worldline by the corresponding transport, which accomplished by requiring that the corresponding total covariant derivative of the frame along the worldline vanish. Suppose {ea} is any spatial frame defined on some fixed hypersurface Σ transversal to the observer congruence. One can extend it to a spatial frame on the spacetime by transporting it along the congruence by one of the three spatial transport conditions

a ∇(tem)(u)ea = 0 ↔ C(tem)(u) b = 0 . (2.227)

One can also spatially transport it along an arbitrary timelike or null worldline from its value at the intersection of that worldline with the hypersurface to deﬁne a spatial frame along that single worldline

D(tem)(P, u)E(P, u, e)a = 0 . (2.228)

The two such frames along the worldline are linearly related

b E(tem)(P, u, e)a = R(tem)(P, u, e) aeb . (2.229)

If {ea} is an orthonormal spatial frame on Σ, then in general only in the ordinary and co-rotating Fermi- Walker cases will it remain orthonormal as it is extended along the observer congruence to a spatial frame on spacetime, as is the case for its corresponding spatial transport along the single given worldline. In these cases the matrix R(fw)(P, u, e) or R(cfw)(P, u, e) respectively will be an orthogonal matrix describing the relative rotation of the orthonormal spatial frame transported along the worldline with respect to the given orthonormal spatial frame on spacetime. a Suppose X = X ea undergoes the corresponding transport along the given worldline

a a a b 0 = D(tem)(P, u)X /dτ(P,u) = dX /dτ(P,u) + ωωω(ν(P, u)) bX (2.230)

or a a b dX /dτ(P,u) = −ωωω(u)(ν(P, u)) bX . (2.231)

Since {ea} is an orthonormal spatial frame, its connection 1-form is antisymmetric in its tensor-valued indices (see section (A.20) of the appendix), so its spatial dual deﬁnes a vector-valued 1-form whose value on the relative velocity deﬁnes the relative angular velocity of the two orthonormal frames {ea} and {E(P, u)a} along the worldline ∗(u) ] ζ(sc)(P, u, e) = ωωω(u)(ν(P, u)) , (2.232)

53 or a 1 abc ζ(sc)(P, u, e) = 2 η(u) ωωω(u)(ν(P, u))bc , a a c ωωω(u)(ν(P, u)) b = η(u) bcζ(sc)(P, u, e) , (2.233) tem = fw,cfw . The transport condition for X then takes the more familiar form

a a b c a dX /dτ(P,u) = η(u) bcζ(sc)(P, u, e) X = [ζ(P, u, e) ×u X] . (2.234)

Letting X = E(tem)(P, u, e)a leads to the relationship between the relative rotation and the relative angular velocity a −1 c a c dR(tem)(P, u, e) c/dτ(P,u) R(tem)(P, u, e) b = η(u) cbζ(sc)(P, u, e) . (2.235)

This determines the ﬁnite rotation of the frame {E(tem)(P, u, e)a} deﬁned along the worldline with 4- momentum P with respect to the frame {ea} restricted to this worldline. Note that relative rotation along the worldline of the co-rotating Fermi-Walker transported spatial frame with respect to the Fermi-Walker transported spatial frame is related to the vorticity vector ~ω(u) in the same way that R(tem)(P, u) is related to the relative angular velocity ζ(tem)(P, u) since

a a a b D(cfw)(P, u)X /dτ(P,u) = D(fw)(P, u)X /dτ(P,u) + ω(u) bX (2.236) and ~ω(u) is the spatial dual of ω(u)]. The interpretation of the rotation R(tem)(P, u) is clear. The initial spatial distribution of the orientation of ea on the hypersurface Σ is completely arbitrary and this arbitrariness is propagated along the observer congruence by the choice of either ordinary or co-rotating spatial Fermi-Walker transport. This rotation removes the arbitrariness along the given worldline. Of course diﬀerent paths between the same pair of points in spacetime will in general lead to diﬀerent results. This is a manifestation of spatial curvature for the given observer congruence.

54 2.4 Relative kinematics: applications

There are two interesting applications of the splitting of the total covariant derivative along an arbitrary timelike or null parametrized curve in spacetime representing the worldline of a test particle of nonzero or zero rest mass m respectively with 4-momentum P and associated parameter λ. The ﬁrst

(4) α D(P )P/dλ = mf(P ) , f(P )αP = 0 , (2.237) is the “acceleration equals force per unit mass” equation for a timelike worldline with 4-velocity U = m−1P

(4) (4) ˜ D(U)P/dτU = f(U) ↔ D(U)U/dτU = f(U) , (2.238) or the aﬃnely parametrized geodesic equation for a null worldline (m = 0)

(4)D(P )P/dλ = 0 . (2.239)

The former reduces to the geodesic equation in the absense of an applied force, which when nonzero must be orthogonal to U in order to maintain the unit character of the 4-velocity

(4) α α(4) α ˜ 0 = D(U)[U Uα]/dτU = 2U DUα/τU = 2U f(U)α . (2.240)

These two variations of original equation will both be referred to as the acceleration equation. The second is the equation for the Fermi-Walker transport of a vector along an arbitrary timelike worldline to which it is orthogonal, which describes the behavior of the spin vector of a torque-free test gyro carried along that worldline or the spin of a spinning test particle following that worldline

α ∇(fw)(U)S = 0 ,UαS = 0 . (2.241)

This will be referred to simply as the spin transport equation.

2.4.1 Splitting the acceleration equation Letting Y = P in subsection (2.2.10) leads to

(4) P (u) D(P )P/dτ(P,u) = E(P, u)D(fw)(P, u)u/dτ(P,u) + D(fw)(P, u)p(Y, u)/dτ(P,u) , (2.242) (4) > [ D(P )P/dτ(P,u)] = dE(P, u)/dτ(P,u) + p(Y, u) ·u D(fw)(P, u)u/dτ(P,u) .

Consider ﬁrst the timelike case, where one can use the unit 4-velocity U instead of the 4-momentum P and (4) D(U)U/dτU ≡ a(U) (2.243) deﬁnes the 4-acceleration. The acceleration vector and hence the force vector must be spatial with respect to U, so their temporal components are determined by the orthogonality condition

a(U) = γ[A(U, u)u + A(U, u)] , A(U, u) = A(U, u) ·u ν(U, u) , (2.244) f(U) = γ[℘(U, u)u + F (U, u)] , ℘(U, u) = F (U, u) ·u ν(U, u) .

Here ℘(U, u) is the power and F (U, u) the spatial force relative to the observer congruence as deﬁned in special relativity [Anderson 1967]. These are deﬁned respectively so that the appropriate time derivatives of the spatial momentum and energy as seen by the observer equal respectively the spatial force and power. Dividing out the mass parameter in the above equations (2.242) describes the splitting of the acceleration vector A(U, u) = γD(fw)(U, u)u/dτ(U,u) + D(fw)(U, u)˜p(U, u)/dτ(U,u) , (2.245) ˜ A(U, u) = dE(U, u)/dτ(U,u) +p ˜(U, u) ·u D(fw)(U, u)u/dτ(U,u) .

55 Having split both the acceleration and the force, equating their spatial and temporal parts leads to the splitting of the acceleration equation. With this in mind, the following preliminary deﬁnition proves useful ˜(G) F (U, u) = −γD(fw)(U, u)u/dτ(U,u) (fw) (2.246) = γ[−a(u) + k(u) ν(P, u)] , so that ˜(G) A(U, u) = D(fw)(U, u)˜p(U, u)/dτ(U,u) − F (fw)(U, u) , ˜(G) (2.247) ≡ D(tem)(U, u)˜p(U, u)/dτ(U,u) − F (tem)(U, u) , tem = fw,cfw,lie . Equating A(U, u) to the spatial force F˜(U, u) per unit mass then leads to the spatial projection of the acceleration equation ˜(G) ˜ D(tem)(U, u)˜p(U, u)/dτ(U,u) = F (U, u) + F (U, u) , (tem) (2.248) tem = fw,cfw,lie . This deﬁnes the ordinary and co-rotating and the Lie spatial gravitational forces per unit mass, related to each other in the same way as the corresponding total spatial covariant derivatives ˜ ˜ F (cfw)(U, u) = F (fw)(U, u) + ω(u) p˜(U, u) (2.249) ˜ = F (lie)(U, u) + θ(u) p˜(U, u) . Since the Lie total spatial covariant derivative does not commute with index lowering, it is convenient to introduce also the Lie covariant spatial gravitational force

[ ] ˜(G) ˜ [D(tem)(U, u)˜p(U, u) /dτ(U,u)] = F(lie[)(U, u) + F (U, u) . (2.250) The four distinct spatial gravitational forces have the explicit form ˜(G) F(fw)(U, u) = γ[−a(u) + ν(U, u) ×u ~ω(u) − θ(u) ν(U, u)] , ˜(G) F(cfw)(U, u) = γ[−a(u) + 2ν(U, u) ×u ~ω(u) − θ(u) ν(U, u)] , (2.251) ˜(G) F(lie)(U, u) = γ[−a(u) + 2ν(U, u) ×u ~ω(u) − 2θ(u) ν(U, u)] , ˜(G) F(lie[)(U, u) = γ[−a(u) + 2ν(U, u) ×u ~ω(u)] . Similarly the temporal projection of the acceleration equation becomes the equation specifying the rate of change of energy ˜ ˜(G) ˜ dE(U, u)/dτ(U,u) = ν(U, u) ·u [F (U, u) + F (U, u)] (fw) (2.252) [ = −γ[a(u) ·u ν(U, u) − θ(u) (ν(U, u), ν(U, u))] + ℘(U, u) . This can be written in terms of all the variously deﬁned spatial gravitational forces as ˜ ˜(G) ˜ dE(U, u)/dτ(U,u) = ν(U, u) ·u [F (tem)(U, u) + F (U, u)] [ (2.253) + (tem)γθ(u) (ν(U, u), ν(U, u)) ,

(tem) ≡ (0, 0, 1, −1) , tem = fw,cfw,lie,lie[ , The relative acceleration deﬁned in section 2.2.9 can be expressed in terms of the spatial projection of the acceleration using equations (2.175) and (2.247)

−1 −1 ˜(G) a(tem)(U, u) = γ Pu(U, u) [F (U, u) + A(U, u)] (tem) (2.254) 1 2 [ − 2 γ D(tem)(U, u)P (u) /dτ(U,u)(ν(U, u), ν(U, u)) ν(U, u) .

56 Using the acceleration equation this becomes

a(tem)(U, u) ≡ D(tem)(U, u)ν(U, u)/dτ(U,u) −1 −1 ˜(G) = γ Pu(U, u) [F (tem)(U, u) + F (U, u)] (2.255) 1 2 [ − 2 γ D(tem)(U, u)P (u) /dτ(U,u)(ν(U, u), ν(U, u)) ν(U, u) . The derivative of the spatial metric is nonzero only in the Lie case where it reduces to an expansion tensor term. However, this term in the relative acceleration is along the relative velocity and so does not contribute to the part orthogonal to the relative motion

−1 ˜(G) ˜ a(tem)(U, u) ×u ν(U, u) = γ [F (tem)(U, u) + F (U, u)] ×u ν(U, u) . (2.256)

2.4.2 Analogy with electromagnetism: gravitoelectromagnetism Comparison of the spatial projection of the acceleration equation with the Lorentz force per unit mass and charge exerted by an electromagnetic field expressed in terms of the electric and magnetic fields leads to analogous electric and magnetic spatial gravitational forces. To appreciate the analogy, for which the term “gravitoelectromagnetism” seems an appropriate description, one must split the Lorentz force associated with the electromagnetic 2-form (4)F [. The prefix (4) will subsequently be used on the electromagnetic field to avoid confusion with the kernel letter for spatial forces. The Lorentz 4-force on a charged particle of charge q, mass m, and 4-velocity U is

(EM) α (4) α β (4) α β (4) α β [f (U)] = q F βU = qγ[ F βu + F βν(U, u) ] , (2.257) f (EM)(U) = q (4)F U = q γ[(4)F u + (4)F ν(U, u)] .

This splits into the spatial force and power

(EM) F (U, u) = q[E(u) + ν(U, u) ×u B(u)] , (2.258) (EM) ℘ (U, u) = qE(u) ·u ν(U, u) , with similar equations for the per unit mass quantities. These expressions may be compared with the spatial gravitational forces which appear in the splitting of the acceleration in terms of either the ordinary or co-rotating Fermi-Walker or Lie total spatial covariant derivative of the covariant spatial momentum. In each case the spatial gravitational force and power have the following form (G) F (U, u) = mγ[~g(u) + H(tem)(u) ν(U, u)] , (tem) (2.259) (G) ℘ (U, u) = mγ[~g(u) ·u ν(U, u) − θ(u)(ν(U, u), ν(U, u))] . Apart from an additional overall gamma factor not present in the electromagnetic case, one has both electric- like and magnetic-like spatial gravitational forces which Thorne [Thorne et al 1986] has called the gravitoelectric and gravitomagnetic forces respectively in slightly diﬀerent contexts. The same terminology will be extended to each of the various points of view and choices of temporal derivatives. Independent of the choice of time derivative one has a unique gravitoelectric vector ﬁeld

~g(u) = −a(u) (2.260)

due to the acceleration of the observer congruence. The gravitomagnetic tensor ﬁeld depends on the choice of time derivative and in the Lie case the contravariant or covariant form of the momentum equation

H(fw)(u) = ω(u) − θ(u) = k(u) , H (u) = 2ω(u) − θ(u) , (cfw) (2.261) H(lie])(u) = 2ω(u) − 2θ(u) = 2k(u) ,

H(lie[)(u) = 2ω(u) .

57 The antisymmetric part of the gravitomagnetic tensor, apart from a factor of two, defines a unique gravitomagnetic vector field H~ (u) = 2~ω(u) (2.262) which is just twice the vorticity vector of the observer congruence. Using these definitions the spatial gravitational forces per unit mass may be written

˜(G) 1 ~ F(fw)(U, u) = γ[~g(u) + 2 ν(U, u) ×u H(u) − θ(u) ν ] , ˜(G) ~ F(cfw)(U, u) = γ[~g(u) + ν(U, u) ×u H(u) − θ(u) ν(U, u)] , (2.263) ˜(G) ~ F(lie)(U, u) = γ[~g(u) + ν(U, u) ×u H(u) − 2θ(u) ν(U, u)] , ˜(G) ~ F(lie[)(U, u) = γ[~g(u) + ν(U, u) ×u H(u)] .

The gravitomagnetic symmetric tensor ﬁeld SYM H(tem)(u), which has no analog in electromagnetism, arises from the temporal derivative of the spatial metric, which is the new ingredient in gravitoelectromagnetism that makes it fundamentally diﬀerent from electromagnetism. The spatial derivatives of the spatial metric also enter the total spatial covariant derivative of the spatial momentum as a “space curvature” force term when expressed in terms of an observer-adapted frame

a D(tem)(U, u)˜p(U, u) /dτ(U,u) a a b = dp˜(U, u) /dτ(U,u) + C(tem)(u) bp˜(U, u) (2.264) a c b + Γ(u) bcp˜(U, u) ν(U, u) .

This “space curvature” force term is quadratic in the spatial velocity. One can thus think of the spatial metric as a potential for these two diﬀerent spatial forces, both associated with the spatial geometry, namely a the relative distances and directions of nearby observers in the observer congruence. The matrix C(tem)(u) b given by equation (2.196) depends on how the spatial frame is transported along the congruence and may be conveniently be chosen to vanish for one of the three temporal derivatives. To handle the case of a lightlike test particle with zero rest mass, one must work with the null 4-momentum P instead of a timelike 4-velocity U. The spatial gravitational force and power then take the form

(G) F (P, u) = E(P, u)[~g(u) + H(tem)(u) ν(P, u)] , (tem) (2.265) (G) [ ℘ (P, u) = E(P, u)[~g(u) ·u ν(P, u) + SYM H(tem)(u) (ν(U, u), ν(U, u)) ,

and measurement of the null geodesic equation yields

(G) D(tem)(P, u)p(P, u)/dτ(P,u) = F (tem)(P, u) , (G) (2.266) dE(P, u)/dτ(P,u) = ℘ (P, u) , tem = fw,cfw,lie,lie[ .

The gravitoelectric force roughly corresponds to the centrifugal force experienced in uniformly rotating coordinates in ﬂat spacetime, while the gravitomagnetic force roughly corresponds to the Coriolis force, but the details of the correspondence will be discussed below in the context of a nonlinear reference frame.

2.4.3 Maxwell-like equations The analogy between the gravitoelectromagnetic vector ﬁelds and the electromagnetic ones shows that the exterior derivative of the observer velocity 1-form corresponds to the electromagnetic 2-form, which is itself locally the exterior derivative of a 4-potential 1-form

d(4)A = u[ ∧ E(u)[ + ∗(u)B(u)[ = (4)F [ , (2.267) du[ = u[ ∧ ~g(u)[ + ∗(u)H~ (u)[ .

58 The observer 4-velocity thus acts as the 4-potential for the gravitoelectromagnetic vector ﬁelds. Similarly in direct analogy with the Lorentz force, the right contraction of the 4-velocity U of a test particle with the mixed form of the 2-form du[ also produces the part of the spatial gravitational force per unit mass ˜(G) F(tem)(U, u) which involves only the gravitoelectromagnetic vector ﬁelds for tem=cfw,lie,lie[, although not for 1 the Fermi-Walker case which involves an additional factor of 2 . The splitting of the identity d(4)F [ = d2(4)A[ = 0 leads to half of Maxwell’s equations

∗(u) (4) [ (M) [d F ] = divu B(u) + H~ (u) ·u E(u) = 0 ,

∗(u) (4) [ (E)] [d F ] = − curlu E(u) + ~g(u) ×u E(u) (2.268) − [£(u)u + Θ(u)]B(u) = 0 ,

which follow from equation (2.114) with S = (4)F [ and making use of the identity (2.100) with S = B(u). Replacing (4)A by (4)u reduces these to the corresponding gravitoelectromagnetic equations

[divu +~g(u)·u]H~ (u) = 0 , (2.269) ~ curlu ~g(u) + [£(u)u + Θ(u)]H(u) = 0 , which are just equations (2.130) and (2.131) rewritten in terms of the gravitoelectromagnetic vector ﬁelds. Splitting the remaining half of Maxwell’s equations

∗d ∗(4)F = 4π(4)J (2.270) using equation (2.114) with S = ∗(4)F [ and the identities (2.24), which become

[ ∗(4)F [](E)(u) = −B(u)[ , [ ∗(4)F [](M)(u) = ∗(u)E(u)[ (2.271) when rewritten in terms of the electric and magnetic ﬁelds, leads to

∗(u) ∗(4) [ (M)] [d F ] = divu E(u) − H~ (u) ·u B(u) = 4πρ(u) ,

∗(u) ∗(4) [ (E)] [d F ] = curlu B(u) − ~g(u) ×u B(u) (2.272) − [£(u)u + Θ(u)]E~ (u) = 4πJ(u) , where (4)J = ρ(u)u + J(u) is the splitting of the 4-current. The remaining Maxwell-like equations for the gravitoelectromagnetic vector ﬁelds arise from the Einstein equations. In order to state them one must ﬁrst introduce appropriate spatial curvature tensors associated with the spatial part of the spatial connection of the observer congruence u, and then split the spacetime curvature. This will be done below. Forward [1961] systematically developed the analogy between general relativity and electromagnetism in the post-Newtonian limit.

59 2.4.4 Splitting the spin transport equation The kinematical tensor k(u) has already been seen to relate spatial Fermi-Walker transport along the observer congruence to the spatial Lie transport. When applied to a spatial vector S this may be interpreted as the equation describing the evolution of the spin of a torque-free gyro carried by an observer. The following are equivalent ways of stating this transport

∇(fw)(u)S = 0 ,

∇(cfw)(u)S = ω(u) S

= −~ω(u) ×u S, (2.273)

∇(lie)(u)S = k(u) S

= −~ω(u) ×u S − θ(u) S.

The spin vector must rotate with the angular velocity of gravitomagnetic precession

1 ~ ζ(gm)(u) = −~ω(u) = − 2 H(u) (2.274) equal to minus the vorticity relative to a spatially Lie dragged spatial frame or a co-rotating Fermi-Walker propagated spatial frame in order to undo the local rotation of such a frame. It must also change in a way that undoes the expansion and shearing of the spatially Lie dragged frame vectors relative to a Fermi-Walker or co-rotating Fermi-Walker propagated spatial frame. One can also split the equation describing the transport of the spin vector of a gyro along an arbitrary worldline with 4-velocity U. Since the spin vector is spatial with respect to U, it has the form

S = [ν(U, u) ·u S~]u + S,~ (2.275) where is the spatial projection S~ = P (u, U)S of the spin. Since Fermi-Walker transport does not change the magnitude ||S|| of the 4-vector, the magnitude of the spatial projection of the spin will change as the inner product ν(U, u) ·u S~ changes 2 2 2 ||S~|| = ||S|| + [ν(U, u) ·u S~] , (2.276) leading to a change in the observed frequency of rotation associated with the spin vector. Since the temporal component of the spin is determined by its spatial projection, one only needs an evolution equation for the spatial projection, which is provided by the spatial projection of the Fermi-Walker transport equation. Because of the spatial condition on S relative to U, the Fermi-Walker transport condition reduces to (4) (4) β 0 = ∇(fw)(U)S = ∇(fw)(U)S = ∇U S − a(U)βS U (2.277) and its spatial projection is

−1 0 = γ P (u)∇(fw)(U)S −1 (4) β = γ P (u)[ ∇U S − a(U)βS U] (2.278) ~ ~ β = D(fw)(U, u)[(ν(U, u) ·u S)u + S]/dτ(U,u) − a(U)βS ν(U, u) ~ ~ β = (ν(U, u) ·u S)D(fw)(U, u)u/dτ(U,u) + D(fw)(U, u)S/dτ(U,u) − a(U)βS ν(U, u) .

Solving this for the Fermi-Walker total spatial covariant derivative and observing that

β a(u)βS = −γ[ν(U, u) ·u A(U, u)][ν(U, u) ·u S~] + γA(U, u) ·u S~ (2.279) [ −1 = γA(U, u) Pu(U, u) S~

using equation (2.57), one gets the result

~ −1 ~ ˜(G) [ −1 ~ D(fw)(U, u)S/dτ(U,u) = γ [ν(U, u) ·u S]F (fw)(U, u) − γ[A(U, u) Pu(U, u) S]ν(U, u) , (2.280)

60 which is easily re-espressed in terms of the co-rotating Fermi-Walker derivative

~ ~ −1 ~ ˜(G) D(cfw)(U, u)S/dτ(U,u) = −~ω(u) ×u S + γ [ν(U, u) ·u S]F (U, u) (fw) (2.281) [ −1 − γ[A(U, u) Pu(U, u) S~]ν(U, u) .

These equations may be decomposed into separate evolution equations for the length ||S~|| and direction Sˆ = ||S~||−1S~ of the spatial projection of the spin vector. The results are

~ −1 ˆ ˜(G) ˆ D(U, u) ln ||S||/dτ(U,u) = γ [ν(U, u) ·u S][F (U, u) ·u S] (fw) (2.282) [ −1 − γ[A(U, u) Pu(U, u) S~][ν(U, u) ·u Sˆ] ,

and ˆ ˆ ˆ D(tem)(U, u)S/dτ(U,u) = Ω(tem)(S, U, u) ×u S, (2.283) where ˆ −1 ˆ ˜(G) ˆ Ω(fw)(S, U, u) = −γ [ν(U, u) ·u S] F (fw)(U, u) ×u S [ −1 + γ[A(U, u) Pu(U, u) Sˆ] ν(U, u) ×u S,ˆ (2.284) ˆ ˆ Ω(cfw)(S, U, u) = Ω(fw)(S, U, u) − ~ω(u) . These formulas describe the precession of the spin vector as seen by the family of diﬀerent observers of the observer congruence along the gyro’s worldline. Note that the angular velocity of the direction of the spatial projection of the spin vector depends on the direction of that spin vector. These formulas describe how the spin vector S changes as seen by the 1-parameter family of observers belonging to the observer congruence which intersect the worldline of the gyro. The observed spin vector S~ = P (u)S does not undergo a rotation with respect to the Fermi-Walker propagated axes of the observer but a more complicated motion required to keep S orthogonal to U. This is superimposed upon the gravitomagnetic precession when compared to co-rotating Fermi-Walker propagated axes, and complicated by the expansion and shear eﬀects when compared to spatial Lie dragged axes.

61 2.4.5 Relative Fermi-Walker transport and gyro precession Another question one can consider is how a single observer with 4-velocity U following the gyro worldline would see the orientation of the gyro change with respect to a spatially co-rotating observer-adapted frame anchored in the given observer congruence u. The observer following the gyro will see these axes to be in relative motion and not even appear orthonormal. However, a basic assumption of special relativity implicit when one discusses pure Lorentz transformations in Minkowski space is that the orientation of a set of orthonormal axes in relative motion is defined to be the orientation they would have if they were not in relative motion. In other words one “brings them to rest” by the inverse relative observer boost in order to define their orientation with respect to a given orthonormal spatial frame in the local rest space of an observer. Suppose {ea} is a spatially co-rotating observer-adapted spatial frame along the observer congruence u. Then B(lrs)(U, u)ea are the axes “momentarily at rest” with respect to the gyro, and the orientation of the spin vector S with respect to them is well-defined and represents the orientation of S with respect to the moving axes ea. However, since the boost is an isometry between the two local rest spaces, the orientation of S with respect to B(U, u)ea is the same as the orientation with respect to ea of the boosted spin vector

S ≡ B(u, U)S (2.285)

in the local rest space of the observer congruence. Again since the boost is an isometry, the lengths of S and S are the same, and since the latter is constant under Fermi-Walker transport, so is the former. In other words S will rotate with respect to the orthonormal spatial frame ea, and its angular velocity with respect to the sequence of observers from the observer congruence along the gyro worldline will be the same as the angular velocity of S with respect to the boosted spatial frame B(U, u)ea, apart from a proper time renormalization. In order to calculate these angular velocities, one must consider the evolution of S along the gyro worldline. This boosted spin vector has the explicit expression

−1 −1 S = S~ − γ (γ + 1) [˜p(U, u) ·u S~]˜p(U, u) , (2.286) which is obtained by rewriting equation (2.55) in terms of the spatial momentum per unit mass. Since the Fermi-Walker total spatial covariant derivatives of S~,p ˜(U, u), and γ = E˜(U, u) are given by equations (2.280), (2.248), and (2.252), it is straightforward to evaluate the corresponding derivative of S. After some algebra one ﬁnds a result which may be expressed in the following equivalent forms

D (U, u)S/dτ = ζ (U, u) ×u S , (tem) (U,u) (tem) (2.287) tem=fw,cfw , where the Fermi-Walker and co-rotating Fermi-Walker “relative angular velocities” are related to each other by by the gravitomagnetic precession

ζ(cfw)(U, u) = −~ω(u) + ζ(fw)(U, u) 1 ~ (2.288) = − 2 H(u) + ζ(fw)(U, u) = ζ(gm)(u) + ζ(fw)(U, u) and the former is deﬁned by

−1 ˜ −1 ˜(G) ζ(fw)(U, u) = −γ(γ + 1) ν(U, u) ×u F (U, u) + (γ + 1) ν(U, u) ×u F (U, u) (fw) (2.289) = ζ(thom)(U, u) + ζ(geo)(U, u) .

This may also be expressed in terms of the relative acceleration

2 −1 ζ(fw)(U, u) = −γ (γ + 1) ν(U, u) ×u a(cfw)(U, u) (2.290) ˜(G) −1 + ν(U, u) ×u [F(cfw)(U, u) − γ(γ + 1) ν(U, u) ×u ~ω(u)] ,

62 a formula first obtained by Massa and Zordan [1975]. These angular velocities, in contrast with the result ˆ for Ω(cfw)(S, U, u), depend only on the relative boost between the local rest spaces. For a gyro at rest relative to the observer congruence, only the gravitomagnetic precession term is nonzero. This is also known as the frame-dragging precession or the Lense-Thirring precession, since the effect was first studied by Lense and Thirring [1918] in the context of weak fields and slow motion. The Thomas precession term

−1 ˜ ζ(thom)(U, u) = −γ(γ + 1) ν(U, u) ×u F (U, u) (2.291)

in the relative angular velocity is due to the applied force acting on the gyro. In Minkowski spacetime choosing u to be a unit timelike rotation-free Killing vector ﬁeld corresponding to a time translation, only this term is nonzero and has the equivalent expression

2 −1 ζ(thom)(U, u) = −γ (γ + 1) ν(U, u) ×u a(cfw)(U, u) −2 (2.292) = −||ν(U, u)|| (γ − 1)ν(U, u) ×u a(cfw)(U, u)

in terms of the relative acceleration

−1 −1 ˜ a(cfw)(U, u) = a(fw)(U, u) = a(lie)(U, u) = γ Pu(U, u) F (U, u) (2.293)

which in this case represents the usual spatial acceleration of special relativity. The acceleration of the gyro in Minkowski space causes it to precess as discussed by Thomas [1927]. In the limit ||ν(U, u)|| 1 and γ → 1 of nonrelativistic relative motion, the Thomas precession under these conditions reduces to

1 1 ˜ ζ(thom)(U, u) → − 2 ν(U, u) ×u a(cfw)(U, u) = − 2 ν(U, u) ×u F (U, u) . (2.294)

For circular motion with angular velocity Ω,~ the precession angular velocity is [γ − 1]Ω,~ as described in exercise 6.9 of Misner, Thorne and Wheeler [1973]. The ﬁrst term in their equation (6.28) is exactly the boosted spin vector S. The second term in the Fermi-Walker angular velocity

−1 ˜(G) ζ(geo)(U, u) = (γ + 1) ν(U, u) ×u F (U, u) (fw) (2.295) −1 1 ~ = γ(γ + 1) ν(U, u) ×u [~g(u) + 2 ν(U, u) ×u H(u) − θ(u) ν(U, u)] is due to the spatial gravitational force alone, which is present even in the absence of an applied force, i.e., for geodesic motion of the gyro. For this reason it is often called the geodetic precession, in addition to the commonly used alternatives de Sitter precession or Fokker precession or even Fokker-de Sitter precession, named after the original investigators of this phenomenon [de Sitter 1916, Fokker 1920, Pirani 1956]. In the nonrelativistic limit γ → 1, neglecting terms of second order in the velocity, this term has the limiting expression 1 ζ(so)(U, u) = 2 ν(U, u) ×u ~g(u) . (2.296) Thorne [1989] describes this nonrelativistic term as an “induced gravitomagnetic precession” or “spin-orbit” precession since it corresponds to the gravitomagnetic precession due to an additional “induced” gravito- ~ magnetic field H(u)(ind) = −ν(U, u) ×u ~g(u) induced by the motion of the gyro in the gravitoelectric field in analogy with the induced magnetic field due to motion in an electric field. These relative angular velocities must still be interpreted in terms of the original discussion of the angular velocity of the relative orientation of the gyro spin and the spatially co-rotating observer-adapted spatial frame. These angular velocities do not depend on the choice of spatial frame but only involve the relative observer boost between the gyro and the observer congruence. They describe the angular velocities of the boosted spin vector S with respect to a spatial frame which is transported along the gyro worldline by the corresponding spatial transport. This transport seeks to eliminate the arbitrariness in the spatial distribution of the orientation of any given spatially co-rotating observer-adapted spatial frame, at least along the given gyro worldline.

63 However, if the gyro returns to a given observer worldline belonging to the observer congruence, the spatial frame transported along its worldline will in general diﬀer from the value of the spatially co-rotating observer-adapted spatial frame there. This is a manifestation of “space curvature” which occurs as long as the spatial metric is not ﬂat. If one is really interested in measuring the relative orientation of S with respect to a particular spatially co-rotating observer-adapted orthonormal spatial frame, one must remove the additional angular velocity of the co-rotating Fermi-Walker transported orthonormal spatial frame with respect to it, leading to a frame-dependent result for the ordinary derivative of the components of S with respect to the former frame

a b c dS /dτ(U,u) = abc[ζ(cfw)(U, u) + ζ(sc)(U, u, e)] S . (2.297)

Thorne [Thorne et al 1986] has referred to the relative angular velocity ζ(sc)(U, u) in the precession formula as the space curvature term. Of course now that ordinary derivatives are being used one has to choose the observer-adapted frame carefully in order that this make some physical sense. Finally, normalizing the proper time to correspond to that of the gyro instead of that of the observer congruence, the angular velocity of the given spatially co-rotating observer-adapted orthonormal spatial frame {ea} as seen by the gyro has components

a a a ζ(gyro)(U, u, e) = γ[ζ(cfw)(U, u) + ζ(sc)(U, u, e) ] (2.298)

a with respect to the boosted frame B(lrs)(U, u)ea. In other words, if S are the components of the spin vector with respect to this boosted frame, then

a b c dS /dτU = abcζ(gyro)(U, u, e) S . (2.299)

The precession formula will have a physical meaning in a given spacetime first only if the observer congruence itself has some physical meaning and second if the spatially co-rotating observer-adapted orthonormal spatial frame has some physical meaning. The obvious candidate satisfying the first condition is an observer congruence associated with a timelike Killing vector field in a stationary spacetime. Since the shear of such an observer congruence vanishes, the co-rotating Fermi-Walker transport along u reduces to spatial Lie transport, and so the spatially co-rotating observer-adapted orthonormal spatial frame is also a stationary frame. This spatial frame should also be as “Cartesian-like” as possible, i.e., the distribution of its orientation should be as aligned as possible, in order that the angular velocity have some meaning at each point along the gyro worldline. For spacetimes which are asymptotically flat at spacelike infinity, Nester [1991a, 1991b] has shown that a preferred spatial orthonormal frame exists on a given asymptotically flat spacelike hypersurface which asymptotically approaches a given inertial frame at spacelike infinity and in some sense has “Cartesian- like properties.” Given a preferred slicing by spacelike hypersurfaces, one could then boost these spatial frames into the local rest space of a preferred observer congruence to define a preferred class of observer- adapted orthonormal frames which might be used in interpreting the angular velocity. Stationary axially symmetric spacetimes have a preferred slicing orthogonal to the zero-angular-momentum observers, and so must possess such preferred frames. Whether or not they are compatible with the spatially-comoving condition is not clear. Alternatively, one can consider worldlines which return to a given observer worldline and attempt to calculate the total rotation of the spin between the initial and final intersection points (corresponding to a closed loop in the space of observers) to avoid this problem. This is the case for circular orbits often considered in gyro precession calculations, where the net rotation per revolution is of interest.

64 2.4.6 The Schiff Precession Formula The classic spin precession formula of Schiff [1960] describes how the spin vector precesses relative to the “distant stars” as seen by an observer carrying the gyro in the gravitational field of an isolated body [Weinberg 1972, Misner, Thorne and Wheeler 1973]. This is evaluated locally by using a spatial frame which is somehow anchored in the asymptotically flat part of the spacetime around such an isolated body. The discussion is cleaner for a stationary spacetime of this type with a stationary frame adapted to the asymptotically nonrotating stationary observers, the so-called static observers. Since spatial Lie transport along u coincides with co-rotating Fermi-Walker transport, the spatial projection of any frame which is comoving with respect to u will yield a spatial frame which is spatially comoving and which undergoes co-rotating Fermi-Walker transport along u. Call this a static spatial frame. Let {u, ea} be a spatially- comoving observer-adapted frame, whose spatial frame is therefore static. One can choose this frame to be orthonormal. For a black hole spacetime in Boyer-Lindquist coordinates, for example, the spatial coordinate differentials are orthogonal to each other and to the asymptotically-nonrotating timelike Killing vector field, and so one may normalize them to obtain such a frame. Light rays from “distant stars” will arrive at a given static observer from fixed directions relative to the static spatial frame. In this way the stationary symmetry anchors the static observer frame to the asymptotically flat part of the spacetime. It can then be used for a local comparison of the spin vector of a gyroscope with the “distant stars.” How can the observer carrying the gyro make this comparison? The static observer’s spatial frame is moving with respect to the gyro. The boost Ea = B(U, u)ea of the static observer’s spatial frame to the rest frame of the gyro is the spatial frame the gyro would see if the static observer were not in relative motion. The angular velocity ζ(gyro)(U, u, e) evaluated above describes the rotation of the spin vector with respect this boosted frame. This angular velocity depends crucially on the spatial distribution of the orientation of the static spatial frame {ea}, i.e., on how the orientation depends on the location of the observer in the observer congruence. If one has a stationary axially symmetric spacetime with an axially symmetric static spatial frame like the one described above for a black hole arising from the normalized Boyer-Linquist spatial coordinate differentials, then the static frame will itself rotate through an angle of 2π for each loop around a circular orbit of the axial symmetry subgroup. Clearly one does not want to measure the gyro precession with respect to such a frame, but rather one with Cartesianlike properties, if such a frame exists. The actual Schiff formula corresponds to the value of the angular velocity ζ(gyro)(U, u, e) in the slow motion weak field limit of general relativity for the linearized gravitational field of an isolated distribution of matter. It has been generalized to the PPN theory as discussed by Misner, Thorne and Wheeler [1973] or Thorne [1989]. The key difference between it and the limiting expression for ζ(cfw)(U, u)

1 ~ 1 ˜ 1 ζ(cfw)(U, u) → − 2 H(u) − 2 ν(U, u) ×u F (U, u) + 2 ν(U, u) ×u ~g(u) (2.300) is the fact that in this limit within general relativity, the space curvature precession has twice the value of 1 3 the spin orbit precession ζ(so)(U, u) → 2 ν(U, u) ×u ~g(u) leading to a total coefficient of 2 . The sum of these two terms is conventionally referred to as the geodetic precession. This will be discussed below in chapter ?? on the post-Newtonian limit. This same result generalizes in a simple way to the parametrized post-Newtonian (PPN) context following the same notation as Misner, Thorne and Wheeler [1973]. The PPN spatial coordinates are orthogonal to lowest order and may be normalized to define an orthonormal spatial frame which may be completed to an orthonormal frame by the addition of the normalization of the time coordinate derivative vector field which is orthogonal to the spatial coordinate differentials to lowest order. This is exactly analogous to the black hole case described above. However, all of the linearized discussions of spin precession are somewhat clouded by the mixing of the linearization process itself with the features of the spin precession. Without the assumption of stationarity, the entire discussion no longer has a very solid foundation. In the PPN context, the PPN coordinate grid is anchored to the asymptotically flat part of the spacetime but one has an entire class of gauge transformations which allow the coordinate grid to change. The measurement of the precession of a torque-free gyro in a free-fall earth orbit is the goal of the long awaited Stanford Gravity-Probe B experiment [Everitt 1979]. Of course to get explicit formulas for the the

65 actual cumulative precession measured, one must consider the actual geometry of the earth-satellite system, as well as the aberration eﬀects involved in the star-tracking telescope. It is this experiment, the proposed LAGEOS experiment [Ciufolini 1990], and others with similar goals [Nordvedt 1988, Mashhoon, Paik, and Will 1989] which have provided much of the motivation for talking about “gravitomagnetism.”

66 2.4.7 The relative angular velocity as a boost derivative

The formula for the co-rotating relative angular velocity ζ(cfw)(U, u) has been obtained in a two-step process, ﬁrst evaluating the evolution of the spatial projection of the spin vector and then of the transformation which converts it to the boosted spin vector. This quantity may be obtained instead directly from the evolution of the relative boost between the local rest spaces along the worldline. Starting from the Fermi-Walker transport equation for the spin vector S

(4) D(U)S/dτU = σ(U ∧ a(U))S, (2.301) one can calculate the spatial projection of the total covariant derivative of the boosted spin vector S = B(lrs)(u, U)S (4) D(fw)(U, u)S/dτ(U,u) = P (u) D(U)S/dτu (4) = P (u) D(U)B(lrs)(u, U)/dτ(U,u) B(lrs)(U, u)S (2.302) (4) + P (u)B(lrs)(u, U) D(U)S/dτ(U,u) , suppressing the contraction notation of the linear transformations involved. The second term vanishes

P (u)B (u, U)(4)DS/dτ = γ(U, u)P (u)B (u, U)σ(U ∧ a(U))B (U, u)S (lrs) (U,u) (lrs) (lrs) (2.303) = γ(U, u)P (u)σ(u ∧ B(lrs)(u, U)a(U))S = 0 since the factor of u is killed by the spatial projection or by contraction with the spatial field S. The first term defines the boost angular velocity tensor

(4) ∗(u) W (U, u) = P (u)[ D(U)B(lrs)(u, U)/dτ(U,u) B(lrs)(U, u)] = − ζ(fw)(U, u) , (2.304) which is an antisymmetric spatial tensor. Its sign-reversed spatial dual therefore deﬁnes the relative angular velocity vector ζ(fw)(U, u). The antisymmetry of W (U, u) follows since this represents the spatial part of the covariant Lie algebra derivative of the 1-parameter family of Lorentz transformations. This leads to the result D(fw)(U, u)S/dτ(U,u) = ζ(fw)(U, u) ×u S (2.305) describing the precession of the boosted spin vector. The relative angular velocity diﬀers from the co-rotating angular velocity by the vorticity of the observer congruence

ζ(cfw)(U, u) = ζ(fw)(U, u) − ~ω(u) , (2.306) i.e., by the gravitomagnetic precession.

67 2.4.8 Relative kinematics: transformation of spatial gravitational fields The spatial gravitational force fields are simply related to the kinematical quantities associated with the observer congruence. If one has two distinct observer congruences with unit tangents u and U, one can describe the transformation law between the spatial gravitational fields observed by each. One need only express the quantities and operators in the expression for the spatial gravitational fields of one in terms of those of the other to obtain such laws, as in the above derivation of the transformation law for the electric and magnetic fields. The acceleration and kinematical field transform as follows a(U) = γ2P (u, U)−1[a(u) − k(u) ν(U, u)] 2 + γ P (U, u)a(fw)(U, u) , (2.307) k(U) = γP (U, u)[k(u) − a(u) ⊗ ν(U, u)[] − γ∇(U)ν(U, u) . The expansion tensor and the rotation tensor and vector then transform as [ 2 [ 1 θ(U) = γ P (U, u)[θ(u) + 2 {a(u) ⊗ ν(U, u) + ν(U, u) ⊗ a(u)] , [ 2 [ 1 ω(U) = γ P (U, u)[ω(u) − 2 a(u) ∧ ν(U, u)] 1 [ (2.308) + 2 γd(U)ν(U, u) , 2 −1 1 ~ω(U) = γ P (u, U) [~ω(u) + 2 ν(U, u) ×u a(u)] 1 + 2 γ curlU ν(U, u) , Converting to the gravitoelectromagnetic symbols leads to 2 −1 H~ (U) = γ P (u, U) [H~ (u) − ν(U, u) ×u ~g(u)]

+ γ curlU ν(U, u) , 2 −1 1 ~ ~g(U) = γ P (u, U) [~g(u) + 2 ν(U, u) ×u H(u) 2 (2.309) − θ(u) ν(U, u)] − γ P (U, u)a(fw)(U, u) 2 −1 = γ P (u, U) [~g(u) + ν(U, u) ×u H~ (u) 2 − θ(u) ν(U, u)] − γ P (U, u)a(cfw)(U, u) , −1 ˜(G) where the expressions in square brackets in the gravitoelectric field transformation laws are just γ F (fw)(U, u) −1 ˜(G) and γ F (cfw)(U, u) respectively, analogous to the Lorentz force and its magnetic analog which appear in the transformation law for the electric and magnetic fields. The terms explicitly involving the gravitoelectromagnetic vector fields in the transformation law for the gravitomagnetic vector field and in the second form of the one for the gravitoelectric vector field are exactly analogous to the corresponding transformation laws for the magnetic and electric fields, apart from the extra gamma factor also present in the force law itself. Apart from the expansion term, the remaining part of the transformation law which breaks this correspondence, namely the relative acceleration and the relative velocity curl, can be further expanded. The relative acceleration, for example, can be re-expressed using equation (2.255). For the relative curl, one can apply the following useful formula −1 curlU X = γP (u, U) {curlu X (2.310) [ ] + ν(U, u) ×u [£(u)uX ] } , valid when X is spatial with respect to u. For an observer-adapted co-rotating Fermi-Walker orthonormal frame, the gravitoelectric and gravitomagnetic fields are related to the 2-form which results from evaluating the tensor-valued connection 1-form on u in a way similar to the way the electric and magnetic fields are related to the electromagnetic 2-form [jancar91]. The homogeneous part of the transformation law for the connection then leads to the terms in the transformation law for the gravitoelectric and gravitomagnetic vector fields which are analogous to those for the electric and magnetic fields.

68 SectionThe hypersurface point of view The zero vorticity case ω(u) = 0 of the congruence point of view essentially defines the hypersurface point of view, described in this way by Zel’manov [1973], Ehlers [1961, 1993] and Ellis [1971]. In this special case one has an integrable family of local rest spaces LRSu which define a spacelike slicing of spacetime with u equal to the unit normal n for this slicing. The spatial metric P (n)[ when restricted to a slice equals the induced Riemannian metric of the slice, while the the kinematical tensor k(n)[ = −θ(n)[ restricts to the extrinsic curvature tensor K[ of the slice. There is a natural isomorphism between spatial tensor fields on a given slice and the tensor algebra of the slice as a manifold itself, under which the spatial covariant derivative operator for spatial tensor fields corresponds to the covariant derivative of the induced metric connection. Because the vorticity vanishes, the covariant normal admits a family of integrating factors, so the acceleration or equivalently the gravitoelectric field admits a family of potentials

[ n = −Ndt , a(n) = gradn ln N = −~g(n) . (2.311)

Thus − ln N serves as an “acceleration potential” [Ehlers 1961, 1993] while ln N serves as the gravitoelectric potential. The integrating factor N is only defined to within a factor whose spatial derivative vanishes, corresponding to a reparametrization of the slicing time function t. Since the vorticity vanishes, the gravitomagnetic vector field is zero. Of course given explicitly the congruence on spacetime in terms of some local coordinates, one would have to integrate partial differential equations (for a time function) to obtain the slices, and conversely given a slicing in terms of local coordinates one would have to integrate ordinary differential equations (for the orthogonal trajectories) in order to obtain the normal congruence. This means that in practice one might have to distinguish the hypersurface and corresponding congruence points of view, but as far as the spacetime geometry of the splitting is concerned, they are the same. In the hypersurface point of view, it is natural to interchange the symbols > and ⊥. Since the slicing is considered primary rather than the implicit normal congruence, it is logical to let ⊥ stand for perpendicular to the slicing and > for tangential to the slicing, compared to the symbols > and ⊥ for tangential and perpendicular to the congruence in the corresponding congruence point of view adapted to the normal 1 congruence. Thus the decomposition of a spacetime 1 -tensor field S corresponding to the ordered family of spatial fields listed above for the equivalent normal congruence point of view becomes instead

0 a 0 a S ↔ {S 0,S 0,S a,S b} ⊥ a ⊥ a ↔ {S ⊥,S ⊥,S a,S b} , (2.312) ⊥ > ⊥ > ↔ {S ⊥,S ⊥,S >,S >} in the dual hypersurface point of view. The commonly used single vertical bar notation for the spatial covariant derivative index notation will also be adopted in place of the double vertical bar

α... α... α... [∇(n)S] β... = ∇(n)γ S β... = S β...|γ . (2.313)

69 2.5 Spatial curvature and torsion 2.5.1 Definitions Having explored the spatial and spacetime connections, the next step is to study the curvatures associated with both. The spatial versions of the usual invariant formulas defining the torsion and Riemann curvature tensors of a connection may be used to define the spatial torsion and curvature tensors associated with the spatial part of the spatial connection. Expressing them in terms of an observer-adapted frame then gives the component definitions. The spatial torsion definition gives no surprises, vanishing under the assumption that the spacetime torsion is zero

∇(u)XY − ∇(u)Y X − [X,Y ](u) = –T (u)(X,Y ) = 0 , a a a (2.314) –T (u) bc = 2Γ(u) [bc] − C bc = 0 ,

where X and Y are spatial vector fields. However, the rotation of the congruence leads to an additional temporal derivative term in the spatial curvature formula which arises from the temporal part of the commutator of two spatial vector fields. Three different curvature tensors then result from the choice of the Lie, Fermi-Walker or co-rotating Fermi-Walker temporal derivative to express the additional term

[ [∇(u)X, ∇(u)Y ] − ∇(u)[X,Y ] Z = R(tem)(u)(X,Y )Z + 2ω(u) (X,Y )∇(tem)(u)Z, (2.315) tem = lie,fw,cfw ,

where X, Y , and Z are spatial vector ﬁelds. These three tensors, the Lie spatial curvature tensor and the Fermi-Walker spatial curvature tensor and the co-rotating Fermi-Walker spatial curvature tensor diﬀer by the same kinematical terms as the temporal derivatives themselves but reversed in sign

[ R(fw)(u)(X,Y )Z = R(lie)(u)(X,Y )Z + 2ω(u) (X,Y )k(u) Z, (2.316) [ R(cfw)(u)(X,Y )Z = R(lie)(u)(X,Y )Z − 2ω(u) (X,Y )θ(u) Z,

Expressing the deﬁning identity in an observer-adapted frame leads to the formula

a a a a f R(tem)(u) bcd = Γ(u) db,c − Γ(u) cb,d − Γ(u) fbC cd (2.317) a f a f a + Γ(u) cf Γ(u) db − Γ(u) df Γ(u) cb − 2C(tem)(u) bω(u)cd .

The three curvature tensors are related to each other by

a a a R(fw)(u) bcd = R(lie)(u) bcd + 2k(u) bω(u)cd , a a a (2.318) R(cfw)(u) bcd = R(lie)(u) bcd − 2θ(u) bω(u)cd .

These are clearly antisymmetric in their ﬁnal pair of indices, but the remaining symmetries characterizing an ordinary curvature tensor must be checked. The formula for the components of these spatial curvature tensors in terms of the spatial connection components and the spatial structure functions diﬀers from the usual one only in the last term. By choosing an observer-adapted frame whose spatial frame undergoes the corresponding transport, this term is set to zero. For the Lie spatial curvature tensor this term is absent in a spatially-comoving spatial frame, and hence in the discussions of Zel’manov and Ferrarese who rely on such frames.

2.5.2 Algebraic symmetries An alternative way to obtain these definitions which reveals the symmetry properties of the first pair of indices relies on the spatial Ricci identity for a spatial vector field or 1-form, which for zero torsion have the

70 form α α [∇(u)γ , ∇(u)δ]Z = 2Z ||[δγ] α β α = R(tem)(u) βγδZ + 2ω(u)γδ∇(tem)(u)Z ,

[∇(u)γ , ∇(u)δ]Zα = 2Zα||[δγ] (2.319) β = −ZβR(tem)(u) αγδ + 2ω(u)γδ∇(tem)(u)Zα , tem = lie,fw,cfw . Since the spatial metric has zero ordinary or co-rotating Fermi-Walker temporal derivative, the two Ricci identities for a vector ﬁeld and a 1-form are clearly related in a consistent way by index shifting as long as the corresponding curvature tensor has the usual antisymmetry property in the ﬁrst pair of indices

(αβ) (αβ) R(fw)(u) γδ = R(cfw)(u) γδ = 0 . (2.320)

This implies that the symmetric part of the Lie spatial curvature tensor is not zero

(αβ) αβ R(lie)(u) γδ = 2θ(u) ω(u)γδ , (2.321) which is exactly the condition that shifting indices in these Ricci identities be consistent when expressed in terms of that curvature and the Lie derivative term. The commutator of the Lie derivative with index shifting generates the expansion tensor term

β β β β £(u)u[P (u)αβZ ] − P (u)αβ£(u)uZ = [£(u)uP (u)αβ]Z = 2θ(u)αβZ . (2.322)

These equivalent conditions on the symmetric part of the curvature tensors are in turn equivalent to the spatial Ricci identity for the spatial metric itself, modiﬁed by the rotation term from the noncommutivity of the spatial derivatives

0 = [∇(u)γ , ∇(u)δ]P (u)αβ = −2R(lie)(u)(αβ)γδ + 2ω(u)γδ£(u)uP (u)αβ (2.323) = −2R(fw)(u)(αβ)γδ = −2R(cfw)(u)(αβ)γδ .

The relation between the curvature tensors is therefore equivalent to

R (u)αβγδ = R (u) + 2ω(u)αβω(u)γδ , (fw) (lie) [αβ]γδ (2.324) R(cfw)(u)αβγδ = R(lie)(u)[αβ]γδ .

The component formula for the Lie spatial curvature tensor is identical with the usual frame component formula for an actual Riemannian 3-manifold, at least in a spatially-comoving frame. Thus any algebraic identity which follows directly from the latter formula with no index shifting (which can lead to additional expansion tensor terms when derivatives are involved) continues to hold for the Lie spatial curvature tensor. The usual cyclic symmetry of the torsionfree Bianchi identity of the ﬁrst kind therefore holds

α α α α 3R(lie)(u) [βγδ] = R(lie)(u) βγδ + R(lie)(u) γδβ + R(lie)(u) δβγ = 0 , (2.325) as one may directly verify by expressing it in terms of the component formula and noting that all terms cancel in pairs. This implies that the other spatial curvature tensors do not satisfy the usual form of the ﬁrst Bianchi identity

α 1 α α α R(fw)(u) [βγδ] = 3 [R(fw)(u) βγδ + R(fw)(u) γδβ + R(fw)(u) δβγ ] α = 2k(u) [βω(u)γδ] , (2.326) α α R(cfw)(u) [βγδ] = −2θ(u) [βω(u)γδ] .

71 2.5.3 Symmetry-obeying spatial curvature One can isolate the kinematical terms in the spatial curvature which arise from the commutator of spatial derivatives by making the second derivative terms in the curvature formula explicit. In this observer-adapted frame component derivation one needs the following relations

0 = ∇(u)chab = ∂chab − Γ(u)acb − Γ(u)bca , ab ab a b b a (2.327) 0 = ∇(u)ch = ∂ch + Γ(u) c + Γ(u) c , reﬂecting the covariant constancy of the spatial metric and its inverse. Then one has e e R(lie)(u)abcd = hae[∂cΓ(u) db − ∂dΓ(u) cb] e e e + Γ(u)aceΓ(u) db − Γ(u)adeΓ(u) cb − C cdΓ(u)aeb − 2Ca>bω(u)cd ef ef e = [∂cΓ(u)adb − ∂dΓ(u)acb] + hae[∂ch Γ(u)fdb − ∂dh Γ(u) cb] + ...

= [∂cΓ(u)adb − ∂dΓ(u)acb] (2.328) e e − [Γ(u)ace + Γ(u)eca]Γ(u) db + [Γ(u)ade + Γ(u)eda]Γ(u) cb + ···

= [∂cΓ(u)adb − ∂dΓ(u)acb] e e e − Γ(u)ecaΓ(u) db + Γ(u)edaΓ(u) cb − C cdΓ(u)aeb − 2Ca>bω(u)cd . Now one can substitute for the covariant gammas using the derivative formulas, but the calculation is much a a easier in a spatially-holonomic (C bc = 0) spatially-comoving (C >b = 0) spatial frame 1 R(lie)(u)abcd = 2 [hab,dc − hbd,ac + hda,bc − hab,cd + hbc,ad − hca,bd] e e − Γ(u) caΓ(u)edb + Γ(u) daΓ(u)ecb 1 1 (2.329) = 2 [∂c, ∂d]hab + 2 [had,bc − hac,bd + hbc,ad − hbd,ac] + ··· 1 = 2θ(u)abω(u)cd + 2 [had,bc − hac,bd + hbc,ad − hbd,ac] + ··· . Now one sees explicitly the expansion rotation factor responsible for the symmetric part of the curvature, but the remainder of the expression (chosen by Cattaneo-Gasperini as the spatial curvature tensor [1961]) still has hidden kinematical terms in the antisymmetric part of the second derivative terms. Making these explicit leads to the expression

R(lie)(u)abcd = 2θ(u)abω(u)cd

+ [−θ(u)adω(u)bc + θ(u)acω(u)bd − θ(u)bcω(u)ad + θ(u)bdω(u)ac] (2.330) 1 + 2 [had,(bc) − hac,(bd) + hbc,(ad) − hbd,(ac)] + ··· . As noted by Ferrarese [1965], the ﬁnal part of the formula following the explicit kinematical terms has all the algebraic symmetries of the corresponding identical coordinate formula on a Riemannian manifold and therefore has all of the algebraic symmetries of the usual covariant Riemann tensor. This symmetry-obeying spatial curvature tensor has the expression 1 R(sym)(u)abcd = 2 [had,(bc) − hac,(bd) + hbc,(ad) − hbd,(ac)] e e (2.331) − Γ(u) caΓ(u)edb + Γ(u) daΓ(u)ecb in a spatially-holonomic spatially-comoving frame, and is related to the Lie spatial curvature tensor and other curvature tensors (independent of the frame) by ab ab ab [a b] R(sym)(u) cd = R(lie)(u) cd − 2θ(u) ω(u)cd − 4θ(u) [cω(u) d] [ab] [a b] = R(lie)(u) cd − 4θ(u) [cω(u) d] (2.332) ab [a b] = R(cfw)(u) cd − 4θ(u) [cω(u) d] ab [a b] ab = R(fw)(u) cd − 4θ(u) [cω(u) d] − 2ω(u) ω(u)cd .

Since both R(lie)(u)abcd and R(sym)(u)abcd satisfy the usual cyclic symmetry of the torsionfree Bianchi identity of the first kind, the two kinematical difference terms together must also, as can be easily verified. Absorbing the first of these into the Lie spatial curvature to yield its antisymmetric part (in the first pair of indices) then explains why that antisymmetric part alone does not satisfy this cyclic symmetry.

72 2.5.4 Spatial Ricci tensors and scalar curvatures Neither one of the two kinematical diﬀerence terms between the Lie and symmetry-obeying spatial curvature tensors contribute to the curvature scalar

ab ab ab ab ab R(sym)(u) ab = R(lie)(u) ab = R(cfw)(u) ab = R(fw)(u) ab − 2ω(u) ω(u)ab (2.333) but they do make a difference in the spatial Ricci and Einstein tensors. It does not make much sense to ab define the Lie spatial Ricci tensor by taking the trace of R(lie)(u) cd by contracting either one or the other of the indices of the first index pair against one from the second antisymmetric pair since two different tensors result which differ by a kinematical term associated with the symmetric part of the first index pair. The trace using instead the antisymmetric part of the first index pair does result in a unique Lie spatial Ricci tensor, as does the trace of the Fermi-Walker spatial curvature as well as the trace of the “symmetry-obeying” spatial curvature tensor, but only the last Ricci tensor is symmetric

a [ca] R(lie)(u) b = R(lie)(u) cb , a a ca R(fw)(u) b = R(lie)(u) b + 2ω(u) ω(u)cb , (2.334) a ca R(sym)(u) b = R(sym)(u) cb .

The relation between the Lie and the symmetry-obeying Ricci tensors is

a a [c a] R(sym)(u) b = R(lie)(u) b − 4θ(u) [cω(u) b] a c a c a ca (2.335) = R(lie)(u) b − θ(u) cω(u) b + θ(u) bω(u) c − θ(u) ω(u)bc , so c c 0 = R(sym)(u)[ab] = R(lie)(u)[ab] − θ(u) cω(u)ab − 2θ(u) [aω(u)b]c , (2.336) hence c c R (u) = θ(u) cω(u)ab + 2θ(u) ω(u) , (tem) [ab] [a b]c (2.337) tem = lie,fw,cfw . This antisymmetric part of the spatial Ricci tensor comes from the lack of pair interchange symmetry in the corresponding spatial curvature tensor. In the case of a nonrotating observer congruence, all four spatial curvature tensors coincide and are equivalent to the Riemann curvature tensor of the Riemannian 3-manifolds which result from the induced metric on the hypersurfaces orthogonal to the congruence. In the general rotating case, one may use any one of the four distinct tensors, although one may be better suited than the others in a particular context in terms of absorbing extra kinematical terms. In the decomposition of the Einstein equations, it is clearly preferable to have a symmetric spatial Ricci or Einstein tensor, so the symmetry-obeying choice of spatial curvature tensor is perhaps the cleanest separation of spatial curvature from kinematical terms which occur there. In the special case of an observer congruence with vanishing expansion tensor, as occurs in stationary spacetimes if the congruence is taken along the Killing trajectories, the Lie, co-rotating Fermi-Walker and symmetry-obeying choices agree but diﬀer from the Fermi-Walker choice by a term quadratic in the rotation.

2.5.5 Pair interchange symmetry The pair interchange symmetry of an ordinary curvature tensor follows from the antisymmetry of the ﬁrst index pair together with the cyclic symmetry of the Bianchi identity of the ﬁrst kind. The symmetry-obeying spatial curvature tensor has all of these symmetries

R(sym)(u)αβγδ = R(sym)(u)γδαβ . (2.338)

The relationship between the symmetry-obeying and Lie spatial curvatures then implies the following

73 identity for the latter curvature

[αβ] αβ αβ αβ R(lie)(u) γδ − R(lie)(u)[γδ] = R(sym)(u) γδ − R(sym)(u)γδ [α β] [α β] + 4θ(u) [γ ω(u) δ] − 4θ(u)[γ ω(u)δ] [α β] = 8θ(u) [γ ω(u) δ] (2.339) αβ αβ = R(fw)(u) γδ − R(fw)(u)γδ αβ αβ = R(cfw)(u) γδ − R(cfw)(u)γδ or equivalently αβ αβ R(lie)(u) γδ − R(lie)(u)γδ (2.340) [α β] αβ αβ = 8θ(u) [γ ω(u) δ] + 2θ(u) ω(u)γδ − 2θ(u)γδω(u) . The trace of the ﬁrst of these relations provides another way to obtain the antisymmetric part of the Lie spatial Ricci tensor.

2.5.6 Spatial covariant exterior derivative The differential form approach to curvature may also be developed in the spatial context. Just as in the spacetime context, the spatial covariant exterior derivative defined for spatial tensor-valued differential forms is very useful in concisely presenting many identities. This operator interpolates between the spatial exterior derivative for scalar-valued (i.e., ordinary) differential forms and the spatial covariant derivative for tensor- valued zero-forms, and is entirely analogous to the corresponding covariant exterior derivative on spacetime, as described for a general manifold in appendix A. Suppose Sα... is a spatial tensor which is antisymmetric in a group of p covariant indices. One β...γ1...γp may think of it as a tensor-valued p-form, i.e., when its p covariant arguments are evaluated, one is left with a tensor with the remaining index structure. One has in invariant notation 1 α... α... γ1 γp S β... = S β...γ ...γ ω ∧ · · · ∧ ω , p! 1 p (2.341) β α... S = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ S β... , and the spatial covariant exterior derivative is then defined by

β α... D(u)S = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ D(u)S β... , β α... = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ d(u)S β... (2.342) β α... + ∇(u) ∧ [eα ⊗ · · · ⊗ ω ⊗ · · ·] ⊗ S β... , where the notation ∇∧ is meant to indicate the wedge product of the additional covariant derivative index with the p-form. The first line of the above definition emphasizes the sloppy classical notation of not distinguishing the components of the covariant derivative from the derivative of the components. The former interpretation is assumed to hold for the occurrence in this formula. In an observer-adapted frame, the component formula is directly analogous to the general one. If S is a spatial tensor-valued differential form of type σ (describing the tensor-valued index structure) then its spatial covariant exterior derivative is

b a... S = ea ⊗ · · · ⊗ ω ⊗ · · · ⊗ S b... , a... a... a... D(u)S b... = d(u)S b... + [σ(ωωω(u)) ∧ S] b... , (2.343) b a... D(u)S = ea ⊗ · · · ⊗ ω ⊗ · · · ⊗ D(u)S b... , where a b a a c ωωω(u) = ea ⊗ ωωω(u) b ⊗ ω , ωωω(u) b = Γ(u) cbω (2.344) 1 is the matrix of spatial connection 1-forms thought of as a 1 -“tensor”-valued 1-form and the wedge product is meant between the connection 1-form and the p-form indices.

74 By introducing the spatial identity tensor as a vector-valued 1-form

a a a ϑϑϑ(u) = P (u) = ea ⊗ ω , ϑϑϑ(u) = ω , (2.345)

one obtains the vector-valued spatial torsion 2-form as its spatial covariant exterior derivative

ΘΘΘ(u) = D(u)ϑϑϑ(u) = d(u)ϑϑϑ(u) + ωωω(u) ∧ϑϑϑ(u) (2.346) where the unusual notation ∧ necessary for the index-free notation indicates a right contraction between the adjacent tensor-valued indices and an exterior product between the diﬀerential form indices. In the index-notation one has explicitly

a a a a b ΘΘΘ(u) = D(u)ϑϑϑ(u) = d(u)ϑϑϑ(u) + ωωω(u) b ∧ ϑϑϑ(u) (2.347) 1 a b c = 2 –T (u) bcω ∧ ω = 0 .

1 Similarly the Lie spatial curvature tensor thought of as a 1 -tensor-valued 2-form has the deﬁnition

[ ΩΩΩ(lie)(u) = d(u)ωωω(u) + ωωω(u) ∧ωωω(u) − 2C(lie)(u)ω(u) a a a c a ΩΩΩ(lie)(u) b = d(u)ωωω(u) b + ωωω(u) c ∧ ωωω(u) b − 2C(lie)(u) bω(u)cd (2.348) 1 a c d = 2 R(lie)(u) bcd ω ∧ ω , where the unfamiliar extra terms are necessary for the spatial Lie derivatives which enter into this geometry in the case of nonzero rotation. The spatial Ricci identities for spatial tensors may then be written in the language of tensor-valued diﬀerential forms as 2 [ D(u) S = σ(ΩΩΩ(lie)(u)) ∧ S + 2ω(u) ∧ £(u)uS. (2.349) The spatial Bianchi identity of the ﬁrst kind is easily derived in this notation

0 = D(u)ΘΘΘ(u) = D(u)2ϑϑϑ(u) (2.350) [ = ΩΩΩ(lie)(u) ∧ϑϑϑ(u) + 2ω(u) ∧ £(u)uϑϑϑ(u) , but since £(u)uϑϑϑ(u) = £(u)uP (u) = 0 , (2.351) one has a b 1 a b c c ΩΩΩ(lie)(u) b ∧ ω = 2 R(lie)(u) [bcd]ω ∧ ω ∧ ω = 0 , (2.352) which when expressed in components gives the usual form of the identity. Similarly the symmetric part of the spatial curvature comes from the Ricci identity for the spatial metric viewed as a “tensor-valued 0-form”

2 0 = D(u) P (u)αβ (2.353) [ = −2ΩΩΩ(lie)(u)(αβ) + 2ω(u) ∧ £(u)uP (u)αβ ,

which leads to the previous result.

75 2.6 The symmetrized curl operator for symmetric spatial 2-tensors

For a spatial vector-valued 1-form, the spatial dual of the spatial covariant exterior derivative leads to a new vector valued 1-form which might naturally be called the spatial curl of the tensor.

α ∗(u) α γδ α [curl(u)T ] β = [ D(u)T ]β = ηβ ∇(u)γ T δ , (2.354)

α ∗(u) α If the spatial tensor X β is antisymmetric, with spatial dual [ X] , then this curl may be re-expressed as

α ∗(u) α α ∗(u) γ [curl(u)X] β = ∇(u)β[ X] − δ β∇(u)γ [ X] (2.355) with trace −2 div ∗(u)X. If one begins with a symmetric such tensor, the result is automatically is tracefree as a tensor α ∗(u) α γδ α [curl(u)S] β = [ D(u)S ]β = η ∇(u)γ S δ , β (2.356) ∗(u) α αγδ [ D(u)S ]α = η ∇(u)γ Sαδ = 0 , α α and for a pure trace such tensor T β = fδ β, the result is the spatial dual of the exterior derivative of the scalar coeﬃcient (dual of the vector “curl grad”), modulo sign and index shifting

αβ αβγ [curl T ] = −η(u) ∇(u)γ f . (2.357)

This operator can also be symmetrized, leading to an operator will be called the Scurl operator.

αβ γδ(α α) [Scurl(u)T ] = η ∇(u)γ T δ , (2.358)

It maps the symmetric spatial vector-valued 1-forms to tracefree symmetric such forms and annihilates their pure trace part αβ γδ(α β) [Scurl(u)S] = η ∇(u)γ S δ (2.359) = [Scurl(u)S(TF)]αβ . The generalized Cotton-York tensor [Cotton 18??, Eisenhart 19??, York 1971] is obtained by applying this operator to the spatial Ricci or Einstein tensors or their tracefree parts

αβ αβ γδ(α β C(tem)(u) = −2[Scurl R(tem)(u)] = −2η(u) ∇(u)γ R(tem)(u) )δ . (2.360)

76 2.7 Splitting spacetime curvature

The spacetime curvature tensor may be split into its representative spatial fields directly or one may first split it algebraically into its irreducible parts: the Weyl curvature tensor, the tracefree Ricci tensor and the scalar curvature, and then split into their representative spatial fields. Both forms of the splitting are convenient for certain applications. These splittings are most easily represented in terms of components with respect to an observer-adapted frame.

2.7.1 Splitting definitions Because of the symmetries of the Riemann tensor, its splitting leads to three independent spatial fields which are more efficiently packaged by taking the spatial dual on the remaining pairs of antisymmetric indices, as (4) discussed in exercises 14.14 and 14.5 of Misner, Thorne and Wheeler [1973]. Since Rαβγδ is symmetric in its two pairs of antisymmetric indices, one can split each pair into its electric and magnetic parts leading to either both electric or both magnetic or one of each for a total of three independent spatial fields. The magnetic spatial index pairs may then be converted into single spatial indices by the spatial dual, resulting in three second rank spatial tensors. The last such field (mixed electric and magnetic parts) transforms as an oriented tensor under a change of spatial basis (due to the single spatial dual) and so defines the magnetic part of the Riemann curvature tensor, while the remaining two transform as ordinary tensors under such transformations (due to an even number of spatial duals) and so define the electric part, in analogy with the electric and magnetic vector fields defined from the electromagnetic 2-form. The two fields which represent the electric part will be called the electric-electric electric part and the magnetic-magnetic electric part in an obvious though somewhat lengthy terminology. These three fields are

a (4) a E(u) b = R >b> , F(u)a = 1 η(u)acdη(u) (4)Rfg = [ ∗ (4)R ∗ ] a = [ ∗ (4)R ∗ ]a , b 4 bfg cd b> > >b> (2.361) a 1 (4) a cd (4) ∗ a H(u) b = 2 R >cdη(u) b = −[ R ] >b> , a E(u)[ab] = 0 = F(u)[ab] , H(u) a = 0 . Here the sign of the electric-electric part has been changed to conform with the convention established by Ehlers [1962, 1993] and Ellis [1971]. The spatial ﬁelds which represent the Ricci tensor, its tracefree part, and the Einstein tensor are related to the electric and magnetic parts of the Riemann curvature tensor in the following way

(4) a a a a c R b = −E(u) b − F(u) b + δ bF(u) c , (4) a a a 1 a c c Q b = −E(u) b − F(u) b + 2 δ b(E(u) c + F(u) c) , (2.362) (4) a a a a c G b = −E(u) b − F(u) b + δ bE(u) c ,

(4) > c (4) c c R > = −E(u) c , R = 2(−E(u) c + F(u) c) , (4) > 1 c c (4) Q > = − 2 (E(u) c + F(u) c) , Q = 0 , (2.363) (4) > c (4) c c G > = −F(u) c , G = 2(E(u) c − F(u) c) , and (4) > (4) > (4) > bc R a = Q a = G a = ηabcH(u) . (2.364) It is also very useful to split the Riemann curvature tensor into its irreducible parts under a change of frame (4) αβ (4) αβ (4) [α β] 1 (4) αβ R γδ = C γδ + 2 Q δ + Rδ , [γ δ] 12 γδ (2.365) (4) α (4) α 1 α (4) Q β = R β − 4 δ β R. (4) α (4) α This formula may be expressed in terms of the Ricci tensor R β or the Einstein tensor G β in place of (4) α the tracefree part of the Ricci tensor Q β, leading to different coefficients for the scalar curvature term. (4) The Weyl tensor Cαβγδ can itself be split into its family of spatial fields in the same fashion as the Riemann

77 tensor, but since it is self-dual − ∗(4)C ∗ = (4)C, the two electric parts of the Riemann tensor collapse to a single electric part of the Weyl tensor

a (4) a 1 (TF)a E(u) b = C >b> = 2 [−E(u) + F(u)] b 1 a a 1 a c c = [−E(u) b + F(u) b + δ b(E(u) c − F(u) c)] , 2 3 (2.366) ab 1 (4) >a bcd ∗(4) >a b (4) ∗ >a b H(u) = 2 C cdη(u) = [ C] > = [ C ] > = H(u)(ab) , where (TF) indicates the tracefree part. These are both symmetric and tracefree. This decomposition is apparently due to Trumper [1961] but the sign convention is diﬀerent.

2.7.2 Spacetime duality and curvature The origin of the decomposition of the Riemann tensor into Weyl, tracefree Ricci and the curvature scalar lies in the behavior of the Riemann tensor under the independent duality transformations one may perform on its two pairs of antisymmetric indices. First note that the dual of the dual of a 2-form yields the sign-reversal of the 2-form due to the Lorentz signature of the spacetime metric

∗ 1 (4) γδ Fαβ = ηαβ Fγδ , 2 (2.367) ∗∗ 1 µν γδ 1 γδ Fαβ = 4 ηαβ ηµν Fγδ = − 2 δ αβFγδ = −Fαβ . For this reason it is convenient to introduce a new duality operation “ ˘ ” called “hook” [Taub 19??] whose repeated action is the identity operation

∗ ˘ Fαβ = i Fαβ , (2.368) ˘ ˘ Fαβ = Fαβ . One can then look for eigenvectors of this new duality operation with eigenvalues 1 and −1 (rather than i and −i for the original duality operation). A 2-form F is then said to be self-dual if ˘ F = F and anti-self-dual if ˘ F = −F , either of which can be true only for a complex-valued 2-form. For example, if F is a real 2-form, 1 then Z = F + ˘ F is self-dual while its complex conjugate Z is anti-self-dual, and F = 2 (Z + Z) is the decomposition of the real 2-form into its complex self-dual and anti-self-dual parts. One can extend the hook operation to rank 4 tensors Bαβγδ like the curvature tensor which are symmetric in two pairs of antisymmetric indices. Such tensors may be interpreted as symmetric bilinear functions on 2-forms (bi-vectors). Any such tensor can be decomposed into its real self-dual and anti-self-dual parts under the two-sided hook operation (a hook on the left pair of indices and another on the right pair)

1 B = S + A,S = 2 (B + ˘ B ˘) , ˘ S ˘ = S, 1 (2.369) A = 2 (B − ˘ B ˘) , ˘ A ˘ = −A, For the Riemann curvature tensor this yields

(4) αβ αβ αβ R γδ = S γδ + A γδ , αβ (4) αβ 1 αβ S γδ = C γδ + 12 Rδ γδ , (2.370) αβ (4) [α β] A γδ = 2 Q [γ δ δ] , deﬁning the Weyl tensor as the tracefree part of the self-dual part of the Riemann tensor and the curvature scalar as the trace (apart from a constant), while the tracefree Ricci tensor determines the anti-self-dual part. The two-sided (ordinary) dual of the Riemann tensor is useful enough to give its own name following the notation of Misner, Thorne and Wheeler [1973]

(4) αβ ∗(4) ∗ αβ (4) αβ (4) αβ (4) αβ -G γδ = [ R ] γδ = −[˘ R ˘] γδ = − S γδ + A γδ (2.371) (4) αβ (4) [α β] 1 (4) αβ = − C γδ + 2 Q [γ δ δ] − 12 Rδ γδ ,

78 or (4) αβ (4) αβ (4) [α β] 1 (4) αβ C γδ = − -G γδ + 2 Q [γ δ δ] − Rδ γδ , 12 (2.372) (4) αβ (4) [α β] 1 (4) αβ = − -G γδ + 2 G [γ δ δ] + 6 Rδ γδ . The Einstein tensor is the trace of the two-sided dual of the Riemann tensor in the same way that the Ricci tensor is the trace of the Riemann tensor itself

(4) αβ (4) β (4) αβ (4) -G αδ = G δ , -G αβ = − R. (2.373)

2.7.3 Evaluation of splitting ﬁelds The actual splitting of the Riemann tensor for the congruence point of view is obtained by simply evaluating the standard component formula for an observer-adapted frame and has been given originally by Cattaneo- Gasperini [1961] and later by Ferrarese [1963, 1965, 1987, 1988]. Using the convenient the abbreviation k(u) k(u) = k(u)2 one has

a (4) a a 2 a a E(u) b = R >b> = £(u)uk(u) b − [k(u) ] b + [∇(u)b + a(u)b]a(u) , a 2 a a = ∇(fw)(u)uk(u) b − [k(u) ] b + [∇(u)b + a(u)b]a(u) , (2.374) ab a bcd a b H(u) = −∇(u)[ck(u) d]η − 2a(u) ω(u) ,

(taking note of (2.141)) and

(4) ab a b ab ab R cd = 2k(u) [ck(u) d] + 2k(u) ω(u)cd + R(lie)(u) cd , a b ab = 2k(u) [ck(u) d] + R(fw)(u) cd , (2.375) a b a b ab ab = {2θ(u) [cθ(u) d]} + {2ω(u) [cω(u) d] + 2ω(u) ω(u)cd} + {R(sym)(u) cd} . Each of the terms delimited by curly brackets in this last expression individually has all of the usual symmetries of a curvature tensor. Taking the two-sided spatial dual of this expression yields the manifestly symmetric magnetic-magnetic electric part of the Riemann tensor

a 2 a 1 a 2 2 a a F(u) b = [θ(u) − Θ(u)θ(u)] b − δ b[Tr θ(u) − Θ(u) ] + 3ω(u) ω(u)b − G(sym)(u) b , 2 (2.376) c 1 2 2 c 1 c F(u) c = − 2 [Tr θ(u) − Θ(u) ] + 3ω(u) ω(u)c + 2 R(sym)(u) c , where the two-sided spatial dual of the spatial curvature yields the corresponding sign-reversed spatial Einstein tensor. The above expression for the electric-electric electric part of the Riemann tensor is not manifestly symmetric 2 2 E(u)ab = £(u)uk(u)ab + [θ(u) − ω(u) ]ab c + 2θ(u)c(aω(u) b) + [∇(u)b + a(u)b]a(u)a , 2 2 E(u)(ab) = −£(u)uθ(u)ab + [θ(u) − ω(u) ]ab c − 2θ(u)c(aω(u) b) + [∇(u)(a + a(u)(a]a(u)b) , (2.377)

E(u)[ab] = £(u)uω(u)ab + ∇(u)[ba(u)a] = 0 , c 2 2 c E(u) c = −[£(u)u + Θ(u)]Θ(u) − Tr θ(u) + Θ(u) + 2ω(u) ω(u)c c (4) > + [∇(u)c + a(u)c]a(u) = − R > , its antisymmetric part vanishing due to the identity satisﬁed by the spatial exterior derivative of the acceleration 1-form. The quadratic term in the rotation has the alternate expression

2 a a a c [ω(u) ] b = ω(u) ω(u)b − δ bω(u) ω(u)c . (2.378) Similarly the magnetic part of the Riemann curvature is not manifestly tracefree

ab a bcd a a b ab c H(u) = ∇(u)[cθ(u) d]η − [∇(u) + 2a(u) ]ω(u) + h ∇(u)cω(u) , a a (2.379) H(u) a = 2[∇(u)a − a(u)a]ω(u) = 0 ,

79 but its trace vanishes due to the divergence identity for the rotation vector. Using these preliminary results, the various spatial ﬁelds which represent the diﬀerent independent pieces of the curvature tensor may now be expressed in terms of the spatial curvature and the kinematic quantities of the congruence. The electric and magnetic parts of the Weyl curvature tensor are

1 c E(u)ab = − 2 {£(u)uθ(u)ab − Θ(u)θ(u)ab + 2θ(u)c(aω(u) b) a + 4ω(u) ω(u)b − [∇(u)(b + a(u)(b]a(u)a) (TF) (2.380) − R(u)ab} , ab (a b)cd (a (a b) (TF) H(u) = {−∇(u)[cθ(u) d]η + [∇(u) + 2a(u) ]ω(u) } .

The scalar curvature and the Einstein tensor are

(4) a R = 2{£(u)u + Θ(u)}Θ(u) − 2[∇(u)a + a(u)a]a(u) 2 2 c c + Tr θ(u) − Θ(u) + 2ω(u) ω(u)c + R(sym)(u) c , (4) > c 2 2 c c 2 G > = −2F(u) c = Tr θ(u) − Θ(u) − 6ω(u) ω(u)c − R(sym)(u) c , (4) > b 2 G a = 4∇(u) k(u) − 4[a(u) ×u ~ω(u)]a [b a] (2.381) b b = −2∇(u)b[θ(u) a − δ aΘ(u)] − 2{[∇~ (u) + 2a(u)] ×u ~ω(u)}a , (4) a a a 1 a 2 2 G b = {£(u)u + Θ(u)}[θ(u) b − δ bΘ(u)] + 2 δ b[Tr θ(u) − Θ(u) ] a) a c − [∇(u)(b + a(u)(b]a(u) + δ b[∇(u)c + a(u)c]a(u) a a c a − 2ω(u) ω(u)b + δ bω(u) ω(u)c + G(sym)(u) b .

Similarly the purely spatial part of the Ricci tensor is

(4) a a a R b = −{£(u)u + Θ(u)}k(u) b − [∇(u)b + a(u)b]a(u) a a c a − 2[ω(u) ω(u)b − δ bω(u) ω(u)c] + R(sym)(u) b , (2.382) (4) 2 Rab = {£(u)u + Θ(u)}θ(u)ab − 2[θ(u) ]ab − [∇(u)(a + a(u)a]a(u)b) a a c − 2[ω(u) ω(u)b − δ bω(u) ω(u)c] + R(sym)(u)ab .??

Recall that the spatial Lie derivative with the expansion scalar Θ(u) = − Tr k(u) added as a multiplication operator has the expression

a... 1/2 −1 1/2 a... [£(u)u + Θ(u)]S b... = (Mh ) (h S b...),0 (2.383) in a spatially-comoving observer-adapted frame. Furthermore 2.220 shows that the acceleration term in the scalar curvature is a spacetime divergence

b α (4) [∇(u)b + a(u)b]a(u) = a(u) ;α = div a(u) . (2.384)

Similarly the acceleration-augmented curl in the mixed part of the Einstein tensor represents the spatial projection of the spacetime divergence of the rotation 2-form. Thus the Hilbert gravitational Lagrangian expressed in a spatially-comoving observer-adapted frame has the form (4) 1/2(4) α 1/2 (4) 1/2 α L(H) = g R α = L(ADM) + 2£(u)e [h Θ(u)] − 2 g a(u) ;α , 0 (2.385) 1/2 2 2 c c L(ADM) = Mh [Tr θ(u) − Θ(u) + 2ω(u) ω(u)c + R(sym)(u) c] .

In order to speak of the e0 derivative term as a “total time derivative” in the usual sense, one must introduce a full splitting of spacetime. Neglecting this term and the divergence deﬁnes the ADM Lagrangian [Arnowit, Deser and Wheeler 1962, Misner, Thorne and Wheeler 1973].

80 2.7.4 Maxwell-like equations The Einstein equations in their Einstein or Ricci form

(4) α (4) α (4) α (4) α α (4) γ G β = κ T β , R β = κ[ T β − δ β T γ ] , (2.386) may easily be split using the above splitting of the lefthand sides, once the energy-momentum tensor is split, leading to a spatial scalar, a spatial vector and spatial tensor equation. The first two are analogous to the source-driven Maxwell equations for the electromagnetic field, while the the spatial scalar and spatial vector identities satisfied by the gravitoelectric and gravitomagnetic fields are analogous to the homogeneous Maxwell equations. The first pair of Maxwell-like equations analogous to the sourcefree Maxwell equations are equations (2.269). The spatial scalar and spatial vector equations which result from the measurement of the Ricci form of the Einstein equations provide the much more complicated analogs of the source driven pair of Maxwell equations (4) > c 1 c R > = [∇(u)c − g(u)c]g(u) − 2 H(u) H(u)c + [£(u)u + Θ(u)]Θ(u) + Tr θ(u)2 − Θ(u)2 (4) > 1 (4) α = κ[ T > − T α] , 2 (2.387) (4) > 2 R a = −{[∇~ (u) − 2~g(u)] ×u H~ (u)}a b b − 2∇(u)b[θ(u) a − δ aΘ(u)] (4) > = 16π T a . The complication arises from the new ingredient described by the spatial metric which has no analog in linear electrodynamics. In the linearized Einstein equations, the analogy of these four equations with Maxwell’s equations becomes not only very close but even useful.

81 2.8 Mixed commutation formulas

The commutation formula for two spatial covariant derivatives acting on a spatial tensor field leads to spatial curvature via the spatial Ricci identities. This formula represents the purely spatial part of the splitting of the corresponding spacetime commutation formula, i.e., of the corresponding spacetime Ricci identity. The remaining members of the family of spatial identities which result from splitting the latter identity are also useful identities. All of the these identities together are equivalent to commutation formulas among various combinations of the spatial covariant derivative and the Fermi-Walker temporal derivative acting on the members of the family of spatial fields which represent a given spacetime tensor field. This provides a more straightforward method of deriving the splitting identities and a more useful packaging of their content. In practice the Ricci identities are used to interchange the order of two successive covariant derivatives in an expression. The spatial identities obtained from the spacetime Ricci identities allow one to interchange the order of two spatial covariant derivatives or of a spatial covariant derivative and a Fermi-Walker temporal derivative. It is equally convenient to be able to interchange the order of a Lie temporal derivative and a spatial covariant derivative. For the Lie temporal derivative this leads to the Lie derivative of the spatial connection, a spatial tensor field which appears in the spatial Bianchi identities of the second kind.

2.8.1 Splitting the Ricci identities The Ricci identity for a vector ﬁeld Z is

(4) (4) α (4) α δ [ ∇γ , ∇δ]Z = R βγδZ . (2.388) Since the operator [(4)∇, (4)∇] itself has an electric and a magnetic part as a 2-form operator, while the vector field Z has both a temporal and a spatial part, four different spatial identities result from splitting this identity. These are equivalent to commutation formulas involving [∇(u), ∇(u)] and [∇(fw)(u), ∇(u)] acting on either a spatial vector field or on u itself. For an arbitrary vector field Z one has

α µ ν α (4) σ (4) ρ [∇(u)γ , ∇(u)δ]Z = 2P (u) [γ P (u) δ]P (u) σ ∇µ[P (u) ν P (u) ρ ∇Z ] µ ν α (4) (4) ρ = P (u) γ P (u) δP (u) ρ[ ∇µ, ∇ν ]Z (2.389) α α ρ + 2ω(u)γδ∇(fw)(u)Z − 2k(u) [γ P (u) δ]uρZ ; and α [∇(fw)(u), ∇(u)β]Z = α δ (4) (4) γ = P (u) γ P (u) βu [ ∇, ∇δ]Z (2.390) α α γ δ + a(u)β∇(fw)(u)Z + a(u) uγ Z ;δP (u) β α γ δ α γ δ + k(u) βuγ Z ;δu + P (u) γ Z ;δk(u) β . Restricting each of these to a spatial vector ﬁeld Z and using the spacetime Ricci identity for the spacetime commutator term leads to the two identities α (4) αβ α β [∇(u)γ , ∇(u)β]Z = ([P (u) R] γδ − 2k(u) [γ k(u) δ])Zβ α (2.391) + 2ω(u)γδ∇(fw)(u)Z and α γα α α δ [∇(fw)(u), ∇(u)β]X = (−Hβγ η(u) δ + a(u) k(u)δβ − k(u) βa(u)γ )Z (2.392) α δ α + a(u)β∇(fw)(u)Z + k(u) β∇(u)δZ . On the other hand letting Z = u one obtains

α αβ α [∇(u)γ , ∇(u)δ]u = η(u)γδβH(u) + 2ω(u)γδa(u) , (2.393) and α α 2 α α [∇(fw)(u), ∇(u)β]u = −E(u) β − [k(u) ] β + a(u) a(u)β . (2.394)

82 From first of these four identities, re-expressing the left hand side with the spatial Ricci identity, one recovers the relationship between the magnetic-magnetic electric part of the spacetime curvature (∼ Fαβ) and the spatial curvature tensor. The last of these identities involving the electric-electric part of the curvature is necessary to finish the discussion of relative acceleration. A relative acceleration vector is the second spatial Fermi-Walker derivative of a spatial “relative position vector” Y which undergoes spatial Lie transport along u. Its expression involves the commutator of the spatial Fermi-Walker derivative and the spatial covariant derivative acting on u itself, leading to the appearance of the electric-electric electric part of the spacetime curvature. Using the appropriate commutation formula in eq. 2.158 leads to 2 α α α β ∇(fw)(u) Y = {−E(u) β + [∇(u)β + a(u)β]a(u) }Y , (2.395) 2 [ or ∇(fw)(u) Y = {−E(u) + ∇(u)a(u) + a(u) ⊗ a(u) } Y, For a geodesic congruence, the acceleration vanishes and this reduces to the geodesic deviation equation involving only the electric-electric electric part of the Riemann tensor, which is interpreted as describing the tidal gravitational field.

2.8.2 Commuting £(u)u and ∇(u) The commutator of the temporal Lie derivative and the spatial covariant derivative of a spatial tensor field is easiest to evaluate in an observer-adapted frame. For a spatial vector field one easily finds

a a a c [£(u)u, ∇(u)b]X = a(u)b£(u)uX + £(u)uΓ(u) bcX , (2.396)

or in any frame α α α γ [£(u)u, ∇(u)β]X = a(u)β£(u)uX + L(u) βγ X , (2.397) where the spatial tensor L(u) is the temporal Lie derivative of the spatial connection, given the overused kernel symbol H by Ferrarese [1987, 1988, 1989]. This tensor is most easily evaluated using the commutation formula for [∂0, ∂a] in a spatially-comoving spatially-holonomic partially-normalized observer-adapted frame. The result

a a L(u) bc = £(u)uΓ(u) bc ad (2.398) = h [θ(u){δβ||γ}− + θ(u){dba(u)c}− ] easily generalizes to the frame-independent formula

α αδ £(u)uΓ(u) βγ = P (u) [θ(u){δβ||γ} + θ(u){δβa(u)γ} ] − − (2.399) αδ µ ν = P (u) θ(u){δµ;ν}− P (u) βP (u) γ . This formula is a closely related to the more familiar result for the variation of the Christoﬀel symbols induced by the variation of a metric.

83 2.9 Splitting the Bianchi identities of the second kind

The next level of splitting addresses the derivatives of curvature, the most important of which occur in the Bianchi identities which hold for both the spacetime curvature tensor and the various spatial curvature tensors.

2.9.1 Spacetime identities The Bianchi identity for the curvature 2-form has the simple index-free form

(4)D(4)ΩΩΩ = d(4)ΩΩΩ + (4)ωωω ∧ (4)ΩΩΩ − (4)ΩΩΩ ∧ (4)ωωω = 0 (2.400)

in terms of the spacetime covariant exterior derivative (4)D, where right contractions are implied between the adjacent tensor-valued indices of the curvature and connection forms, or in index form

(4) α R β[γδ;] = 0 . (2.401) Taking the two-sided dual of this equation leads to

(4) αβγδ -G ;δ = 0 , (2.402)

which has the contracted form (4) β Gα ;β = 0 (2.403) called the contracted Bianchi identity. These identities simplify the expression for the divergence of the Weyl tensor (4) αβ ;δ (4) αβ ;δ [α (4) β] ;δ 1 αβ (4) ;δ C γδ = − -G γδ + 2δ [γ G δ] + δ γδ R . 6 (2.404) (4) [α ;β] 1 αβ (4) ;δ = G γ + 6 δ γδ R , enabling it to be expressed in terms of the covariant derivatives of the Einstein tensor and the scalar curvature. The identities for the divergence of the Weyl and Einstein tensors can each be split to yield useful information about the curvature. The spacetime split of the Weyl divergence identity yields the analog of Maxwell’s equations for its electric and magnetic parts as discussed by Tr¨umper [1967], but the results are more accessible in work by Hawking [1966] and Ellis [1971], while the spacetime split of the Einstein tensor divergence identity provides the framework for the discussion of the preservation of the constraints in the evolution of solutions of the “initial value problem”. Introduce the Einstein ﬁeld equation tensor

(4) α (4) α (4) α E β = G β − κ T β (2.405)

which vanishes when the Einstein equations are satisfied. The Einstein tensor has identically vanishing divergence. The -momentum tensor of the source of the gravitational field must therefore have vanishing divergence. This is either a consequence of the source field equations or, in the case of a perfect fluid with an explicit equation of state, is equivalent to them. When this divergence is zero, the Einstein field equation tensor has vanishing divergence (4) α E β;α = 0 . (2.406) The splitting of this equation takes the following form when expressed in a spatially-comoving observer- adapted frame (4) α 1/2 −1 1/2(4) > 0 = E >;α = (Lh ) £(u)e0 (h E >) −2 c 2(4) > d (4) c − L ∇(u) (L E c) + θ(u) c E d , (2.407) (4) α 1/2 −1 1/2(4) > 0 = E a;α = (Lh ) £(u)e0 (h E a) (4) > c (4) > (4) c 2 E ck(u) a − a(u)a E > + [∇(u)c + a(u)c] E a . One can divide the Einstein field equations into “dynamical equations”

(4) a (4) E b = 0 or P (u) E = 0 (2.408)

84 and “constraint equations” (4) > E > = 0 (2.409) (4) > (4) > β E a = 0 or E βP (u) α = 0 . The significance of the above divergence equations is then the following. Suppose one considers a “tubular region” of spacetime with respect to the congruence, namely a part of spacetime obtained by starting with an “initial hypersurface” transversal to the congruence with or without boundary and then Lie dragging it into the future a fixed proper time interval along the congruence. If the “dynamical equations” are satisfied within the tubular region swept out in this manner, then the above equations clearly show that the “constraint equations”, if satisfied at the initial hypersurface, remain satisfied along the congruence within this tubular region. By the same process used above to evaluate divergences with a spatially-comoving observer-adapted frame, one may split the divergence of the Weyl tensor and then split the Bianchi identities involving the covariant derivatives of the Einstein tensor and the scalar curvature. If the Einstein equations are satisfied, these latter terms in the Bianchi identities may be replaced by the corresponding derivatives of the energy-momentum tensor which then provide source terms for the Maxwell-like equations for the electric and magnetic parts of the Weyl tensor. This was first done by Trümper [unpublished, see also Trümper 1967] and discussed by Ellis [1971]. To emphasize the similarity with Maxwell’s equations, one considers the α α spatial fields E β and H β as vector-valued 1-forms, but since they are symmetric upon index shifting, the operations on the vector-valued 1-forms must be symmetrized to respect that symmetry. A divergence and curl (the Scurl operator of section 2.6) can then be defined for these fields in a natural way using the spatial covariant exterior derivative D(u)

α β α α [div(u)E(u)] = ∇(u) E β = −[δ(u)E(u)] ,

δα βγ(α δ) ∗(u) (δα) [Scurl(u)E(u)] = η(u) ∇(u)βE γ = [ D(u)E(u)] . (2.410)

Similarly the spatial cross product operation can be applied to their 1-form indices to reﬂect taking the spatial dual of the wedge product of a spatial 1-form with the vector-valued 1-form in a way that makes the curl operation just “del cross”

δα (δ α)βγ [X ×u E(u)] = XβE(u) γ η(u) . (2.411)

(4) ;δ In the splitting of the Weyl divergence Cαβγδ one must recall the tracefree condition on the Weyl tensor and the ﬁrst Bianchi identity for the Weyl tensor, which imply

(4) αγ ;δ (4) ;δ C γδ = 0 , C[αβγ]δ = 0 . (2.412)

δ Taking these into account, the splitting of the Weyl divergence Cαβγ;δ may be represented by the following independent spatial ﬁelds

(4) δ (4) δ ∗(4) δ ∗(4) δ { C >a>;δ, C (ab)>;δ, [ C] >a>;δ, [ C] (ab)>;δ} , (2.413) where the traces of the second and fourth ﬁelds vanish, i.e., by two spatial vectors and two tracefree symmetric tensors. These are explicitly

(4) δ a a b cd a c C >a>;δ = −[div(u)E(u)] + η(u) bcH(u) dσ(u) + 3H(u) cω(u) , (4) δ ab (a (a b)c 2 ab ab cd C (ab)>;δ = −£(u)uE(u) [σ(u) c + ω(u) c]E(u) − 3 ??Θ(u)E(u) − h σcdE(u) ab + {[curl(u) + 2a(u)×u]H(u)} , ∗(4) δ a a b cd a c [ C] >a>;δ = −[div(u)H(u)] + η(u) bcE(u) dσ(u) + 3E(u) cω(u) , (2.414) ∗(4) δ ab (a (a b)c 2 ab ab cd [ C] (ab)>;δ = −£(u)uH(u) [+σ(u) c + ω(u) c]H(u) − 3 ??Θ(u)H(u) − h σcdH(u) ab +??{[curl(u) + 2a(u)×u]E(u)} , CHECKDETAILS; h??

85 The form of these equations given by Tr¨umper or Ehlers [1961,1993] and later by Hawking [1966] and Ellis [1971] uses the Fermi-Walker temporal derivative instead of the Lie temporal derivative. This change is easily made resulting in the equations

(4) δ ab (a (a b)c ab ab cd C >a>;δ = ∇(fw)(u)uE(u) + [−3σ(u) c + ω(u) c]E(u) −??Θ(u)E(u) + h σcdE(u) ab + {[curl(u) + 2a(u)×u]H(u)} , (4) δ ab (a (a b)c ab ab cd C (ab)>;δ = ∇(fw)(u)uH(u) + [−3σ(u) c + ω(u) c]H(u) −??Θ(u)H(u) + h σcdH(u) (2.415) ab − {[curl(u) + 2a(u)×u]E(u)} , stilltocorrect the form given by Ellis and Bruni [1989]. The divergence of the Weyl tensor is speciﬁed by the once contractec Bianchi identities of the second kind as explained in the appendix. FIX UP

86 2.9.2 Spatial identities The form taken by the Bianchi identities of the second kind for each of the three spatial curvature tensors is most easily obtained by ﬁrst evaluating the spatial covariant exterior derivative of the Lie spatial curvature 2-form D(u)ΩΩΩ(lie)(u) = d(u)ΩΩΩ(lie)(u) + ωωω(u) ∧ΩΩΩ(lie)(u) − ΩΩΩ(lie)(u) ∧ωωω(u) = ... (2.416) = d(u)2ωωω(u) = 2ω(u)[ ∧ £(u)uωωω(u) , or in index form αβ α ||β R(lie)(u) [γδ||] = 2L(u) [γ ω(u)δ] , (2.417) yielding the result of Ferrarese [1965]. Deﬁning the nonsymmetric Lie spatial Einstein tensor as the sign-reversed two-sided spatial dual of the spatial curvature tensor

α 1 αγδ µν G(lie)(u) β = − 4 η(u) η(u)βµν R(lie)(u)γδ , [αδ] 1 α γδ (2.418) = R(lie)(u) βδ − 2 P (u) βR(lie)(u) γδ α 1 α γ = R(lie)(u) β − 2 P (u) βR(lie)(u) γ , leads to the usual formula in terms of the Ricci tensor. Then taking the two-sided spatial dual of the previous identity on each group of antisymmetric indices results in the divergence of the Lie spatial Einstein tensor

αβ αβγ δ G(lie)(u) ||β = −η(u) ω(u) θ(u)δβ||γ . (2.419)

The symmetric part of the identity is instead

(αβ) αβ R(lie)(u) [γδ||] = 2θ(u) ||[γ ω(u)δ] . (2.420)

More complicated versions of these formulas may be obtained for the remaining two spatial curvature tensors by considering the derivatives of the diﬀerence tensors relative to the Lie spatial curvature.

87 2.10 “Time without space deﬁnes space without time” and vice versa

The congruence point of view defines “time” as the proper time measured by the test observers but establishes no correlation between the proper times of different observers, which is instead a property of “space”. One can define a useful space, however, namely the quotient (4)M/ Flow(u) of the spacetime by the 1-parameter group of diffeomorphisms generated by u, i.e., the space of observer worldlines. The term “observer space” is a convenient label for this space, which is without any Riemannian structure or any notion of time, precisely because of the lack of correlation between different observers. For example, in order to integrate the Einstein equations in this point of view one needs an initial value hypersurface to get started, where contraints must be satisfied by certain initial data. This initial hypersurface then leads to a slicing of the spacetime under dragging along by the observer congruence, yielding a nonlinear reference frame, in terms of which the initial data may then be evolved. Without such additional structure one can only consider “propagation equations” describing the evolution of the gravitational and source fields along each individual observer worldline, and certain “constraint” equations, as described by Ellis [1971]. Similarly in the hypersurface point of view, one can take the quotient of the spacetime by the slicing to obtain a time line but there is no explicit notion of “points of space” which may be associated with the time line. Of course there is implicitly because of the existence of the normal congruence, but explicitly introducing it leads to a full splitting. In order to convert the split tensor equations involving spatial Lie derivatives along u and spatial covariant derivatives into explicit partial differential equations, one must introduce local coordinates. In the interest of simplification it seems clear that these coordinates should be adapted to the congruence in some way. This essentially means introducing a full splitting of spacetime. For a rotating congruence this means introducing a slicing and an associated time coordinate function t (i.e., a parametrized slicing) and the equivalence class of comoving spatial coordinate systems for the congruence. For a nonrotating congruence it is clear that the integrable family of orthogonal hypersurfaces should be taken as the slicing, allowing the freedom to use spatial coordinates which are not comoving along the normal congruence. The spatial Lie derivative by u can then be represented as a rescaled time coordinate derivative, plus a spatial Lie derivative if the spatial coordinates are not comoving. However, the spatial covariant derivative in the case of a nonintegrable connection, i.e., associated with a rotating congruence, cannot be represented in terms of spatial coordinate derivatives alone and is necessarily accompanied by a time derivative. This means that the “constraint” equations involving spatial derivatives of the kinematical quantities and only first order spatial Lie derivatives along u necessarily involve second coordinate time derivatives, as does the spatial curvature. Only in the case of a nonrotating congruence, with the slicing taken to be the orthogonal slicing to the congruence, is it possible to represent the constraint equations in terms of only first order coordinate time derivatives. Only then is it meaningful to speak of the contraints as defining an “initial value problem” in the usual sense.

88 Chapter 3

The slicing and threading points of view

3.1 Introduction

In order to solve tensor field equations on spacetime, one has to deal either with local coordinates or certain equivalence classes of local coordinates reflecting some kind of geometrical properties assumed to hold for the spacetime, i.e., reflecting either spacetime symmetries or algebraic properties of curvature or something of this nature. Without any such assumptions one must tie the geometry to the differential structure of the spacetime manifold to have actual partial differential equations to solve. The nonlinear reference frame provides a way of anchoring the congruence or hypersurface point of view to the spacetime manifold and makes available a complete set of “potentials” whose various derivatives yield each of the pieces of the spacetime connection. In a sense it establishes an “integrable” structure on the spacetime manifold which is used to represent the (in general) “nonintegrable” structure of the congruence alone. The hypersurface point of view is integrable, but the nonlinear reference frame must be adapted to this integrable structure or a situation similar to the nontrivial congruence point of view results. Perhaps the most important function of the nonlinear reference frame is that it converts the temporal Lie derivative into an ordinary partial derivative with respect to the time coordinate in the congruence point of view and the integrable spatial derivatives in the hypersurface point of view into ordinary partial derivatives with respect to the spatial coordinates in an adapted coordinate system. The remaining derivatives in each point of view involve both spatial and time derivatives, except in the special case of an orthogonal nonlinear reference frame where no mixing occurs. The action of the gravitational field on test bodies is described in the context of a partial splitting of spacetime by the spatial gravitational force fields and the spatial part of the spatial connection, all of which together represent the spacetime connection. In a certain sense the spatial metric P (u)[ is a potential for the spatial part of the spatial connection ∇(u) ◦ P (u) and for the expansion tensor, while u[ itself acts as a potential for the vector spatial gravitational force fields through equation (2.267). However, the latter relationship does not involve the spatial or temporal derivatives of spatial quantities, like the scalar and vector potentials that result from the splitting of the electromagnetic 4-potential. One needs a full splitting in order to introduce a 4-potential for the gravitoelectromagnetic vector fields in a way analogous to the electromagnetic case.

3.2 Algebra 3.2.1 The nonlinear reference frame Suppose spacetime or a portion of spacetime (4)M has a nonlinear reference frame as described in the introduction. This amounts to an equivalence class of adapted coordinate systems {t, xa}, the time coor-

89 Figure 3.1: The relationship between the slicing and threading computational 3-spaces and the nonlinear reference frame.

Figure 3.2: Reference decomposition of tangent and cotangent spaces.

dinate t parametrizing the slicing and the spatial coordinates {xa} parametrizing the threading. One may reparametrize the time function t 7→ f a(t) and change the spatial coordinates in a time independent way xa 7→ f a(x) without changing the nonlinear reference frame to which these adapted coordinates correspond. This is an implicit representation of the nonlinear reference frame. On the other hand one may explicitly represent the nonlinear reference frame by a parametrization map I : R × Σ → (4)M which is a diffeomorphism from the “parameter spacetime” R × Σ onto (4)M, where Σ is a 3-manifold diffeomorphic to the quotient (4)M/T hd of (4)M by the threading congruence. This may be (4) thought of as a 1-parameter family of imbeddings It :Σ → M or as a 3-parameter family of imbeddings (4) Ix : R → M yielding respectively the slices Σt = It(Σ) and the parametrized threading curves Ix(R), where It(x) = Ix(t) = I(t, x) and (t, x) ∈ R×Σ. A particular parametrization map defines a parametrized nonlinear reference frame, while the original nonlinear reference frame is the equivalence class of such parametrized nonlinear reference frames under repararametrization of the time parameter. A given parametrization map defines in a natural way a time function t on (4)M, whose value on a given slice equals the value of the time parameter which describes it, and selects naturally a choice of parametrization for each curve in the threading congruence (by restriction of the time function to the curve). One may introduce two “computational 3-spaces”: 1) the slicing computational 3-space Σ associated with an explicit parametrization map and 2) the threading computational 3-space (4)M/T hd which is the quotient of (4)M by the threading congruence. Let π : (4)M → (4)M/T hd be the projection map associated (4) with this quotient and let πt :Σt → M/T hd be its restriction to the slice Σt. The composed maps (4) π ◦ It = πt ◦ It :Σ → M/T hd provide a natural identification between the two computational 3-spaces (which is of course independent of t), each of which can be identified in turn with the hypersurface Σt for each value of t. Any one of these may be thought of as a representation of an abstract computational 3-space. This space will be understood when the qualifier slicing or threading is omitted. Figure 3.1 illustrates the relationship of the various spaces and maps. The nonlinear reference frame induces a direct sum decomposition of the tangent and cotangent bundles (and all higher tensor bundles) called the reference decomposition. Each tangent space is decomposed into the 1-dimensional threading subspace tangent to the threading congruence and the 3-dimensional slicing subspace tangent to the slicing. The differential dt = ω0 of a time function for the slicing defines the “slicing 1-form” ω0 whose kernel determines the distribution of slicing subspaces. The parametrization of the threading curves induced by this time function defines a tangent vector field e0 called the “threading vector field”, whose 1-parameter group of diffeomorphisms has the threading congruence as its family of orbits. Given an explicit parametrization map I, this group corresponds to translation in the parameter t in the parameter spacetime R × Σ. This group provides the realization of the concept of “evolution” with respect to the nonlinear reference frame. The dual role of t as a time function and as a parameter along 0 the integral curves of e0 is represented by the compatibility condition ω (e0) = 1. Figure 3.2 illustrates the reference decomposition of the tangent and cotangent spaces.

3.2.2 Measurement and the lapse function The causality property of the nonlinear reference frame introduces the equally important concept of measurement. In each point of view, three-dimensional quantities must be interpreted in terms of the orthogonal decomposition of the tangent space induced by the 4-velocity of a family of test observers and their associated local rest spaces, which have yet to be identiﬁed. When both points of view hold, the case of a spacelike slicing and a timelike threading, then a well-deﬁned transformation between the two families of observers will exist.

90 In the slicing point of view, the slicing 1-form is timelike and can be normalized

N −2 = −(4)g−1(ω0, ω0) , ω⊥ = Nω0 , (4)g−1(ω⊥, ω⊥) = −1 , (3.1)

⊥ ] reversed in sign, and its index raised to obtain the future-pointing unit normal n ≡ e⊥ = −(ω ) to the slicing. This may be interpreted as the 4-velocity field of a family of test observers whose worldlines coincide with the normal congruence. The unit 1-form ω⊥ = −n[ has as its kernel the integrable distribution of local rest spaces LRSn of these observers. This 1-form also measures the differential of proper time along their individual worldlines. All of the notation from the hypersurface point of view for the normal vector field n may be taken over intact. Recall that the “perp” index “⊥” indicates perpendicularity to the slicing in this context. In the threading point of view, the threading vector field is timelike and can be normalized

2 (4) −1 (4) M = − g(e0, e0) , e> = M e0 , g(e>, e>) = −1 , (3.2) to obtain a unit tangent vector field e> ≡ m which may be interpreted as the 4-velocity of a family of test > [ [ observers. The sign-reversed index-lowered 1-form ω = −m = −(e>) has as its kernel the (in general) nonintegrable family of local rest spaces of this family of observers, to be denoted by LRSm. This 1-form measures the differential of proper time along their worldlines. The identification u = m enables all of the notation from the congruence point of view to be taken over intact. Recall that the “tan” index symbol “>” indicates tangency to the threading. The normalization factors N and M, expressable as

⊥ −1 0 N = ω (e0) ,M = ω (e>) , (3.3)

or equivalently as −1 0 > N = ω (e⊥) ,M = ω (e0) , (3.4) will both be referred to as the lapse function, with the clarifier “slicing” or “threading” preceding this term when necessary to distinguish the two points of view. The slicing point of view notation for the lapse and shift is that of Arnowit, Deser and Misner [1962], while the terminology is due to Wheeler [1964]. The lapse function in each case describes the rate of change with respect to the coordinate time of the observer proper time measured along the observer congruence. To allow a simultaneous discussion of both points of view, let o stand respectively for n and m. Cor- respondingly let L(n) = N and L(m) = M so that L(o) denotes the lapse function in each point of view. Then the proper time along each curve in the the observer congruence is related to the parametrization by the ccordinate time function by dτo/dt = L(o) . (3.5) Given the family of test observers in each point of view, one can decompose all tensor fields on the spacetime using the orthogonal decomposition of the tangent space induced by their unit 4-velocity o. This measurement process has already been discussed at length for the congruence point of view. By making the identification u = o, one can carry over that discussion to the slicing and threading points of view. Of course in the slicing point of view, it is technically the hypersurface point of view that describes the measurement process, and the notation will reflect this distinction. Thus the observer decomposition (respectively slicing decomposition and threading decomposition) of the tangent space is accomplished by two projections

X =T (o)X + P (o)X, α α β α β [T (o)X] = T (o) βX = (−o oβ)X , (3.6) α α β α α β [P (o)X] = P (o) βX = (δ β + o oβ)X ,

where X is an arbitrary vector field. The temporal projection T (o) projects along the local time direction of the observer with 4-velocity o, while the spatial projection P (o) projects into the local rest space LRSo associated with that observer. A single vector field X can either be represented as a sum of the vector fields (o) β T (o)X and P (o)X, or as a pair consisting of the scalar X = −oβX and the vector P (o)X. This scalar

91 Figure 3.3: Two diﬀerent parametrization maps lead to a relative shifting of the points of the computational 3-space.

will be denoted respectively by X⊥ and X> in the slicing and threading points of view, suppressing the argument o for a slight simplification of the notation. Thus the family of spatial fields which represents the vector field is {X⊥,P (n)X} in the slicing point of view and {X>,P (m)X} in the threading point of view. The notation and discussion is identical for the measurement process in the hypersurface and congruence points of view respectively. It is most easily described in terms of an observer-adapted frame, but with a nonlinear reference frame it is more natural to consider first frames which are adapted to the nonlinear reference frame. These are partially-observer-adapted frames, which may then be projected to obtain observer-adapted frames.

3.2.3 The Shift Splitting the metric tensor in each point of view still only yields the spatial metric P (o)[ as the only nontrivial spatial field in the family of spatial fields into which it decomposes. The missing pieces come from the decomposition of the “spatial-gauge field” of the parametrized nonlinear reference frame: e0 in the slicing point of view and ω0 in the threading point of view. These fields contain information about the tilting of the threading subspace and slicing subspace respectively away from the local time direction and the local rest space of the family of test observers respectively in the two points of view, which are determined instead by 0 the timelike “causality fields” ω and e0 respectively. Although it is common to see the spatial gauge field assumed to timelike as well, this is not necessary. In this way one recovers the lapse function already discussed as the normalizing factor for the slicing 1-form and threading vector field respectively from the scalar part and one obtains the slicing shift vector field or threading shift 1-form respectively from the rank one part

e0 = T (n)e0 + P (n)e0 = Nn + N~ = Ne⊥ + N~ ↔ {N, N~ } , = = = (3.7) ω0 = T (m)ω0 + P (m)ω0 = M −1(−m[) + M = M −1ω> + M ↔ {M −1, M} . Again the modifier “slicing” or “threading” will precede the term “shift” when necessary to distinguish the two points of view. In slicing point of view the shift is most naturally considered as a vector field N~ , and it determines the tilting of the threading curves away from the normal direction n. In the threading point of view the shift = is most naturally considered as a 1-form M, and it determines the tilting of the threading local rest spaces = = [ ] LRSm away from the directions tangent to the slicing. Let N = N~ and M~ = M denote the slicing shift 1-form and the threading shift vector field respectively. The vector oversymbol and double overbar notation is necessary when an index-free notation is used in order to conform with the identical kernel symbols for the lapse and shift established by the slicing point of view conventions. The choice of the word “shift” in the slicing point of view refers to a “shifting of the points of space” between the normal congruence and the threading congruence. In that point of view the threading serves only as an identification map between the different slices in the slicing, and differing threadings lead to different identifications which are described by a diffeomorphism on each slice. For example, if one has two different parametrization maps I and I which correspond to the same parametrized slicing of (4)M but −1 different threading curves, then as illustrated in Figure 3.3, the map It ◦ It :Σt → Σt shifts the points −1 of the slice Σt from the old points to the new points, while the map It ◦ It :Σ → Σ does the same on the computational 3-space Σ. If I describes the orthogonal threading and I any other threading with shift vector field N~ , then the time-dependent diffeomorphism on Σ representing the relative shifting of the points −1 ~ of space is just the 1-parameter group of diffeomorphisms of the time-dependent vector field It ∗N there. In the threading point of view the “shift” connotation is less appropriate, although the alternative term “tilt” applies equally well to both points of view. There is no natural slicing with respect to which the slices are shifted for a given threading congruence (except in the nonrotating case), and different slicings correspond to a relative shifting of the time parameters along each threading curve (rather than a shifting of the “spatial parameters” along each slice as occurs in the slicing point of view).

92 If one interprets the threading and slicing points of view as the mathematical realizations of the two complementary notions of time, then it is natural to associate the remaining freedom in the nonlinear reference frame for a given choice of “time” (slicing in the slicing point of view and threading in the threading point of view) with the “spatial gauge freedom” in the nonlinear reference frame. It is in this sense that 0 the fields e0 and ω respectively in the slicing and threading points of view have been referred to as spatial gauge fields. For a given parametrized slicing or threading respectively, it is the shift vector field and 1-form respectively whose variation corresponds to this spatial gauge freedom.

3.2.4 Computational frames and the reference decomposition Of course indices can be very useful and a particular choice of frame used to introduce them is wise if one is interested in viewing the spacetime from the point of view of the nonlinear reference frame, as indeed we are. Given an explicit parametrization of the nonlinear reference frame, it is natural to complete the threading vector field e0 to a frame adapted to both the threading and slicing and which may be thought of as a linear extension of the given parametrized nonlinear reference frame. It suffices to choose a frame {ea} for the 3-dimensional subspace of the tangent space which is tangent to the slicing and which is Lie dragged along e0 c [ea, eb] = C abec , [e0, ea] = 0 . (3.8) An unambiguous term for such a frame already exists, named a “computational frame” by York [1979], a c generalizing the coordinate frame of adapted coordinates {t, x } which occurs as the special case C ab = 0. Adapted coordinate frames often prove useful for local computations. Note that only the “spatial” structure γ γ a b c functions of the computational frame are nonzero: C αβ = δ cδ αδ βC ab. In terms of the congruence point of view terminology, the computational frame is a partially-observer-adapted frame which is comoving with the threading vector field e0. The dual frame {ω0, ωa} consists of the slicing 1-form (whose kernel is the slicing tangent subspace) and three 1-forms which annihilate the threading vector field. Expressing a tensor field in terms of components with respect to a computational frame leads to a “reference decomposition” of the field according to the reference temporal index 0 and the reference spatial indices 1,2,3, corresponding to the (in general) nonorthogonal decomposition of the tangent space into the direct sum of the slicing subspace and the threading subspace. For example, the reference decomposition of a vector field and a 1-form defines the “reference temporal projection” T and the “reference spatial projection” P for one-index tensors

0 a X = X e0 + X ea = [T X] + [PX] , (3.9) [ 0 a [ [ X = X0ω + Xaω = [T X ] + [PX ] .

One can even speak of a collection of “reference spatial fields” into which a tensor decomposes as in the orthogonal decomposition associated with the observers. For a given type of tensor field, there is in fact an isomorphism between the orthogonal decomposition and certain pieces of the reference decomposition of all the tensors related by index raising and lowering. To see this it is enough to project the computational frame as necessary to obtain a frame adapted to the observer orthogonal decomposition, which then establishes the appropriate isomorphism with the reference decomposition. This observer-adapted frame will be called the projected computational frame. In the slicing point of view, the reference spatial vector fields ea are already adapted to the observer local rest space LRSn 0 and it suffices to project e0 along the normal direction, while the slicing 1-form ω is proportional to the covariant unit normal n[ and one need only project the reference spatial 1-forms ωa orthogonally to the normal direction

a a slicing deﬁnitions: 0 = T (n)e0 , θ = P (n)ω ,

projected computational frame: {0, ea} , (3.10) dual frame: {ω0, θa} .

In the threading point of view, the threading vector ﬁeld e0 is already tangent to the threading subspace and it suﬃces to project the reference spatial vectors ea into the local rest space LRSm, while the reference

93 Figure 3.4: The relationship between the computational frame and the projected computational frame. spatial 1-forms ωa are already to orthogonal to the local time direction and one need only project the slicing 1-form along it 0 0 threading deﬁnitions: a = P (m)ea , θ = T (m)ω ,

projected computational frame: {e0, a} , (3.11) dual frame: {θ0, ωa} . In each case the isomorphisms from the reference projection eigenspaces to those of the observer orthogonal projection spaces are just the corresponding orthogonal projections themselves. The inverse isomorphisms are just the reference projections T and P, whose nontrivial values on the projected computational frame and dual frame are a a slicing: T 0 = e0 , Pθ = ω , 0 0 (3.12) threading: T θ = ω , Pa = ea . Thus one can continue to work with the computational frame using these isomorphisms rather than explicitly introducing new components with respect to the projected computational frame, although for certain purposes like diﬀerentiation the latter is essential. Figure 3.4 illustrates the relationships between the computational frame vectors and 1-forms and those of the projected computational frame. By duality the lapse function in each point of view also normalizes the temporal projection of the spatial gauge ﬁeld 2 (4) −1 N = − g(0, 0) , e⊥ = n = N 0 , (3.13) M −2 = −(4)g−1(θ0, θ0) , θ> = −m[ = Mθ0 .

3.2.5 Decomposing the metric The orthogonal projections of the computational frame vectors and 1-forms deﬁne the shift components in each point of view; alternatively the matrix of the linear transformation between the two frames determines the shift components a a a a a 0 slicing: 0 =T (n)e0 = e0 − N ea , θ = P (n)ω = ω + N ω , a P (n)e0 = N ea = N,~ 0 0 0 a (3.14) threading: a =P (m)e0 = ea + Mae0 , θ = T (m)ω = ω − Maω , = 0 a P (m)ω = Maω = M. The two choices of shift correspond to the two possible ways of completing the square in the metric to reduce it to block-diagonal form, which is the form it takes in the projected computational frame. The lapse then determines the metric on the 1-dimensional space along the time direction and the spatial metric on the 3-dimensional spatial subspace. Indices on components taken with respect to the projected computational frame can then be shifted with the appropriate subblock of the metric. Given these deﬁnitions, the spacetime metric and inverse metric in the computational and projected computational frame in the slicing point of view are

(4) (4) α β 2 0 0 a a 0 b b 0 g = gαβω ⊗ ω = −N ω ⊗ ω + gab(ω + N ω ) ⊗ (ω + N ω ) 2 0 0 a b ≡ −N ω ⊗ ω + gabθ ⊗ θ , (3.15) (4) −1 (4) αβ −2 a b ab g = g eα ⊗ eβ = −N (e0 − N ea) ⊗ (e0 − N eb) + g ea ⊗ eb −2 ab ≡ −N 0 ⊗ 0 + g ea ⊗ eb , i.e., in components

(4) 2 c (4) (4) g00 = −(N − NcN ) , g0a = Na , gab = gab , (3.16) (4)g00 = −N −2 , (4)g0a = N −2N a , (4)gab = gab − N −2N aN b ,

94 ab where (g ) is the matrix inverse of the positive-deﬁnite matrix (gab) and the indices on the shift vector a ab ﬁeld N~ = N ea are lowered and raised using gab and g . The single independent component of the volume 4-form, i.e., the (absolute value of the) square root of the metric determinant, is (4)g1/2 = Ng1/2. In the threading point of view they are instead given by

(4) 2 0 a 0 b a b g = −M (ω − Maω ) ⊗ (ω − Mbω ) + γabω ⊗ ω 2 0 0 a b ≡ −M θ ⊗ θ + γabω ⊗ ω , (3.17) (4) −1 −2 ab g = −M e0 ⊗ e0 + γ (ea + Mae0) ⊗ (eb + Mbe0) −2 ab ≡ −M e0 ⊗ e0 + γ a ⊗ b , i.e., in components

(4) 2 (4) 2 (4) 2 g00 = −M , g0a = M Ma , gab = γab − M MaMb , (3.18) (4) 00 −2 c (4) 0a a (4) ab ab g = −(M − McM ) , g = M , g = γ .

ab Here the spatial metric matrix (γab) is positive-definite, with inverse (γ ). These are used to lower and raise (4) 1/2 1/2 indices on the shift 1-form. Letting γ = det(γab) > 0, one has g = Mγ . For the slicing and threading parametrizations of the spacetime metric, it is precisely the two explicit terms in the final representation of (4)g and (4)g−1 above which correspond to the covariant and contravariant form of the orthogonal projections along the local time and space directions. The spatial metric in each case is just the covariant form of the projection. In the slicing point of view, its restriction to a slice yields the induced metric on the slice submanifold making it into a Riemannian manifold, while in the threading point of view it yields the projected metric on the slice submanifold, making the slice into a different Riemannian manifold representing the projected geometry rather than the induced geometry, which is not necessarily Riemannian in the threading point of view without an additional causality assumption. In each point of view, however, this spatial metric describes the relative distances of the worldlines of nearby observers at a given time t. Spatial tensor fields in each approach have only spatially indexed components nonzero when expressed in the projected computational frame. These spatial indices may be raised and lowered with the respective spatial metric matrix to yield the same result as index shifting with the spacetime metric on the computational frame indices. This equivalence follows from the relations

[ b a ] ab ea = gabθ , θ = g eb , (3.19) [ b a ] ab a = γabω , ω = γ b , which hold for the bases of spatial vector ﬁelds and 1-forms in each point of view. In fact on the respective spatial subspace of the tangent space, the bases {θa ]} and {ωa ]} are respectively reciprocal to the bases {ea} and {a} in the classical terminology [Sokolnikoﬀ 1964]. The spatial metric matrices are projected computational frame components of the fully covariant and fully contravariant forms of the spatial projection tensors, themselves spatial tensors in each point of view

[ a b ] ab P (n) = gabθ ⊗ θ ,P (n) = g ea ⊗ eb , (3.20) [ a b ] ab P (m) = γabω ⊗ ω ,P (m) = γ a ⊗ b .

Similar remarks apply to index raising and lowering of purely temporal ﬁelds using the lapse function

[ 2 0 0 ] −2 0 = −N ω , ω = −N 0 , (3.21) [ 2 0 0 ] −2 e0 = −M θ , θ = −M e0 .

However, the measurement process reduces all ﬁelds to families of spatial ﬁelds so this is not needed. The projected computational frame in each point of view is an observer-adapted frame for the corresponding congruence of test observers. As such one can take over all of the results derived in such a frame.

95 Figure 3.5: A pictorial representation of the various decompositions of a vector ﬁeld.

One need only make the appropriate identiﬁcations between the present notation and the congruence point of view notation. This correspondence is as follows n ← u = o → m {0, ea} ← {e0, ea} → {e0, a} {ω0, θa} ← {ω0, ωa} → {θ0, ωa} gab ← hab → γab N ← L → M. (3.22)

3.2.6 Relationship between the reference and observer decompositions The explicit isomorphisms between the reference decomposition subspaces and the orthogonal decomposition subspaces can easily be seen from the following explicit form of both projections themselves and their eﬀect on a vector ﬁeld X and its corresponding 1-form

Id = T + P = T (n) + P (n) = T (m) + P (m) , α β 0 a 0 a 0 a δ βeα ⊗ ω = e0 ⊗ ω + ea ⊗ ω = 0 ⊗ ω +ea ⊗ θ = e0 ⊗ θ +a ⊗ ω , | {z } | {z } ⊥ > e⊥ ⊗ ω e> ⊗ θ 0 a 0 ba −2 a X = X e0 + X ea = X 0 +Xbg ea = −M X0e0 +X a , (3.23) | {z } | {z } ⊥ > X e⊥ X e> [ 0 a 2 0 0 a 0 b a X = X0ω + Xaω = −N X ω +Xaθ = X0θ +X γbaω . | {z } | {z } ⊥ > X⊥ω X>θ where ⊥ α 0 > α 0 a X = −X nα = NX ,X = −X mα = M(X − MaX ) , (3.24) α −1 a α −1 X⊥ = Xαn = N (X0 − N Xa) ,X> = Xαm = M X0 . Figure 3.5 illustrates the three decompositions of a vector ﬁeld X in a suggestive way. In the slicing point of view, the covariant reference spatial components parametrize the spatial part of the vector ﬁeld X or 1-form X[, while in the threading point of view it is the contravariant reference spatial components which do this. Similarly the contravariant reference temporal component parametrizes the temporal part in the slicing point of view, while the covariant reference temporal component does this in the threading point of view. To recover those reference components which are not adapted to the observer orthogonal decomposition, one needs the transformation

a −1 a ⊥ ab a 0 ab X = −N N X + g Xb = −N X + g Xb , b b Xa = MMaX> + γabX = MaX0 + γabX , (3.25) 0 −1 > a −2 a X = M X + MaX = −M X0 + MaX , a 2 0 a X0 = NX⊥ + N Xa = −N X + N Xa .

This generalizes in an obvious way to higher rank tensors. The orthogonal projection via computational frame indices is accomplished in the slicing point of view by choosing all possible sets of components with the zero indices up and the spatial indices down, while these positions are reversed in the threading point of view, as emphasized by Zel’manov [1956, 1973]. The lapse in each case can be used to rescale the zero-indexed components to obtain partially-orthonormal components along the given time direction. For the sake of an

96 1 explicit example, the reference splitting, slicing splitting, and threading splitting of a 1 -tensor ﬁeld S are parametrized by the original computational frame components in the following way

0 0 0 a a 0 a b S =S 0e0 ⊗ ω + S ae0 ⊗ ω + S 0ea ⊗ ω + S bea ⊗ ω 0 0 a a a b ↔ {S 0,S aω ,S 0ea,S bea ⊗ ω } , [ 2 2 00 0 0 2 0 0 a 2 0 a 0 a b S =(−N ) S ω ⊗ ω + (−N )S aω ⊗ θ + (−N )Sa θ ⊗ ω + Sabθ ⊗ θ ⊥⊥ ⊥ ⊥ ⊥ ⊥ a ⊥ a ⊥ a b =S ω ⊗ ω + (−1)S aω ⊗ θ + (−1)Sa θ ⊗ ω + Sabθ ⊗ θ (3.26) ⊥⊥ ⊥ a ⊥ a a b ↔ {S , −S aθ , −Sa θ ,Sabθ ⊗ θ } , ] −2 2 −2 a −2 a ab S =(−M ) S00e0 ⊗ e0 + (−M )S0 e0 ⊗ a + (−M )S 0a ⊗ e0 + S a ⊗ b a a ab =S>>> ⊗ > + (−1)S> e> ⊗ a + (−1)S >a ⊗ e> + S a ⊗ b a a ab ↔ {S>>, −S> a, −S >a,S a ⊗ b} .

Once the tensor is split into a collection of spatial fields in the slicing and threading points of view, their “spatial” indices may be shifted using the spatial metric as follows from 3.20, while the normalized temporal indices can be shifted by a sign change. Zel’manov has called the collection of spatial fields which represents a spacetime tensor as “kinematically invariant quantities” (invariant under a change of threading”) in the slicing point of view [Zel’manov 1973] and as “chronometrically invariant” (invariant under a change of slicing”) in the threading point of view [Zel’manov 1956]. Note that in the special case of a tensor which is already a spatial field in a given point of view, the projected computational frame components are directly equal to the reference spatial components. The remaining computational frame components are parametrized by these reference spatial components in such a way that index shifting the reference spatial components with the spatial metric matrices exactly reproduces index shifting of the tensor with respect to the spacetime metric taking into account its remaining components. For a tensor which is not spatial, the projected computational components differ from the reference spatial components by terms involving the temporal components and the shift.

3.2.7 The slicing, threading and reference representations The splitting of spacetime into space plus time is done to be able to work with time-dependent fields on a 3-dimensional space rather than the original spacetime fields on the spacetime manifold. This may be done in several ways. In either point of view one may consider either a “slicing representation” or a “threading representation” by mapping the family of spatial fields which represents a given spacetime field onto a family of time- dependent fields on the slicing or threading computational 3-space respectively. In the slicing representation, ∗ one may pullback the covariant family of spatial fields to Σ using It . In the threading representation, one (4) may pushforward (down) the contravariant family of spatial fields to M/T hreading using πt ∗. In each case the spatial metric P (o)[ maps onto a time-dependent Riemannian metric which may be used to shift indices back to their original positions. The slicing representation of the slicing point of view is used by the extended Fischer-Marsden group [Fischer and Marsden 1979, 1980, Gotay et al 1990]. One may also stay on the spacetime and project the original family of spatial fields representing a given a a spacetime tensor field using the reference spatial projection P which reduces θ to ω and a to ea. This is often what is done when working in adapted coordinates where one frequently makes many convenient identifications in notation which obscure the fine distinctions between the two computational 3-spaces and the slices in spacetime. [ a b For example, in adapted local coordinates, one often sees the reference spatial field PP (n) = gabdx ⊗dx [ a a b b rather than P (n) = gab(dx +N dt)⊗(dx +N dt) referred to as the slicing spatial metric, and indeed if one interprets {xa} as the restrictions of the spatial coordinates to the slices, then the induced metric on those slices is the former expression, which may also be identified with the metric on the slicing computational ∗ a 3-space if one instead interprets these coordinates as their pullbacks It x to that manifold. Durrer and Straumann [1988] have chosen this option for the slicing point of view, referring to the reference spatial fields as “horizontal fields”. Similarly one might choose to use PP (m)] = γab∂/∂xa ⊗ ∂/∂xb as a representative

97 ] ab a b for P (m) = γ (∂/∂x + Ma∂/∂t) ⊗ (∂/∂x + Mb∂/∂t) in the threading point of view. If one interprets the spatial coordinates instead as the coordinates on the threading computational 3-space which they naturally induce, then this is the correct expression for the inverse metric there. Note that the restriction to a slice of either the spacetime metric, or the reference spatial metric, or the spatial metric yields the same induced metric on the slice in the slicing point of view, or the same pullback to the slicing computational 3-space for a given value of t. Similarly in the threading point of view, the projection onto a slice of the spacetime inverse metric, the reference inverse metric, or the inverse spatial metric all yield the inverse spatial metric when restricted to the slice, or the same pushforward down to the threading computational 3-space for a given value of t. Given any spatial tensor ﬁeld S, the reference spatial projection

a... b PS = S b...ea ⊗ · · · ⊗ ω ⊗ · · · (3.27)

may be used to as a representative of the original spatial ﬁeld in the “reference representation” of the given point of view. This isomorphism gives a representation of the computational 3-space and its geometry on [ ] ab ab the spacetime itself. The reference spatial metric PP (o) (gab or γab) has the “inverse” PP (o) (g or γ ) in the sense that their contraction yields the reference spatial identity tensor

PP (o)] PP (o)[ = PP (o) = P . (3.28)

Index-shifting of reference spatial fields with the reference spatial metric then corresponds to the reference projection of index-shifting of spatial fields with respect to the spacetime metric or equivalently with respect to the spatial metric. The reference representation is the only way one can interpret in terms of fields on spacetime the manipulations one does when working in classical coordinate notation and partitioning components according to their temporal and spatial index structure. The style in which Landau and Lifshitz discusses the threading point of view typifies this classical approach. Durrer and Straumann have adopted the explicit modern reference representation for the slicing point of view. Using the slicing point of view in the black hole context, Thorne [Thorne and MacDonald 1984] originally referred to the computational 3-space for a stationary nonlinear reference frame with its time-independent slicing spatial metric as “absolute space”, but later decided to use this term for the observer space, namely the space of observer worldlines (normal congruence) rather than threading curves, preferring to think of the grid of an adapted spatial coordinate system as moving relative to this space [Thorne et al 1986]. Gerosh [1971] while studying solution generation techniques for stationary spacetime explored the geometry of the computational 3-space in the threading point of view. In this case stationary fields may be represented by families of time-independent fields on the threading computational 3-space, as discussed below.

3.2.8 Transformation between slicing and threading points of view In many applications both n and m are timelike on some region of spacetime, and so one may consider the relative observer boost between them as discussed in section 2.1.3. Introduce the following abbreviated notation to handle this application of those formulas

a a v = ν(m, n) ,V = V(m, n) = −ν(n, m) , γL = γ(m, n) = γ(n, m) . (3.29)

Then one has

a a m = B(m, n)n = γL(n + v ea) , or n = B(n, m)m = γL(m − V a) , a a a −1 a a v = N /N , V = γL v = MM ,Va = γLva , 2 a b −2 a 2 a a b 2 (3.30) v = gabv v = N N Na = M M Ma = γabV V = V , 1/2 1/2 2 −1/2 γL = N/M = γ /g = (1 − v ) .

Then a −1 ~v = v ea = N N~ = v(m, n) (3.31)

98 Figure 3.6: The 2-plane of the Lorentz boost between the slicing and threading observers.

is the spatial velocity associated with this boost as seen from the local rest space associated with the unit normal, while a V~ = V a = MM~ = −v(n, m) (3.32) is the one as seen from the local rest space orthogonal to the time lines. If either m or n (but not both) changes causal character by becoming null, this boost becomes singular, with v → 1. Note also that the −1 inverse relative projection P (m, n) : LRSm → LRSn is just equal to the reference spatial projection P, since by deﬁnition the latter map undoes the projection into the threading local rest space of m. Any remarks in what follows involving relationships between the slicing and threading perspectives will be understood to apply only when both n and m exist as timelike unit vectors, the case of a spacelike slicing and a timelike threading. When this is the case, the diﬀerent representations of the observer orthogonal decomposition of a vector or a tensor as given in Section 3.2.6 then become a relationship between the two points of view which may be re-expressed in terms of this Lorentz boost. Figure 3.6 illustrates the 2-plane in which the Lorentz boost takes place, making use of the unit vectors along the relative velocity vectors

vˆa = v−1va , Vˆ a = V −1V a . (3.33)

Instead of analyzing the relationship between the two points of view in geometrical terms, one can treat the a problem as a transformation of gravitational variables. Either set of variables (N,N , gab) or (M,Ma, γab) may be used to parametrize the spacetime metric and express the ﬁeld equations. The transformation between these two sets is easily derived by comparing the two parametizations of the spacetime metric and its inverse. The result is

2 c 1/2 M = (N − N Nc) , 2 c −1 a −2 a Ma = (N − N Nc) Na M = N N , (3.34) 2 c −1 ab ab −2 a b γab = gab + (N − N Nc) NaNb , γ = g − N N N ,

with inverse −2 c −1/2 N = (M − M Mc) , a −2 c −1 a 2 a N = (M − M Mc) M Na = M M , (3.35) 2 ab ab −2 c −1 a b gab = γab − M MaMb , g = γ + (M − M Mc) M M . This transformation can then be pushed up to the connection and curvature levels. The transformation of the Einstein tensor is very instructive in considering how the two formulations of the Einstein equations relate to each other.

3.2.9 So far: 0 0 To summarize, the pair of fields (e0, ω ) with the compatibility condition ω (e0) = 1 represent the parametrized nonlinear reference frame at the linear level. In each point of view the causal assumption about the nonlinear reference frame leads one of these fields to introduce a local notion of time which is represented by a unit timelike vector field o, interpretable as the 4-velocity of a field of test observers, while the other field represents the gauge freedom remaining in the nonlinear reference frame holding that family of observers fixed, a freedom which is natural to call the spatial gauge freedom. This observer 4-velocity field o is the unit normal n = e⊥ to the slicing in the slicing point of view, obtained by normalizing and index-lowering the 0 sign-reversed slicing 1-form −ω , and it is the unit tangent m = e> to the threading in the threading point of view, obtained by normalizing the threading vector field e0. The orthogonal decomposition of the tangent space into the local time direction of the observer and the local rest space LRSo may be used to “measure” a spacetime field, leading to a collection of “spatial” fields of different ranks obtained by contraction with −o rather than projection along o in the tensor product of the orthogonal decomposition. The “measurement” of the spacetime metric gives only one nontrivial piece, the “spatial metric” which is the metric on

99 0 the local rest space LRSo, while the decomposition of the spatial gauge field, respectively e0 and ω , leads to the remaining pieces of the spacetime metric. The spatial projection gives the shift vector field and 1-form respectively, while the temporal contraction yields the lapse and the negative reciprocal of the lapse respectively. In each case the lapse function is the normalizing function (with the appropriate power) for the parametrized nonlinear reference frame field which determines the local time direction, and determines the rate of change of the observer proper time with respect to the time parameter of the parametrized nonlinear reference frame. All of the spatial fields may be interpreted as time-dependent fields on either of the two computational 3-spaces in the slicing or threading representation. The reference projections also allow one to represent these fields on the spacetime in terms of fields which instead are “spatial” with respect to the reference decomposition of the tangent space associated with the nonlinear reference frame, and these reference fields lead to the same time-dependent fields on the computational 3-spaces as the original spatial fields. Finally, in the event that both points of view hold, one has a unique Lorentz boost which transforms between the two points of view. The principal difference between the two points of view is that the test observers are fixed in the computational 3-spaces in the threading point of view, but move with respect to those spaces in the slicing point of view, the position of a given observer at time t determined by an integral curve of the time-dependent sign-reversed shift vector field. This makes the slicing point of view a hybrid type of formalism based on two distinct congruences, while the threading point of view is entirely determined by a single congruence.

100 3.3 Derivatives

The slicing point of view, with its somewhat ugly but in general necessary separation of the observer and evolution congruences, wins as soon as one considers spatial derivatives, which are integrable in this point of view, i.e., representable in terms of partial derivatives with respect to spatial coordinates only plus linear transformations. It is the nonintegrability of the spatial derivatives which sabotages the somewhat more aesthetically pleasing threading point of view, where the measurement and evolution are both determined by a single congruence. Each of the derivative operators which act on spatial tensor fields may be expressed in terms of corresponding operators in the slicing, threading, and reference representations of a given point of view. These operators are defined in the latter point of view by reference spatial projection of the corresponding spatial operators. The reference representation is just a way of modeling the computational 3-space on the spacetime itself. For the natural differential operators, this leads to a reference spatial Lie derivative

[P£]XS = P[£XS] (3.36) and a reference spatial exterior derivative

[Pd]S = P[dS] . (3.37)

3.3.1 Evolution Once a given tensor field has been decomposed into a collection of spatial tensor fields in each point of view, corresponding to the notion of measurement, one can discuss the evolution of these “measured quantities”, namely how they change along the threading congruence relative to the structure imposed on spacetime by the nonlinear reference frame. Given a parametrization of this nonlinear reference frame, the 1-parameter group of diffeomorphisms of the threading vector field e0 provides a natural way of comparing fields at different “times” relative to that reference frame itself, namely by Lie dragging. “No evolution” corresponds to Lie invariance. However, if one evolves spatial fields rather than spacetime fields, one must compose the Lie dragging with the spatial projection in order to compare spatial fields since in general, spatial fields will not remain spatial under Lie dragging. At the differential level, one therefore needs the spatial Lie derivative rather than the Lie derivative along e0 £(o)e0 S (3.38) in order to discuss the evolution of a spatial tensor field S. This Lie derivative will be referred to as the evolutionary Lie derivative. It is this derivative which will correspond to the ordinary time derivative on the computational 3-space. In the threading point of view this is just a rescaling of the observer Lie temporal derivative when acting on spatial fields since in that case o = Me0 and the observer and evolution congruences coincide. However, in the slicing point of view the observer congruence and threading congruence are in general distinct so the measurement projection is not orthogonal to the evolution congruence, slightly complicating the situation.

3.3.2 Natural time derivatives The time derivative relevant for the evolution is in good shape in both points of view, representable in terms of an “ordinary time derivative”. By assumption the computational frame vectors and 1-forms are Lie dragged along e0 so a a £(n)e0 ea = £e0 ea = 0 , £(n)e0 ω = £e0 ω = 0 . (3.39)

The spatial Lie derivatives along e0 of the remaining projected computational frame spatial vectors and 1-forms are instead a a 0 a £e θ = (e0N )ω , £(n)e θ = 0 , 0 0 (3.40) £e0 a = (e0Ma)e0 , £(m)e0 a = 0 ,

so the projected computational spatial frame and dual frame are spatially comoving along e0. This means that the spatial Lie derivative of a spatial tensor by e0 is obtained simply by diﬀerentiating the projected

101 computational components by e0, which amounts to a partial derivative with respect to t in adapted coordinates. In either the slicing or threading representation, this corresponds to the time-derivative of the time-dependent field on the computational 3-space to which a given spatial field is mapped, i.e., the derivative of the 1-parameter family of fields with respect to its parameter ˙ S 7→ St , £(o)e0 S 7→ d/dt St ≡ St . (3.41) A dot will be used to indicate the time derivative of a time-dependent field on the computational 3-space.

The evolution time derivative £(o)e0 is simply related to the observer spatially-projected Lie derivative £(o)o. In the threading point of view one has for any spatial ﬁeld S

−1 £(m)e0 S = M£(m)mS or £(m)mS = M £(m)e0 S, (3.42) so one need only rescale the observer Lie temporal time derivative operator used in the congruence point of view discussion. In the slicing point of view, one has instead for a spatial ﬁeld S

£(n) S = £(n) S + £(n) S = N£(n) S + £(n) S, (3.43) e0 Nn N~ n N~ or £(n) S = N −1[£(n) S − £(n) S] = N −1£(n) S n e0 N~ 0 −1 (3.44) 7→ Nt [S˙t − £(n) St] , N~t so in addition the shift Lie derivative appears, i.e., a spatial derivative necessarily accompanies the ordinary time derivative in the observer time derivative if the threading is tilted. In other words the evolution time derivative is simpler in the threading point of view since the it coincides with the reference time derivative, while a correction term is in general necessary in the slicing point of view.

3.3.3 Natural spatial derivatives The situation is reversed for the spatial derivatives in comparison with the temporal derivatives. The slicing spatial derivatives are simpler since the reference spatial projection is the observer spatial projection in the slicing point of view, but a correction term is necessary for the spatial derivatives in the threading point of view. To see this in detail, one must discuss how the spatial derivatives are represented in terms of the nonlinear reference frame and the spatial metric, i.e., in terms of derivative operators which make sense on the computational 3-space with its time-dependent Riemannian metric and its tensor algebra representation of the algebra of spatial tensor fields. For the slicing point of view this is simple and well known. The spatial exterior derivative is equivalent to the restriction to the slice of the exterior derivative, while the spatial Lie derivative for spatial fields is equivalent to the intrinsic Lie derivative on the slice itself and the spatial connection is equivalent to the connection of the induced metric on the slice. For the threading point of view two differences arise. First, the direction of spatial differentation, whether in the spatial exterior derivative, the spatial Lie derivative along a spatial vector field, or in the spatial covariant derivative, is no longer along the slicing subspace of the tangent space and the spatial derivatives decompose into a part along the slicing subspace and a part along the threading using the reference decomposition. This decomposition closely parallels the splitting of the total spatial covariant derivative in the congruence point of view using the observer decomposition. Second, the spatial part of the spatial connection itself is distinct from the metric connection associated with the projected metric on the slice, so a difference tensor arises between the two connections. a The first question is understood by examining the natural spatial derivatives. Let [Pd]f ≡ P[df] = ω eaf define the reference spatial exterior derivative which differentiates a function f along the slicing directions. The same relation [Pd]S = P[dS] (3.45) defines this operator for any differential form S and one easily sees that [Pd]2 = 0. For the slicing point of view, the slicing spatial exterior derivative projects to the reference spatial exterior derivative P ◦ d(n) = [Pd] . (3.46)

102 In the threading point of view one has instead for a function f

= a a d(m)f = ω af = ω [ea + Mae0]f = [Pd]f + M£(m)e0 f , (3.47) which is easily generalized to a spatial diﬀerential form S by including the wedge product

= Pd(m)S = d(m)S = {[Pd] + M ∧ £(m)e0 }S. (3.48) (Note that reference spatial and spatial covariant tensor ﬁelds in the threading point of view coincide.) a In an adapted coordinate frame, the reference spatial derivatives eaf = ∂f/∂x are just ordinary partial a derivatives, but the threading spatial derivatives af = (∂/∂x + Ma∂/∂t)f are not, and are often called “transverse partial derivatives”. On a computational 3-space, the reference spatial exterior derivative becomes the ordinary exterior derivative while the threading spatial exterior derivative acquires a correction term involving the time derivative S 7→ St ,

[Pd]S 7→ dSt , (3.49) = d(m)S 7→ dSt + M t ∧ S˙t . Similar formulas hold for the spatial Lie derivative of spatial tensor ﬁelds with respect to a spatial vector ﬁeld X For a function f one has

£(n)Xf = [P£]PXf = PXf 7→ Xtft , = (3.50) £(m)Xf = [P£]PXf + M(X)£(m)e0 f = ˙ 7→ Xtft + M t(Xt)ft , while for a spatial vector ﬁeld Y one has

£(n)XY = [P£]PXPY 7→ £ Y , Xt t = = (3.51) £(m)XY = [P£]PXPY + M(X)£(m)e0 Y − M(Y )£(m)e0 X = = 7→ £ Y + M (X )Y˙ − M (Y )X˙ . Xt t t t t t t t

3.3.4 Gauge transformations of the nonlinear reference frame For a given partial splitting of spacetime, one can consider all of the compatible full splittings, i.e., the equivalence class of nonlinear reference frames with one of its two components fixed by the partial splitting. For a fixed threading, the nonlinear reference frame is only defined up to a choice of slicing (spatial gauge freedom) and its parametrization (the remaining temporal gauge freedom). For a fixed slicing, the nonlinear reference frame is only defined up to a choice of threading (spatial gauge freedom) and its parametrization (the remaining temporal gauge freedom). It is therefore of interest to consider what happens to the gauge variables, namely the lapse and shift, under such gauge transformations. The spatial metric is gauge invariant of course, as are all quantities which only depend on the local time direction of the point of view. 0 One may use the pair (e0, ω ) to discuss these gauge transformations on the linear level. In the threading point of view, the threading vector field can only be scaled, while the slicing 1-form may be changed arbitrarily consistent with the transversality condition

−1 0 0 00 e0 = M e0 , ω = Mω . (3.52)

The lapse and shift then behave in the following manner

= = = M 0 = MM, M 0 = M−1(M − M) . (3.53)

103 The condition M = 1 ﬁxes the parametrization in the case that a preferred parametrization exists. In the slicing point of view, the slicing 1-form can only be scaled, while the threading vector ﬁeld may be changed arbitrarily consistent with the transversality condition

0 00 −1/2 0 ω = N ω , e0 = M (e0 − N ) . (3.54)

The lapse and shift then behave in the following manner

N 0 = N N, N~ 0 = N N~ + N . (3.55)

The condition N = 1 ﬁxes the parametrization. Most discussions of these gauge transformations take place in the context of an adapted coordinate system, in which case the above relations interpret the corresponding class of coordinate transformations for either a given threading or slicing. TO DO.

104 3.3.5 Observer-adapted frame structure functions and kinematical quantities In order to examine spatial and spacetime covariant derivatives in relation to the nonlinear reference frame, it is enough to extend the correspondence (3.22) between the projected computational frame in each point of view and the observer-adapted frame of the congruence point of view associated with the respective observer congruence to include the structure functions of the frame. Denote the projected computational structure α α functions by C(o) βγ and those of the partially normalized projected computational frame by D(o) βγ . The purely spatial structure functions in all cases are the same as those of the original computational frame

a a a C(o) bc = D(o) bc = C bc . (3.56)

In the slicing point of view the only other nonzero structure functions are

C(n)b = −θb(£(n) e ) , (3.57) 0a N~ a which describe the failure of the frame to be spatially comoving along 0, or equivalently along n, and the corresponding functions for the partially-normalized frame

D(n)b = −N −1θb(£(n) e ) , (3.58) ⊥a N~ a plus the functions ⊥ D(n) ⊥a = a(n)a = [d(n) ln N]a (3.59) which describe the acceleration of the observer congruence. It is convenient to re-express the structure a functions D(n) ⊥b in terms of the spatial covariant derivative using the comma-to-semicolon formula for the Lie derivative a −1 a a c D(n) ⊥b = N [N |b − Γ(n) cbN ] −1 a a c (3.60) = N N |b − Γ(n) cbv , where ~v = N −1N~ is the relative velocity of the threading curves with respect to the test observers. In the threading point of view one has

= 0 C(m) 0a = e0Ma = [£(m)e0 M]a , = (3.61) 0 c −1 C(m) ab = aMb − bMa − McC ab = [d(m)M]ab = 2M ω(m)ab , and = > D(m) >a = e0Ma + a ln M = [d(m) ln M]a + [£(m)e0 M]a = a(m)a ≡ Ca , = (3.62) > c D(m) ab = M[aMb − bMa − McC ab] = M[d(m)M]ab = 2ω(m)ab ≡ Ω˜ ab , the latter of which coincide with the projected computational frame components of the acceleration and twice the rotation of the observer congruence. The symbols Ca and Ω˜ ab were used by Cattaneo [1958, 1959a,b] in his formalism based on adapted coordinates. a a The vanishing of C(m) 0b and D(m) >b means that the projected computational frame is spatially comoving in the threading point of view. This is not the case in general in the slicing point of view where instead the frame is spatially comoving not along the observer congruence but along the threading congruence, i.e., the spatial Lie derivatives along e0 of the spatial frame and dual frame are zero. The only kinematical quantity associated with the observer congruence which does not enter into the structure functions is the expansion tensor, which arises from the temporal Lie derivative of the spatial metric. Gathering the above results together with the evaluation of the expansion tensor leads to the following expressions for the kinematical quantities of the observer congruence and their representations on

105 the computational 3-space

[ a(n) = d(n) ln N 7→ d ln Nt , = = [ ˙ a(m) = d(m) ln M + £(m)e0 M 7→ d ln Mt + M t ,

ω(n)[ = 0 , = [ 1 ω(m) = 2 Md(m)M, 1 ~ ~ ~˙ (3.63) ~ω(m) 7→ 2 [Mt curl Mt + Mt × M t] ,

θ(n)[ = 1 N −1£(n) P (n)[ = 1 N −1[£(n) − £(n) ]P (n)[ , 2 0 2 e0 N~ −1 θ(n)ab 7→ (2Nt) [g ˙t ab − £ gt ab] , N~t [ 1 [ θ(m) = 2 M£(m)e0 P (m) −1 θ(m)ab 7→ (2Mt) γ˙ t ab . The expansion tensor components enter into the structure functions if one passes to an orthonormal spatial frame from the projected computational frame. Such an orthonormal frame is used by Durrer and Straumann [1988] in the slicing point of view.

3.3.6 Spatial covariant derivative Given the correspondence with the results of the congruence point of view, one can immediately evaluate the components of the spatial part of the spatial connection in the projected computational frame. It is convenient to introduce the symbols

a a 1 ad Γ[g] bc = Γ(P, n) bc = g (g{db,c} + C(n){dbc} ) , 2 − − (3.64) a a 1 ad Γ[γ] bc = Γ(P, m) bc = 2 γ (γ{db,c}− + C(m){dbc}− ) , a a a where C(n) bc and C(m) bc indicates that the indices on C bc are shifted with the respective spatial metrics. a a The notation Γ(P, o) bc allows simultaneous reference to both points of view, while the notation Γ[g] bc and a Γ[γ] bc is more suggestive of the symmetric connection of the Riemannian spatial metric on a computational 3-space. Then Eq.(2.204) with hab = gab yields a a Γ(n) bc = Γ[g] bc , (3.65) while with hab = γab it yields a a a Γ(m) bc = Γ[γ] bc + ∆Γ(P, m) bc , a 1 ad ∆Γ(P, m) bc = 2 γ £(m)e0 γ{db Mc}− (3.66) ad = γ Mθ(m){db Mc}− . On a computational 3-space, the diﬀerence tensor becomes

a 1 ad ∆Γ(P, m) bc 7→ 2 γ γ˙ {db Mc}− , (3.67) suppressing the index t representing the time dependence of the fields on this space. One can introduce reference spatial covariant derivatives associated with the induced metric on the slices in the slicing point of view and the projected metric on the slices in the threading point of view in each case modeling the metric connection of the Riemannian computational 3-space at a given value of the time t. For a... a a reference field S = S b...ea ⊗ · · · ω ⊗ · · ·, define the reference spatial components of its reference spatial covariant derivative by

a... a... a d... d a... ∇(P, o)cS b... = S b...,c + Γ(P, o) cdS b... + · · · − Γ(P, o) cbS d... + ··· . (3.68)

106 If S is a spatial tensor in a given point of view, then the reference spatial covariant derivative of its reference spatial projection ∇(P, o)PS maps onto the covariant derivative of St on the computational 3-space using the metric connection at time t on this space. For the slicing point of view, this is an isomorphism which then maps the spatial covariant derivative on the spatial tensor algebra onto the metric covariant derivative on the computational 3-space at corresponding times t P[∇(n)XS] = ∇[g]PXPS. (3.69) For the threading point of view, it is not. Instead there are two correction terms, each due to the tilting of the local rest spaces of the observers away from the slices, one involving the direction of differentiation of S and a second involving the derivatives occurring in the connection components themselves = P[∇(m) S] = ∇[γ] PS + M(X)£(m)e S + σ(∆Γ(X))S, X PX 0 (3.70) a a c ∆Γ(X) b = ∆Γ(P, m) cbX , where X is a spatial vector field and σ is the Lie algebra tensor representation appropriate to the index = structure of S. The first correction term maps onto M t(Xt)S˙t on the computational 3-space, while the coefficients in the second correction term have already been described above.

3.3.7 Spatial vector analysis To discuss spatial vector analysis, let diff stand for any of the operator symbols grad, curl and div. Then one may introduce operators diff(P, o), with alternative symbols diff[g] for o = n and diff[γ] for o = m, for the corresponding reference spatial operators. These operators represent on the spacetime the corresponding operators defined on the computational 3-space in terms of its Riemannian connection and may be used to express the spatial derivative operators diffo in the reference representation. One also needs a reference spatial dot product and cross product between two reference spatial vector fields X and Y a b X ·(P,o) Y = [PP (o)]abX Y , (3.71) 1/2 ] ad b c X ×(P,o) Y = p(o) [PP (o) ] dbcX Y ea , [ with p(o) = det([PP (o) ]ab) standing respectively for g and γ. Similarly one has the alternative notation X ·[g] Y and so on. This notation may also be extended to the spatial duality operation, though admittedly it becomes more and more cumbersome. However, the point is not to develop a notation that is the simplest to use in a specific application but one which is capable of describing unambiguously all possible applications. For the various gradient operators one has

] ab grad(P, o)f = [PP (o) ] [Pd]f = [P (o) ebf]ea , (3.72) or equivalently ab ab grad[g]f = [g ebf]ea , grad[γ]f = [γ ebf]ea . (3.73) a For a reference spatial vector ﬁeld X = X ea one deﬁnes

−1/2 abc [ d curl(P, o)X = p(o) ∇(P, o)b [PP (o) ]cdX ea (3.74) and a −1/2 1/2 a div(P, o)X = ∇(P, o)aX = p(o) /∂a[p(o) X ] , (3.75) where a b a /∂aX ≡ (ea − C ab)X . (3.76)

Given these operators one can then express the spatial derivative operators diﬀo in the reference representation, with the corresponding images on the computational 3-space. The results are

P gradn f = grad[g]f 7→ grad[gt]ft , ~ P gradm f = grad[γ]f + [£(m)e0 f] PM (3.77) ˙ 7→ grad[γt]ft + ftM~ t ,

107 for the gradient, P curln X = curl[g] PX 7→ curl[gt]Xt , ~ P curlm X = curl[γ] PX + PM ×[γ] [P£]e0 PX (3.78) ~ ˙ 7→ curl[gt]Xt + Mt ×[γt] Xt for the curl, and P divn X = div[g] PX 7→ div[gt]Xt , ~ −1/2 1/2 P divm X = div[γ] PX + PM p(o) [P£]e0 p(o) PX (3.79)

−1/2 1/2 7→ div[γt] Xt + M~ t p(o)t p(o)t Xt ˙ for the divergence, where in each occurrence X is assumed to be a spatial vector field of the appropriate type. It is also of interest to relate the spatial divergences of spatial vector fields to the spacetime divergences. Here one finds a −1 a [∇(n)a + a(n)a]X = N divn NX α = X ;α , a −1 a a (3.80) [∇(m)a + a(m)a]X = M divn MX + [£(m)e0 Ma]X α = X ;α . The spatial derivative operators that appear in the text of Landau and Lifshitz [1975] are just the reference spatial derivative operators appropriate for the threading point of view. Hanni [1977] has noted the complementary definitions which correspond to the slicing point of view.

3.3.8 Partially-observer-adapted frames: connection components Given the correspondence with the results of the congruence point of view, one can immediately express the splitting of the tensor representing the connection components in the projected computational frame. It is also of interest to split the tensor representing the connection components with respect to the computational frame itself. Many of the spatial fields from these two families coincide, since the transformation matrix between the two frames, whose derivative is responsible for the difference tensor, is of a very simple form. To examine the relationship between the two frames, a more efficient notation is needed. Let

β e(o)α = A(o) αeβ (3.81) denote the projected computational frame vectors in each point of view. The transformation matrix has a very simple form

0 0 a a a a A(n) 0 = 1 ,A(n) a = 0 ,A(n) 0 = −N ,A(n) b = δ b , 0 0 a a a (3.82) A(n) 0 = 1 ,A(n) a = Ma ,A(n) 0 = 0 ,A(n) b = δ b .

Let (4)ωωω and (4)ωωω(o) denote the tensor-valued connection 1-forms in the original computational frame and in the projected computational frame. They are related by the well known inhomogeneous transformation law

(4) α −1 α (4) γ δ −1 α γ ωωω(o) β = A(o) γ ωωω δA(o) β + A(o) γ dA(o) β , (3.83)

or (4)ωωω(o) = (4)ωωω + ∆(4)ωωω(o) . (3.84) The diﬀerence term is rather simple, having only the following components nonzero in the projected computational frame −1 a a a 0 a b [A(n) dA(n)] ⊥ = −dN = −0N ω − ebN θ , (3.85) −1 > 0 b [A(m) dA(m)] a = dMa = e0Maθ + bMaω .

108 It leads to correction terms for the projected computational frame components of the connection coeﬃcients with respect to the computational frame with respect to those of the projected one

(4) α −1 α (4) α δ (4) α Γ(o) βγ = A(o) γ Γ βγ A(o) β + ∆ Γ(o) βγ , (3.86) where (4) α α (4) Γ βγ = ω ( ∇eβ eγ ) (3.87) are the computational frame components of the spacetime connection. The projected computational components of the nonzero correction terms

(4) a (4) a ∆ Γ(n) ⊥⊥ , ∆ Γ(n) ⊥b , (3.88) (4) > (4) > ∆ Γ(m) ⊥a , ∆ Γ(m) ba , may be read oﬀ from the above derivatives of the shift components. In particular the following component expressions for the kinematical quantities hold in either frame

(4) ⊥⊥ a (4) a a(n)a = Γ a , a(m) = Γ >> , (3.89) (4) ⊥ ab (4) ab k(n)ab = − Γ ab , k(m) = − Γ > .

3.3.9 Total spatial covariant derivatives The Lie and Fermi-Walker total spatial covariant derivatives have been introduced in the congruence point of view so they are available for the observer congruence in both the slicing and threading points of view with the identification u = o and U = γ(U, o)[o + ν(U, o)]. The discussion of the congruence point of view is relevant to the threading point of view, while that of the hypersurface point of view is relevant to the slicing point of view. Because two congruences are involved, the entire qualifier (U, o) cannot be suppressed but only abbreviated to (o). When differentiating a spatial tensor field along the worldline with 4-velocity U, the Lie total spatial covariant derivative has the following expressions in the two points of view

D(lie)(U, o)S/dτo = [£(o)o + ∇(o)ν(o)]S  M −1£(m) S + ∇(m) S  e0 ν(m) =  N −1[£(n) − £(n) ]S + ∇(n) S e0 N~ ν(n) (3.90)  −1 Mt S˙t + ∇(m) St  ν(m)t 7→ −1  Nt [S˙t − £(n) St] + ∇(n) St . N~t ν(n)t Explicit component expressions for these operators may be obtained in the projected computational frame using Eq.(2.166) once the correspondence n ← u = o → m a a D(n) ⊥b ← C >b → 0 (3.91) with the congruence observer-adapted frame is made. The Lie total spatial covariant derivative of a spatial vector then has the expression

a a a b c D(lie)(U, m)X /dτm = dX /dτm + Γ(m) bcν(m) X , (3.92) a a a b b c −1 a b D(lie)(U, n)X /dτn = dX /dτn + Γ(n) bc[ν(n) − v ]X + N N |bX .

The projected computational frame is a spatially-comoving observer-adapted frame in the threading point of view, and the entire discussion of the congruence point of view applies directly to the threading point of view. The simplifying assumption of a spatially-comoving frame convenient for many calculations is also

109 valid. However, in the slicing point of view, the projected computational frame is instead spatially comoving along e0 rather than 0, or equivalently n, and this leads to the two additional shift terms in the expression. However, in the slicing point of view it is more appropriate to use the Lie derivative £(n)e0 along the threading congruence than the temporal Lie derivative £(n)n along the observer congruence, in order to measure evolution with respect to the nonlinear reference frame. Making this change leads to still another Lie total spatial covariant derivative which will be designated at the slicing point of view operator, while the original one will be referred to as the hypersurface operator. This slicing Lie total spatial covariant derivative is an operator which uses the reference decomposition 0 U = U e0 +PU of the 4-velocity vector rather than the slicing decomposition as in the hypersurface operator, constructed with the spatial Lie derivative along the e0 and the spatial covariant derivative along P (n)U

0 D(lie)(U, P, n)/dτU = U £(n)e0 + ∇(n)PU , −1 (3.93) D(lie)(U, P, n)/dτn = N £(n)e0 + ∇(n)ν − ~v . This leads to a diﬀerence term with respect to the corresponding hypersurface point of view operator

−1 D(lie)(U, P, n)/dτn − D(lie)(U, n)/dτn = [N £(n)e0 + ∇(n)ν − ~v] − [£(n)n + ∇(n)ν] (3.94) = N −1[£(n) − ∇(n) ] . N~ N~ The diﬀerence operator represented by the last equation is a linear transformation operator, not a derivative operator. The comma-to-semicolon formula for the Lie derivative, spatially projected, expresses this linear transformation in terms of the covariant derivative of the shift N~ . If S is a tensor of type σ then

£(n) S − ∇(n) S = −σ(∇N~ )S. (3.95) N~ N~ Thus for a spatial vector ﬁeld X or its covariant form, the diﬀerence operator is

a a −1 a b D(lie)(U, P, n)X /dτn = D(lie)(U, n)X /dτn − N N |bX , (3.96) −1 b D(lie)(U, P, n)Xa/dτn = D(lie)(U, n)Xa/dτn + N N |aXb , which leads to the formulas

a a a b b c D(lie)(U, P, n)X /dτn = dX /dτn + Γ(n) bc[ν(n) − v ]X , (3.97) c b b D(lie)(U, P, n)Xa/dτn = dXa/dτn − Γ(n) ba[ν(n) − v ]Xc .

The change from the hypersurface to the slicing point of view introduces the shift derivative term in the total spatial covariant derivative in order to change to the spatial velocity ~ν − ~v relative to the threading curves. This term must therefore appear explicitly in the spatial gravitational force deﬁned using the new operator. The relationship of these various Lie operators is nicely represented in Figure 3.3.9. As with the previous Lie total spatial covariant derivative, index shifting with the spatial metric generates Lie derivative terms

−1 −1 D(lie)(U, P, n)gab/dτn = N £(n)e gab = 2θ(n)ab + 2N N(a|b) , 0 (3.98) ab −1 ab ab −1 (a|b) D(lie)(U, P, n)g /dτn = N £(n)e0 g = −2θ(n) − 2N N . These terms must be taken into account in comparing index-shifted formulas. The relative velocity may be represented in terms of the time derivatives of the spatial coordinates

a a −1 a ν(m) = dx /dτm = M dx /dt −1 a 7→ Mt x˙ t , a a a −1 a a (3.99) ν(n) = dx /dτn + v = N [dx /dt + N ] −1 a a 7→ Nt [x ˙ t + Nt ] .

110 u

γ(u, n)n ...... ∇(m) ...... ~ν(m) ...... γ(u, m)m ...... m ...... ∇(n) ∇(n) ...... γ(m, n)n ...... ~ν(m) ~ν(n) ~ν(n) − ~v ...... e0 ...... n ...... ~ν(n) −.... ~v ...... ~v ...... −.. 1 ...... N .. e0 ...... −1 ...... £(n)n .... N £(n)e ...... £(m)m ...... 0 ......

D(⊥)(u)/dτn D(sl)(u)/dτn D(th)(u)/dτm

γ(U, n)−1U = n + ~ν(n) γ(U, m)−1U = m + ~ν(m) −1 −1 γ(U, n) U = N e0 + [~ν(n) − ~ν]

Figure 3.7: A suggestive representation of the various total Lie spatial covariant derivatives for the hypersurface, slicing and threading points of view, normalized to correspond to the proper time of the appropriate observers. Each covariant derivative is the sum of a temporal Lie derivative and a spatial covariant derivative along the temporal and spatial parts of the relevant multiple of the 4-velocity U represented in the triangle, each vector decomposition given under the symbol for the corresponding covariant derivative.

This leads to analogous expressions for the slicing and threading Lie total spatial covariant derivatives

a −1 a a b c D(lie)(U, m)X /dτm = M [dX /dt + Γ(m) bcdx /dt X , (3.100) −1 c b D(lie)(U, P, n)Xa/dτn = N [dXa/dt − Γ(n) badx /dt Xc .

This leads to an analogous gravitomagnetic force in the slicing point of view even though the gravitomagnetic force in the hypersurface point of view vanishes.

3.3.10 Spatial gravitational forces The discussion of spatial gravitational forces depends on the choice of derivative operator one uses to express the acceleration equation. The discussion for the congruence point of view applies to the observer congruence in both the slicing and threading points of view. However, the change from the hypersurface to the slicing Lie total spatial covariant derivative leads to another representation of spatial gravitational forces. The diﬀerence term in this derivative operator compared with the hypersurface operator in the decomposition of the acceleration equation leads to a Lie gravitomagnetic force in the slicing point of view which is absent in the hypersurface point of view due to the vanishing of the rotation. The various Lie gravitomagnetic tensors are

−1 H(lie[)(m)ab = 2MM[b||a] ,H(lie])(m)ab = 2MM[b||a] − 2M £(m)e0 γab , −1 H(lie[)(n)ab = 0 ,H(lie])(n)ab = −N [£(n)e0 gab − 2∇(n)(aNb)] , (3.101) −1 −1 −1 H(lie[)(P, n)ab = N Nb|a ,H(lie])(P, n)ab = N Nb|a − N £(n)e0 gab ,

111 and the gravitomagnetic vectors are ~ ~ a abc H(lie)(m) = M curlm M,H(lie)(m) = Mη(m) ∇(m)bMc , ~ a H(lie)(n) = 0 ,H(lie)(n) = 0 , (3.102) ~ −1 ~ a −1 abc H(lie)(P, n) = N curln N,H(lie)(P, n) = N η(n) ∇(n)bNc . With these definitions the newly defined Lie total spatial gravitational forces are then ~ (G) ↔ F(lie[)(P, n) = γ(U, n)[~g(n) + ν(n) ·n H (lie[)(P, n)] 1 ~ ↔ = γ(U, n)[~g(n) + ν(n) ×n H(lie)(P, n)] + ν(n) ·n SYM H (lie[)(P, n)] , 2 (3.103) ~ (G) ↔ F(lie])(P, n) = γ(U, n)[~g(n) + ν(n) ·n H (lie])(P, n)] 1 ~ ↔ = γ(U, n)[~g(n) + 2 ν(n) ×n H(lie)(P, n)] + ν(n) ·n SYM H (lie])(P, n)] . Note that the same factor of one half which appears in the gravitomagnetic vector force term in the Fermi- Walker spatial gravitational force in the congruence point of view also appears in the Lie spatial gravitational forces in the slicing point of view. The energy equation (??) may also be expressed in the slicing and threading points of view. The total spatial covariant derivatives all agree for a scalar but the spatial gravitational forces differ. Using the covariant Lie spatial gravitational forces one has the result ~ (G) ~ −1 a b D(U, m)E(U, m)/dτm = [F(lie[)(m) + F (m)] ·m ν(m) − [2M] £(m)e0 γabν(m) ν(m) , (3.104) ~ (G) ~ −1 a b D(U, P, n)E(U, n)/dτn = [F(lie[)(P, n) + F (n)] ·n ~ν(n) − [2N] £(n)e0 gabν(n) ν(n) . The gravitoelectric and gravitomagnetic force fields have been discussed in the black hole case by Thorne et al [1986] using the hybrid total spatial covariant derivative operator in the slicing point of view. Both Zel’manov [1956] and Cattaneo [1958] have discussed them from the threading point of view, while Landau and Lifshitz [1975] discuss only the stationary case in the threading point of view. Möller[1952, 1972] discusses them both from his parametrization-dependent description of the threading point of view as well as for the threading point of view. All of these threading point of view discussions introduce the Lie spatial gravitational forces. Massa [1974b] has re-expressed the Cattaneo approach in a somewhat more modern framework, using the spatial co-rotating Fermi-Walker total spatial covariant derivative.

3.3.11 Second-order acceleration equation INCORPORATE MATERIAL FROM MFGREV.TEX: In the context of a nonlinear reference frame which enables one to represent the spacetime geometry in terms of time-dependent fields on a computational 3-space, one can re-express the first order spatial acceleration equation for the spatial momentum as a second order equation describing the evolution of the spatial coordinates along the worldline under consideration. Assume a projected computational frame obtained from adapted coordinates. The rates of change of the spatial coordinates with respect to the coordinate time t then define the reference spatial components of the coordinate spatial velocity ( Mν(m)a = Γ(U)−1p(n)a U a = dxa/dt = . (3.105) Nν(n)a − N a = Γ(U)−1p(m)a − N a This may be interpreted as a spatial vector in each point of view

a a U(n) = U ea , U(m) = U a . (3.106) The projected computational frame components of the appropriate Lie total spatial covariant derivative of this spatial vector then yield the corresponding second derivatives of the spatial coordinates

2 a 2 a 2 a b c D(lie)(U, m) x /dt D(lie)(U, m)U(m) /dt d x a dx dx 2 a 2 = a = 2 + Γ(o) bc . (3.107) D(lie)(U, P, n) x /dt D(lie)(U, P, n)U(n) /dt dt dt dt

112 These derivatives may then be related either to the relative acceleration equation and logarithmic derivatives of the lapse or to the spatial momentum equation and logarithmic derivatives of the coordinate gamma factor. For example, in the threading point of view one has

a 2 −1 a a D(lie)(U, m)U(m) /dt = M γ(U, m) D(lie)(U, m)p(m) /dτm − U D(U, m) ln Γ(U)/dt 2 a a (3.108) = M D(lie)(U, m)ν(m) /dτm + U(m) D(U, m) ln M/dt . Since the relative acceleration has already been evaluated in equation (2.255), the second expression is easier to evaluate. It only leads to an extra term along the relative velocity of the form

a (G) b U(m) {£(m)e0 ln M + [−F(lie])(m)b − F (m)b + a(m)b − £(m)e0 Mb]U(m) } . (3.109) The rest is just the rescaling of the total spatial force

2 a M [~g(m) + U(m) ×m H~ (m) + F~ (m)] . (3.110) Thus apart from the term along the direction of motion due to the change of parametrization of the worldline, the terms in the second-order acceleration equation directly correspond to the spatial force terms. In the slicing point of view, something more interesting happens because of the additional shift term in the coordinate relative velocity

a 2 a D(lie)(U, P, n)U(n) /dt = N D(lie)(U, P, n)ν(n) /dτn a a − D(lie)(U, P, n)N /dt + U(n) D(U, n) ln N/dt 2 a a a b = N D(lie)(U, P, n)ν(n) /dτn − N[£(n)e0 N + ∇(n)bN U(n) ] a b (3.111) + U(n) [£(n)e0 N + a(n)bU(n) ] 2 −1 ~ ~ ~ = N [~g(n) − N £e0 N + {U(n) ×n H(n) + F (n)} 1 ~ a a − 2 {v × H(n)}] + U(n) [...] . The additional shift spatial covariant derivative term combines with the existing term in the spatial gravitational force to form twice its antisymmetric part, eliminating the contribution of the symmetric part of the shift tensor gravitomagnetic force term and doubling the contribution of the gravitomagnetic vector force to yield a term exactly analogous to the threading point of view expression, modulo a term quadratic in the shift vector field. A shift Lie derivative term also adds to the gravitoelectric field to form an expression analogous to the one in the threading point of view. These changes to the slicing point of view discussion are not surprising since these expressions must result from a transformation of the threading point of view expressions, and in the nonrelativistic limit the lapse and shift of the two points of view coincide. Thorne et al [1986] discuss these changes in the slicing point of view second-order acceleration equation for black hole spacetimes in the weak field slow motion limit. Forward [1961] briefly discusses a reference decomposition of the second-order acceleration equation before linearizing to go to the same limit for an isolated body.

3.3.12 The spin transport equation The gravitomagnetic force ﬁelds govern the evolution of the spin vector of a gyro carried by the observers of the observer congruence. In the present context the spin transport equation along the observer congruence can be expressed as follows 1 ~ D(lie)(m, m)S/dτm = − H(m) ×m S + θ(m) S = H(lie])(m) S, 2 (3.112) 1 ~ D(lie)(n, P, n)S/dτn = − 2 H(n) ×n S + θ(n) S = H(lie])(P, n) S.

Thus the gravitomagnetic tensor force H(lie]) controls the evolution of the spin vector in the slicing and threading points of view, as it does in the congruence and hypersurface points of view. The splitting of the general spin transport equation is not so interesting since the physical information most useful is described by the hypersurface point of view discussion as far as it relates to the observations of the observer carrying the gyro itself. This topic will therefore be omitted.

113 3.3.13 Transformation of spatial gravitational fields In the case of a spacelike slicing and a timelike threading, both the slicing and threading points of view hold simultaneously and one can consider transforming between the two. The unique boost discussed above relates the two orthogonal decompositions and may be used to transform between the two families of spatial tensors which represent a given spacetime field. However, in the case of the spatial gravitational fields, their definitions in each point of view determine the transformation which relates them to each other. The gravitoelectric and gravitomagnetic vector fields turn out to be related in a way which is closely connected to the Lorentz boost of the electric and magnetic fields. In comparing the two points of view it is helpful to recall that ~v = N −1N~ and V~ = MM~ define the relative velocity of the threading observers with respect to the normal observers as measured by the normal and threading observers respectively. This explains the reciprocally related factors of the lapse and dual lapse which appear in the gravitomagnetic fields and in many other contexts. A simple calculation yields the following formulas for the transformation of the the gravitomagnetic vector field and the gravitoelectric 1-form h = i] 1 ~ 1 ~ ~ ~ 1 2 −2 P (m) 2 H(n) = 2 H(m) + V ×m ~g(m) + V ×m 2 M £(m)e0 (M M) , = = 2 ~ [ ~ [ 2 (3.113) P g (n) = γL [~g(m) − H(m) V ] + V [£m ln(MγL)] + γL £(m)e0 M 2 ~ 1 ~ ~ [ = γL [~g(m) − V ×m 2 H(m) − SYM H(m) V ] + .... King and Ellis[1973] represent the threading kinematical quantities in terms of those of the slicing, which is relevant to the inverse transformation h i 1 ~ 2 1 ~ 1 ~ P 2 H(m) = γL 2 H(n) + ( 2 H(n) ·n ~v)~v + ~v ×n (~g(n) + SYM H(n) ~v) h = i] + ~v × 1 £(n) (M −2N) , n 2 e0 (3.114) = = 2 [ [ −2 P (n) g (m) = γL [~g(n) + H(n) ~v] − γL~v £e0 ln(M) − £(n)e0 (M N) 2 1 ~ [ = γL [~g(n) + ~v ×n 2 H(n) + SYM H(n) ~v] + .... The relations a a a a P (m)(Y ea) = Y a , P(σaθ ) = σaω , a a a a (3.115) P(Y a) = Y ea ,P (n)(σaω ) = σaθ , enable one to convert the above index-free equations for the transformation of the gravitomagnetic vector field and the gravitoelectric 1-form back into component form 1 a 1 a 2 H(n) = H(m) + . . . , g(n)a = γL [g(m)a + ..., 2 2 (3.116) 1 a 2 1 a 2 2 H(m) = γL [ 2 H(n) + . . . , g(m)a = γL [g(n)a + ..., Since the gravitomagnetic vector field and the gravitoelectric 1-form only involve the exterior derivative, their transformation laws are relatively simple and closely analogous to the electromagnetic case, especially in the stationary case when all the Lie derivative terms vanish. The transformation formulas for the gravitomagnetic vector field may be directly compared with the transformation of the magnetic field under the boost between the two points of view, but the formulas for the gravitoelectric 1-form must be compared with the boost of the electric 1-form field. The symmetric part of the gravitomagnetic tensor on the other hand involves the spatial connection and has a much more complicated transformation law. In order to compare the above equations with the familiar orthonormal component form of the Lorentz transformation of the electric and magnetic fields, one must bridge the gap between the four-dimensional spatial notation and the usual orthonormal frame component notation which is so familiar. This is done in appendix B. However, one thing is clear independent of these details. The gravitomagnetic vector field transforms with an additional factor of one half with respect to the magnetic field. This is contrary to the usual analogy used in weak field slow motion arguments between the transformation laws for the gravitoelectric and gravitomagnetic fields and those for the electric and magnetic fields under a boost. In fact in those contexts one has a transformation between two different nonlinear reference frames associated with nearly inertial coordinate systems, not a transformation between two sets of observers associated with the same nonorthogonal nonlinear reference frame. Such a transformation law has not really been carefully studied.

114 3.4 Spatial curvature

Having expressed the spatial part of the spatial connection in terms of the metric connection on the computational 3-space or its equivalent reference representation representative on the spacetime itself, one can now relate the spatial curvature tensor to the curvature tensor of the Riemannian computational 3-space or its reference representative representative. This explains in a more direct way the unusual symmetries that occur and the additional kinematical terms that litter the classical equations involving the spatial curvature in the congruence point of view for a rotating congruence. In the slicing point of view there is instead a direct isomorphism, which has made it so useful in computations in so many problems over the years. The complications in the threading point of view have instead apparently discouraged anyone from even exploring the geometry in detail. The symmetry-obeying spatial curvature is the most closely connected with the Riemann curvature tensor of the spatial metric on a computational 3-space. The two curvature tensors coincide in the slicing point of view but diﬀer in the threading point of view

R (n)abcd =R[g]abcd , (sym) (3.117) R(sym)(m)abcd =R[γ]abcd + ∆R(P, m)abcd , where the diﬀerence term has the following representation on the computational 3-space

1 ∆R(P, m)abcd 7→ 2 {MbMcγ¨ad − MbMdγ¨ac + MaMdγ¨bc − MaMcγ¨bd} n 1 ˙ + (2M(b∇[γ]c) + ∇[γ](bMc) + M(bMc))γ ˙ ad 2 (3.118) −(c ↔ d) + (a ↔ b) − (c ↔ d) } 1 mn − 4 g γ˙ {mcMa}− γ˙ {ndMb}− − γ˙ {mdMa}− γ˙ {ncMb}− . The three terms in curly brackets in the threading point of view expression each have all the usual symmetries of a curvature tensor, leading to the same properties for the symmetry-obeying spatial curvature.

3.5 Initial value problem? 3.5.1 Hypersurface and slicing points of view The geometry of the initial value problem for the gravitational variables in the hypersurface point of view is well known, often referred to as the Cauchy problem. The question one asks is, can an initial spacelike hypersurface be imbedded in some spacetime with certain sources present? Of the gravitational variables only the intrinsic metric of the hypersurface and its extrinsic curvature are required to state and resolve this question. One must solve the gravitational constraints, which consist of the three components of the (4) ⊥ (4) ⊥ supermomentum contraint E a = 0 and the scalar super-Hamiltonian contraint E ⊥ = 0. In these contraints only hypersurface derivatives appear, so they represent a set of tensor equations within the hypersurface itself. These may be expressed entirely in terms of the spatial metric gab and the extrinsic curvature tensor Kab (and their spatial derivatives) at an initial time on the computational 3-space of an adapted nonlinear reference frame. The resolution of these equations via conformal methods has a long history starting with Lichnerowicz and Choquet-Bruhat and ending with the work of York and his contemporaries. A useful summary of this history may be found in a survey article by Choquet-Bruhat and York[1980]. Once an initial data set is found, then one must choose a nonlinear reference frame to evolve this data and determine the spacetime which contains the initial hypersurface. This amounts to a choice of the lapse and shift variables, and this may be done arbitrarily, but certain useful conditions have been found which lead to preferred choices of nonlinear reference frames. Smarr and York discuss this question in detail [Smarr and York 1978a,b??, York 1979].

115 3.5.2 Thin sandwich problem In the slicing point of view, one may reinterpret the initial value problem in terms of the gravitational variables and their time derivatives on the computational 3-space at a given moment of time. The same a equations become constraints on the variables (gab, g˙ab,N,N ). One method of solving these contraints is to specify the metric and its time derivative arbitrarily and attempt to ﬁnd values of the lapse and shift which solve them. This problem, known as the thin sandwich problem, is much more complicated and seems to have no deﬁnitive answer, as discussed by Belasco and Ohanian [1969].

3.5.3 Congruence and threading points of view The initial value problem for the congruence point of view is fundamentally different from the hypersurface point of view. Instead of initiating a slicing of a spacetime by considering a problem confined to an initial hypersurface involving only hypersurface derivatives, one is trying to initiate a timelike congruence in a spacetime. The basic gravitational variable is a unit vector field defined on an initial hypersurface which no longer requires a spacelike causality assumption, together with a metric on the its orthogonal complement within the tangent space to the hypersurface at each point. The fundamental difficulty with the corresponding gravitational contraint equations in the congruence point of view is that they involve spatial derivatives of the gravitational variables at the initial hypersurface, but the spatial derivatives do not coincide with the hypersurface derivatives. Since the congruence itself is the object one is trying to get started, the lapse and shift variables (M,Ma) (or variables equivalent to them) must be specified in order to consider the initial value problem. In fact there is little point in not passing immediately to the threading point of view and discussing the contraints in their representation on the computational 3-space. In this sense the problem is much closer related to the thin sandwich problem than the hypersurface initial value problem. To evaluate spatial derivatives of spatial fields in their representation on the computational 3-space, one must specify not only the field but its time derivative. Both the first and second time derivatives of the spatial metric (γab) are necessary to consider the threading constraints. First time derivatives appear in the components of the spatial connection, while second time derivatives occur in the spatial curvature and in spatial derivatives of the expansion tensor. The expansion tensor is now so closely related to the first −1 time derivative of the spatial metric θab = (2M) γ˙ ab that there seems little point to using a first order formalism.?? How can one deal withγ äb creeping into the initial value problem? Does it make sense to try to solve the threading evolution equations for this quantity and then replace it in the constraints? This is equivalent to taking new linear combinations of the Einstein equations related to the hypersurface directions. Should one therefore abandon the threading splitting of the Einstein equations for the initial value problem? To investigate these questions one can look at two extremes: stationary spacetimes where the time derivatives drop out, and Bianchi type I cosmological models, where the slicing spatial derivatives drop out. A third possibility is linearization, but this seems to eliminate some of these ugly questions which start at second order. Finally if one actually solves the initial value problem, to what extent can one specify (M,Ma) (consistent with the causality assumption) to evolve γab? How does the spatial gauge freedom enter into the evolution in this description? And if one satisfies a hybrid initial value problem, what does one do with the actual threading constraints and the evolution?

116 3.5.4 Perfect fluids TO DO: The threading point of view lapse and shift play an important role in the dynamics of an isoentropic perfect fluid in Taub’s adapted comoving coordinate system [Taub 1969, Taub and MacCallum 1972]. Taub chooses the threading point of view adapted to the fluid in the following way

−1 −1 e0 = µ u , M = µ ,Ma = va = µua , (3.119) where u is the unit four-velocity of the fluid, µ is the chemical potential of the fluid in the notation of Misner, Thorne and Wheeler [1973], and in this section only, va are the spatial components of the circulation 1-form [ ⊥ v = µu . (The time component has the value v0 = −1.) Note also the relation γL = u . This identifies the threading congruence with the fluid flowlines, but chooses the parametrization to simplify the fluid equations of motion. The choice of initial slice for the slicing is still arbitrary. This choice of parametrized threading of the perfect fluid spacetime makes the fluid equations of motion β very simple since the normal and tangential projections of the conservation equations Tα ; β = 0 are just

(4) div nu = 0 = £e0 v , (3.120) where in this section only, n refers to the baryon number density in the notation of Misner, Thorne and Wheeler [1973]. The tangential equation just says that the circulation components are time-independent in this computational frame, and the normal equation says that the spatial scalar density

1/2 ⊥ 1/2 1/2 1/2 ` ≡ ng u = γLg n = Ng µn = Mγ µn (3.121) is time-independent as well. (Note that µn = ρ + p, where ρ is the energy density and p the pressure.) The Taub approach to isoentropic perfect fluids fits nicely into threading point of view and allows an elegant geometrization in terms of a principal fiber bundle in a way similar to the stationary case described below. The extension of Taub’s work to the slicing point of view by Bao, Marsden and Walton [1985] might also benefit somewhat from a comparison with the present framework; the alternative fluid gauge variables of Walton are in fact related to a reference decomposition of the fluid circulation vector.

117 Chapter 4

Maxwell’s equations

4.1 Introduction

(4) [ 1 (4) α Maxwell’s equations and their immediate consequences for the electromagnetic 2-form F = 2 Fαβω ∧ β (4) ω , or simply Fαβ in index notation, can be written covariantly in many ways. In the notation of diﬀerential forms one has d(4)F [ = 0 , → (4)F [ = d(4)A, (4.1) −δ(4)F [ = 4π(4)J [ , → − δ(4)J = 0 , or in terms of covariant derivatives

∗(4) αβ (4) (4) F ;β = 0 , → Fαβ = 2 A[β;α] , (4.2) (4) αβ (4) α (4) α F ;β = 4π J , → J ;α = 0 , or in terms of ordinary derivatives and components with respect to a frame

(4) (4) (4) Fαβ,γ + Fβγ,α + Fγα,β (4) δ (4) δ (4) δ − Fδγ C αβ − FδαC βγ − FδβC γα = 0 , (4.3) (4) −1/2 (4) (4) αβ (4) αβ γ 1 α (4) βγ (4) α g ( g F ),β − F C βγ − 2 C βγ F = 4π J , and (4) (4) (4) γ Fαβ = 2 A[β,α] − Aγ C αβ , (4.4) (4) −1/2 (4) (4) α (4) α γ g ( g J ),β − J C αγ = 0 , α the latter four of which reduce to the more familiar coordinate component expressions when C βγ = 0. The equivalence of these various forms of Maxwell’s equations depends on the fact that the metric covariant derivative is symmetric, since otherwise torsion terms lead to diﬀerences between the covariant derivative form and the exterior derivative forms. The existence of a vector potential (4)A for the electromagnetic ﬁeld is of course local, and is associated with the identity d2 = 0. The “conservation law” for the current 4-vector (4)J instead follows from the identity δ2 = 0.

REWRITE FROM HERE!

4.2 Splitting the electromagnetic field 4.2.1 Congruence point of view The decomposition of the electromagnetic 2-form into a 1-form electric part and a 2-form magnetic part is the first step in splitting the electromagnetic field into an electric field and a magnetic field as observered by a

118 test observer with 4-velocity u. The electric part The spatial dual of the magnetic part of the electromagnetic 2-form deﬁnes the magnetic ﬁeld 1-form [Lichnerowicz 1967]. In index-free form one has E(u)[ = (4)F [(E) , (4.5) B(u)[ = ∗(u)F (u)[ ,F (u)[ = (4)F [(M) ,

in terms of which the 2-form has the expression

(4)F [ = u[ ∧ E(u)[ + ∗(u[ ∧ B(u)[) , (4.6) ∗ (4)F [ = −u[ ∧ B(u)[ + ∗(u[ ∧ E(u)[) .

In index form one has instead β α ∗ βα 1 αβγ E(u)α = −u Fβα ,B(u) = uβ F = η(u) Fβγ , 2 (4.7) γ δ (4) F (u)αβ = P (u) αP (u) β Fγδ . and γδ 1 γδ Fαβ = δ uγ E(u)δ + η αβuγ B(u)δ , αβ 2 (4.8) ∗ γδ 1 γδ Fαβ = −δαβuγ B(u)δ + 2 η αβuγ E(u)δ . It is sometimes useful to introduce the complex ﬁeld Z which is an eigenvector of the duality operation

(4)Z[ = (4)F [ + i ∗ (4)F [ , ˘ (4)Z[ = (4)Z[ , (4.9) (4) (4) ∗ (4) γδ(4) (4) Zαβ = Fαβ + i Fαβ , iηαβ Zγδ = Zαβ , and its spatial projections Z⊥α = E(n)α − i B(n)α = Z(n)α , (4.10) Z>α = E(m)α − i B(m)α = Z(m)α .

4.2.2 Slicing and threading points of view The congruence point of view formulas with the replacement u → o define the slicing and threading decompositions of the electromagnetic field for u = n and u = m respectively. The most familiar way of stating this result is by specifying the observer-adapted components of the electric and magnetic fields, both spatial with respect to the observer. For the slicing point of view one has

(4) a ∗(4) ⊥a ∗(4) 0a 1 abc(4) E(n)a = − F⊥a B(n) = − F = −N F = 2 η(n) Fbc , a ⊥a (4) 0a ∗(4) E(n) = F = N F ,B(n)a = F⊥a , (4.11) 1/2 a 1/2 a E(n)a = g E(n)a B(n) = g B(n) . Thus the spacetime ﬁelds are

a a E(n) = E(n) ea ,B(n) = B(n) ea , (4.12) [ a [ a E(n) = E(n)aθ ,B(n) = B(n)aθ . For the threading point of view one has

a (4) >a ∗(4) −1 ∗(4) 1 (4) bc E(m) = F B(m)a = F>a = M F0a = η(m)abc F , 2 (4.13) (4) −1(4) a ∗(4) >a E(m)a = − F>a = −M F0a B(m) = − F , with a a E(m) = E(m) a B(m) = B(m) a , (4.14) [ a [ a E(m) = E(m)aω B(m) = B(m)aω .

119 4.2.3 Observer Boost The relationship between the electric and magnetic ﬁelds measured by the two families of observers is rather simple P (m)E(n) = γL[E(m) − V~ ×m B(m)] , (4.15) P (m)B(n) = γL[B(m) + V~ ×m E(m)] . The inverse of the map P (m) restricted to the space orthogonal to n is just the natural projection P, so another way of stating this relationship is

E(n) = PγL[E(m) − V~ ×m B(m)] , (4.16) B(n) = PγL[B(m) + V~ ×m E(m)] . The simplicity of these formulas compared to the usual special relativistic formulas for the transformation of electric and magnetic ﬁelds [Anderson 1967, Misner, Thorne and Wheeler 1973] is due to the fact that the boost B(m, n) which maps n onto m diﬀers by an additional factor of γL (associated with the Lorentz contraction of lengths) along the direction of the relative velocity compared to the projection P (m) when acting on the local rest space of n. The transformation of these formulas into the more familiar ones is discussed in the appendix.

4.2.4 Reference representation (Landau-Lifshitz-Hanni) Landau and Lifshitz [1975] and several related discussions instead decompose the covariant and contravariant electromagnetic field tensor using the reference decomposition splitting, modulo scale factors, leading to a hybrid approach involving a pair of electric and magnetic fields from each of the observers m and n in the case of a timelike threading and a spacelike slicing. Landau and Lifshitz make the following component definitions (corrected for signature difference)

(4) E(LL)a = − F0a = ME(m)a , ∗(4) 1 (4) 1/2 (4) bc H(LL)a = F0a = MB(m)a = g abc F , 2 (4.17) a (4) 0a −1 a D(LL) = M F = γL E(n) , a ∗(4) 0a −1 a 1 −1/2 abc(4) B(LL) = −M F = γL B(n) = 2 γ Fbc . The reference spatial vector ﬁelds

a −1 a −1 D(LL) = D(LL) ea = γL E(n) ,B(LL) = B(LL) ea = γL B(n) (4.18)

are proportional to the ﬁelds associated with n, while the reference spatial 1-forms

[ a [ [ a [ E(LL) = E(LL)aω = ME(m) ,H(LL) = H(LL)aω = MB(m) (4.19)

are proportional to the ﬁelds associated with m. The corresponding vector ﬁelds are

a a E(LL) = E(LL) a = ME(m) ,H(LL) = H(LL) a = MB(m) . (4.20)

a b Spatial indices are shifted with the reference spatial metric γabω ⊗ ω and its reference spatial inverse as discussed in section 3.2.7 for the reference representation of the threading point of view. The interpretation of the Landau-Lifshitz fields in terms of the slicing or threading observers is valid only when the appropriate causality properties of the nonlinear reference frame hold, which is often the case in applications. However, even when neither causality property holds, one can make this decomposition, the interpretation of which one must then investigate. The initial definitions of Landau and Lifshitz, before shifting indices, lead to a decomposition of the electromagnetic 2-form into a pair of reference spatial 1-forms E(LL)[ and H(LL)[ coming from the covariant components of the electric part of the 2-form and its dual (equivalently from the covariant components of Z) and a pair of reference vector fields D(LL) and B(LL) coming from the contravariant components of the

120 electric part of the 2-form and its dual (equivalently from the contravariant components of Z), as expressed in the computational frame. On the other hand the alternative pairing (E(LL)[,B(LL)) corresponds to the electric and magnetic parts of the 2-form alone and (D(LL),H(LL)[) to its contravariant components or to the covariant components of its dual. The raising of indices on the 2-form and its dual with respect to the spacetime metric (equivalently taking Z to Z]) can then be interpreted as a linear map from the ﬁrst pair (E(LL)[,H(LL)[) to the second pair (D(LL),B(LL)) in the ﬁrst pairing

a −1 ab −1/2 abc D(LL) = M γ E(LL)b − γ MbH(LL)c , (4.21) a −1 ab −1/2 abc B(LL) = M γ H(LL)b + γ MbE(LL)c .

These linear relations between the spatial fields (E,H) and (D,B) represent, apart from normalization factors, just the Lorentz transformation from the local rest space associated with m to the one associated with n, when both m and n are timelike. Several variations of this hybrid scheme appear in the literature. All use the same definitions of the covariant spatial components of E and H, but differ on the contravariant spatial components of D and B by a scale factor, and use different spatial metrics. Closely related to the Landau-Lifshitz approach is the one of Plebanski [1960] who, following earlier work by Skrotsky [1957], uses instead the normally projected densities Ea and Ba D(P )a = (4)g1/2(4)F 0a = Ea = γ1/2D(LL)a , (4.22) B(P )a = −(4)g1/2 ∗(4)F 0a = Ba = γ1/2B(LL)a . He then uses natural contractions of contravariant and covariant indices in dealing with Maxwell’s equations and the energy-momentum tensor, thus simulating a Euclidean metric. He also introduces a different threading shift convention g(P )a = −g(LL)a = −Ma . (4.23) Hanni [1977] uses the normally projected vector fields

a a a a a D(H) = E(n) = γLD(LL) ,B(H) = B(n) = γLB(LL) , (4.24) but uses the slicing spatial metric to shift indices from their natural positions and the shift vector rather than the threading shift to describe the mixed spacetime metric components, with the notation g(H)a = −Na. This corresponds to the reference representation of the slicing point of view for the electromagnetic field. However, the spatial curl and divergence operators in both the Landau-Lifshitz and Hanni approaches are rescalings of the same natural differential operators, and natural contractions and tensor products of dual fields must in any case be used to obtain physically interesting quantities (like energy-momentum). The relationship among the various fields in this picture is given by

b 1/2 b c E(H)a = NgabD(H) + g abcN B(H) , (4.25) b 1/2 b c H(H)a = NgabB(H) − g abcN D(H) , which represents the inverse Lorentz transformation from the local rest space of n to that of m, when n and m are both timelike. By working with the densities and natural contractions in a coordinate frame, Plebanski is able to reduce the Maxwell equations to their form in flat spacetime cartesian coordinates, appropriate for use with asymptotically cartesian coordinate systems. The other approaches use a curved 3-space, but one in which the spatial metric really plays a superficial role since natural contractions replace metric contractions. In each case the Lorentz boost/projection which relates the pair (E,H) to (D,B) allows an interpretation as the constituitive relations of an equivalent material medium in motion [Tamm 1924, Sommerfeld 1964], although this analogy has never really been satisfactorily discussed. A review of this idea in connection with the Sagnac effect was given by Post [1967], while a more recent discussion appears in the text by Van Bladel [1984].

121 4.3 Splitting the 4-current

To split Maxwell’s equations one must first split the spacetime current-density vector field (4)J (not to be confused with a vector density). This 4-current field decomposes into the the charge density ρ(u) and the current density J(u) measured by the test observers with four-velocity u in the congruence point of view

(4)J = ρ(u)u + J(u) , (4)J α = ρ(u)uα + J(u)α . (4.26)

The reference, slicing and threading decompositions and the rescaled reference decomposition used by Landau and Lifshitz are instead

(4) (4) 0 (4) α J = J + J eα a = ρ(n)e⊥ + J(n) ea a (4.27) = ρ(m)e> + J(m) a −1 a = M [ρ(LL)e0 + J(LL) ea] , where the charge densities are deﬁned by

ρ(n) = (4)J ⊥ = N (4)J 0 , (4) > −1(4) ρ(m) = J = −M J0 , (4.28) (4) 0 −1 ρ(LL) = M J = γL ρ(n) , while the spatial current vectors are deﬁned by

a ab(4) a J(n) = g Jb ,J(n) = J(n) ea , a (4) a a J(m) = J ,J(m) = J(m) a , (4.29) a (4) a a J(LL) = M J ,J(LL) = J(LL) ea = MPJ(m) .

For the Landau-Lifshitz definitions it is assumed that both n and m are timelike. Note that in the Landau-Lifshitz approach, the fields associated with the observer n are scaled by the −1 inverse gamma factor γL , while those associated with m are scaled by the dual lapse M. Absorbing the factor of M into their definitions is a matter of convenience so that it does not appear explicitly in Maxwell’s equations. Apart from this factor, the spatial current density projects onto the one measured by the threading observers, while the charge density is defined so that the Lorentz contraction factor of the spatial volume is exactly compensated for when integrating against the spatial volume three-form η(m) on a time slice ρ(LL)γ1/2 = ρ(n)g1/2 , (4.30) ρ(LL)η(m) |slice = ρ(n)η(n) |slice . One cannot in general integrate the charge density orthogonally to m since a vector field does not admit a family of orthogonal hypersurfaces unless its rotation vanishes.

Charge conservation The vanishing of the divergence of the 4-current can be expressed either using the formula for the splitting of the operator δ for a 1-form or for the covariant divergence of a vector ﬁeld. In the former case only the magnetic part of the resulting 0-form exists by default. One ﬁnds

(4) (M) (4) α α −[δ J] = J ;α = [£(u)u + Θ(u)]ρ(u) + [∇(u)α + a(u)α]J(u) = 0 . (4.31)

To express charge conservation in the remaining points of view, one must manipulate these formulas or instead decompose the component formula for the divergence of a vector ﬁeld

α (4) −1/2 (4) 1/2 α (4) −1/2 (4) 1/2 0 (4) −1/2 (4) 1/2 a X ;α = g /∂α( g X ) = g /∂0( g X ) + g /∂a( g X ) , (4.32)

122 where the slashed partial notation for the computational frame derivatives stands for

β a b /∂α ≡ eα − C αβ = eα − δ αC ab . (4.33)

One easily derives the following three versions of this formula corresponding to the slicing, threading and Landau-Lifshitz rescaled reference decompositions

(4) (4) −1 −1/2 1/2 ⊥ div X = N [g £0 (g X ) + div (NP (n)X)] −1 −1/2 1/2 > a = M [γ £e0 (γ X ) + divm (MP (m)X) + MX ∂0Ma] (4.34) −1 −1/2 1/2 0 = M [γ £e0 (γ MX ) + divLL (MPX)] , where the symbol £ = £ − £ must be interpreted as acting on a spacetime or spatial scalar density. 0 e0 N~ This distinction is irrelevant for the Lie derivative £e0 , which simply produces the e0 derivative of the scalar density expression, but the shift derivative has the following form

£ F = (N a∂ + /∂ N a)F . (4.35) N~ a a In the threading expression for the divergence, the second two terms inside the square brackets may be re-arranged as (4) (divm −~g(m)·m) P (m)X = M div P (m)X. (4.36) which is proportional to the spacetime divergence of the spatial projection of the vector field. With the above definitions, one finds the results

(4) (4) −1 −1/2 1/2 0 = div J = N [g £0 (g ρ(n)) + divn NJ(n)] −1 −1/2 1/2 = M [γ £e0 (γ ρ(m)) + M(divm −~g(m)·m) J(m)] (4.37) −1 −1/2 1/2 = M [γ £e0 (γ ρ(LL)) + div(P, m) J(LL)] . Since the divergence may be expressed in terms of the exterior derivative, the slicing version of this equation corresponds exactly to the congruence point of view associated with the normal congruence.

4.4 Splitting Maxwell’s equations 4.4.1 Congruence point of view The splitting of the exterior derivative and its adjoint divergence operator for a 2-form together with the definitions of the electric and magnetic fields immediately leads to the splitting of the differential form version of Maxwell’s equations in the congruence point of view. Equivalently one can use the covariant divergence form of Maxwell’s equations and the formulas of Eq.(2.219) for the splitting of the covariant divergence of a 2-form to obtain the same result. The source equations are

(4) >β (4) > (4) (E) F ;β = 4π J :[δ F ] = div(u)E(u) − 2~ω(u) ·u B(u) = 4πρ(u) , (4.38) (4) aβ (4) a (4) (M) F ;β = 4π J :[δ F ] = [curl(u) + a(u)×u]B(u) − [£(u)u + Θ(u)]E(u) = 4πJ(u) , while the source-free equations are

∗(4) >β ∗(4) (E) F ;β = 0 : [δ F ] = −[div(u)B(u) + 2~ω(u) ·u E(u)] = 0 , (4.39) ∗(4) aβ ∗(4) (M) F ;β = 0 : [δ F ] = [curl(u) + a(u)×u]E(u) + [£(u)u + Θ(u)]B(u) = 0 .

In an observer-adapted frame, the Lie derivative term in Maxwell’s equations may be re-expressed using the following formula for a spatial vector ﬁeld X

a 1/2 −1 1/2 a −1 a b [£(u)u + Θ(u)]X = (Mh ) [h X ],0 + M C 0bX . (4.40)

123 4.4.2 Slicing and threading points of view

REWRITE REST:

124 Divergence equations The spatial divergence equations for the slicing and Landau-Lifshitz variables result from the normal projection of Maxwell’s equations

0α (4) −1/2 (4) 1/2 0α −1 h −1/2 a i −1 h −1/2 1/2 a i F ;α = g /∂α( g F ) = N g /∂a(E ) = M γ /∂a(γ D(LL) ) (4.41) = 4πJ 0 = 4πN −1ρ = 4πM −1ρ(LL) .

The expressions in square brackets are just the slicing and Landau-Lifshitz covariant spatial divergences of the vector ﬁelds E = E(n) and D(LL), respectively

g−1/2 div E = div E = 4πρ , div(P, m) D(LL) = 4πρ(LL) . (4.42)

Similarly the dual equation leads to

g−1/2 div B = div B = 0 , div(P, m) B(LL) = 0 . (4.43)

On the other hand, the projection along m leads to the spatial divergence equation in the threading point of view −1 α − M F0 ;α = divm E(m) − H~ (m) ·m B(m) = 4πρ(m) , (4.44) −1 ∗ α M F0 ;α = divm B(m) + H~ (m) ·m E(m) = 0 . Replacing the spatial divergence of the spatial vector ﬁelds in terms of the spacetime divergence leads to the Ellis form of these equations based on the congruence point of view [Ellis 1973].

Curl equations The ﬁelds E(LL) and H(LL) share a common scaling factor of the threading lapse relative to the projections orthogonal to m. These enter into the remaining equations in the Landau-Lifshitz approach which result from the projection of Maxwell’s equations orthogonal to m

aα (4) −1/2 (4) 1/2 aα 1 a bc MF ;α =M[ g /∂α( g F ) − 2 C bcF ] (4) −1/2 (4) 1/2 a0 (4) −1/2 (4) 1/2 ab 1 a bc =M[ g ∂0( g F ) + g /∂b( g F ) − 2 C bcF ] (4.45) −1/2 1/2 a −1/2 abc 1 a bdc a = − γ ∂0(γ D(LL) ) + [γ ( /∂bH(LL)c − 2 C bd H(LL)c] = 4πJ(LL) . where the expression in square brackets on the last line may be shown to be equivalent to

−1/2 abc 1 d −1/2 abc γ (∂bH(LL)c − 2 C bcH(LL)d) = γ (dH(LL))bc , (4.46) which coincides with the Landau-Lifshitz curl operator. This leads to

−1/2 1/2 −γ £e0 (γ D(LL)) + curl(P, m) H(LL) = 4πJ(LL) , (4.47)

where the Lie derivative is equivalent to the e0 derivative of the computational components. Similarly the dual equation leads to −1/2 1/2 γ £e0 (γ B(LL)) + curlLL E(LL) = 0 . (4.48) Expressing these same equations in terms of the threading variables leads instead to the equations

= −1 −1/2 1/2 −1 ] − M γ £e0 (γ E(m)) + M curlm MB(m) + (£(m)e0 M) ×m B(m) = 4πJ(m) , = (4.49) −1 −1/2 1/2 ] −1 M γ £e0 (γ B(m)) + (£(m)e0 M) ×m E(m)M curlm ME(m) = 0 or equivalently −1 −1/2 1/2 − M γ £e (γ E(m)) + (curlm −~g(m)×m) B(m) = 4πJ(m) , 0 (4.50) −1 −1/2 1/2 M γ £e0 (γ B(m)) + (curlm −~g(m)×m) E(m) = 0

125 These are the Lie form of the evolution equations For a threading spatial vector ﬁeld as in these equations, the Lie derivative coincides with the projected Lie derivative. Replacing this projected Lie derivative by the equivalent expression in terms of the spatial Fermi-Walker derivative leads to the Fermi-Walker form of these same equations given by Ellis [1973]. (?? TO DO) The slicing versions of these equations come from the tangential projection of Maxwell’s equations. These equations take the form

(4) 1/2 ab(4) α −1 −1/2 1/2 −1 4π J =g g Fb ;αea = −N g £ (g E(n)) + N curln(NB(n)) , 0 (4.51) 1/2 ab ∗(4) α −1 −1/2 1/2 −1 0 =g g F (n)b ;αea = N g £0 (g B(n)) + N curln(NE(n)) , a where the relevant Lie derivative of a spatial density Y = Y ea is defined by a £0 Y = (∂0Y )ea − £ ~ Y , N (4.52) ~ a £N~ Y = [N, Y] + (/∂aN )Y . ?? The factors of the lapse which enter into Maxwell’s equations arise essentially because of the difference in the spatial and spacetime metric duality operation on differential forms which use the volume 3-form and 4-form respectively. The threading approach leads to equations first expressed by Benvenuti [1960] which are equivalent to the congruence approach of Ellis [1973] when the latter version of Maxwell’s equations are expressed in terms of projected Lie derivatives with respect to a choice of parametrization for the congruence and in terms of the spatial divergence and curl. The slicing approach to Maxwell’s equations is equivalent to the Ellis equations for the normal congruence with similar qualifications.

4.5 Vector potential

Maxwell’s equations imply that the electromagnetic ﬁeld 2-form is closed and therefore locally representable as the exterior derivative of a potential 1-form (4)A. The splitting of this simple four-dimensional relation in the various points of view is easily performed. First the potential 1-form is split (4)A = φ(u)u[ + A(u) , (4.53) which leads to the slicing and threading decompositions, together with the reference and Landau-Lifshitz decompositions (4) ⊥ a A = −φ(n)ω + A(n)aθ > a = −φ(m)θ + A(m)aω (4.54) (4) 0 (4) a = A0ω + Aaω 0 a = −φ(LL)ω + A(LL)aω , where the scalar potentials are deﬁned by

(4) (4) 0 φ(n) = − A⊥ = N A , (4) −1(4) φ(m) = − A> = −M A0 , (4.55) (4) φ(LL) = − A0 = Mφ(m) , and the spatial vector potentials by

(4) a A(n)a = Aa ,A(n) = A(n)aθ , (4) b a A(m)a = γab A ,A(m) = A(m)aω , (4.56) (4) a A(LL)a = Aa ,A(LL) = A(LL)aω = PA. Then one splits the exterior derivative in the respective approaches. In this calculation it is convenient to a temporarily assume a spatial coordinate frame where C bc = 0 in order not to worry about the complications of which occur in formulas expressed in frames.

126 For example, the electric field in the slicing point of view is (4) −1 (4) b(4) E(n)a = − F⊥a = −N ( F0a − N Fba) (4.57) −1 (4) (4) b(4) = −N ( Aa,0 − A0,a − N Fba) , the last equation being equivalent to equation (21.103) of Misner, Thorne and Wheeler, who use the reference decomposition of the 1-form potential rather than the slicing decomposition, a poor choice initially followed by Arms [1979]. The slicing decomposition of the potential is in fact important in the Hamiltonian formulation of the coupled Einstein-Maxwell equations, as discussed by Arms, Marsden, and Moncrief [1982] for the more (4) general Einstein-Yang-Mills equations. Re-expressing A0 in terms of the normal component and using the Lie derivative formula in a coordinate frame referring to components along ωa b b b b b £ ~ Aa = Aa,bN + AbN ,a = Aa,bN + (AbN ),a − Ab,aN N (4.58) (4) b b = FbaN + (AbN ),a , one finds −1 −1 E(n)a = −N [−NA⊥],a − N [£e0 − £N~ ]Aa −1 (4.59) = −N [−NA⊥],a − £nAa , or E(n)[ = −N −1(grad[Nφ])[ − [P (n)£]nA (4.60) and E(n) = −N −1 grad[Nφ] − ([P (n)£]nA)] (4.61) for the electric 1-form field and vector field respectively. The calculation of the magnetic field is much simpler, giving easily B(n) = curl A] . (4.62) Note that the above Lie derivative coordinate frame formula is just a special case of the expression

£X σ = d(X σ) + X dσ (4.63) for the Lie derivative of a differential form σ, where X σ refers to contraction of the differential form on the left by the vector field X, also written iX σ in an alternative notation. A similar calculation for the threading point of view leads to the equations = −1 ] E(m) = −M gradm[Mφ(m)] − φ(m)£e0 M − (£(m)mA(m)) , ] = (~g(m) − gradm)φ(m) − (£(m)mA(m)) , (4.64) ] B(m) = curlm A(m) + φ(m)H~ (m) , while the Landau-Lifshitz approach is the simplest [ E(LL) = −dLLφ(LL) − £e A(LL) , 0 (4.65) B(LL) = curlLL A(LL) .

4.6 Wave equations

Expressing Maxwell’s equations in terms of the vector potential leads to second order equations. Using the diﬀerential form approach one immediately ﬁnds (4) (4) (4) δd A = ∆(dR) − dδA = 4π J. (4.66) Imposing the Lorentz gauge condition δ(4)A = 0 leads to the wave equation (4) (4) = ∆(dR)A = 4π J (4.67) involving the deRham Laplacian. The equations for the 2-form itself also lead to a wave equation involving the deRham Laplacian with the exterior derivative of the 4-current 1-form as a source (4) [ (4) [ (4) [ (4) (4) [ δ F = −4π J , d F = 0 → ∆(dR) F = −4πd J . (4.68)

127 4.7 Computational 3-space representations

Each of the above decompositions of the electromagnetic ﬁeld and its ﬁeld equations may be explicitly represented on the appropriate computational 3-space. ?? TO DO

4.8 Lines of force

A very useful way of representing electric and magnetic fields in flat spacetime is in terms of their lines of force, namely the integral curves of these vector fields on space at a given moment of time. Similar representations hold on the various computational 3-spaces for a curved spacetime. ?? TO DO

128 Chapter 5

Stationary spacetimes

5.1 Stationary nonlinear reference frame

For stationary spacetimes the parametrized nonlinear reference frame can be adapted to the isometry group by choosing the parametrized threading to coincide with the flow of a timelike Killing vector field e0 associated with the stationarity. It is natural to call this a “stationary nonlinear reference frame”. One is still free to choose some initial hypersurface for the slicing as well as consider affine transformations of the threading parametrization. For such a nonlinear reference frame, the tensor algebra of stationary spatial tensor fields on spacetime, including the covariant derivative, spatial Lie derivative, and exterior derivative operators on such fields, is isomorphic to the tensor algebra of fields on the computational space with its time-independent Riemannian metric and its associated symmetric covariant derivative, Lie derivative, and exterior derivative. It is exactly this case which motivated Landau and Lifshitz to use the reference representation of the computational space geometry for the threading point of view.

For a stationary nonlinear reference frame, the Lie derivatives £e0 of all stationary ﬁelds vanish, so the spatial Lie derivatives £(m)m of all stationary spatial ﬁelds also vanish. The expansion tensor θ(m) of the threading congruence is therefore zero

−1 θ(m) = 0 , θ(n)ab = −N N(a|b) , (5.1)

leading to a direct correspondence between the components of the spatial part of the spatial connection in the threading point of view and the components of the connection of the projected spatial metric on each slice or on the threading computational 3-space. The Lie and symmetry obeying spatial curvature tensors in this point of view also coincide and in turn correspond directly to the Riemann tensor of the quotient spacetime or of the time-independent Riemannian geometry of the computational 3-space. REWRITE: The spatial covariant derivative of a stationary spatial field when projected to the quotient space agrees with the covariant derivative with respect to the spatial metric on the quotient space. One can also raise and lower indices on spatial quantities without generating additional Lie derivative terms in the total spatially covariant derivative, and the Lie derivative terms drop out in the Fermi-Walker derivative and in the transformation formulas for the gravitoelectric and gravitomagnetic fields. The latter formulas for the gravitoelectric and gravitomagnetic vector fields are then almost conformally related to the Lorentz boost formulas for the electric and magnetic parts of a 2-form, except for the symmetric part of the gravitomagnetic tensor in the slicing point of view. 1 The 1 -tensor-valued connection 1-form with respect to the threading projected computational frame, evaluated on m (4) [ (4) α β [ = 1 [ − ωωω(m)(m) = Γα>βω ⊗ ω = m ∧ g (m) + 2 H(m) (5.2) defines a 2-form whose electric part is the gravitoelectric 1-form and whose magnetic part is the spatial dual of half the gravitomagnetic vector field in the threading point of view. In the slicing point of view the

129 connection 1-form evaluated on n has a symmetric part like the gravitomagnetic tenosr but is not exactly the analogous expression in terms of these two ﬁelds. However, the transformation between the two sets of connection components has the form

−1 ωωω(n)(n) = γL ωωω(m)(m) + ··· (5.3) which explains why it so closely resembles a Lorentz transformation with an overall gamma factor missing as discussed in Appendix B. ??MORE HERE. Maxwell’s equations simplify since the presence of the spatial metric determinant in the e0 Lie derivative term is eliminated. ??MORE HERE Since the lapse M is a stationary function, the proper time parameters along each threading curve are affinely related to the coordinate time parametrization. Again due to stationarity, the dual lapse alone serves as the potential for the dual gravitoelectric field g(m)[ = −d(m)(ln M) = −d(ln M) (5.4) The threading gravitoelectric field measures the spatial variation in the affine relationship between the coordinate parametrization (equivalently Killing parametrization) of the threading congruence and the proper time parametrization, vanishing only when the two are globally affinely related, the case of a constant lapse function.

5.2 Synchronization gap and Sagnac eﬀect

= One also has a nice interpretation of the threading shift M in the stationary case. Its integral along a curve in a slice represents the amount of simultaneous translation along the threading congruence necessary to maintain the orthogonality to the threading congruence of the tangent of the resulting curve. This represents a local synchronization of clocks along the new curve [Landau and Lifshitz 1975]. The integral of the dual shift around a closed curve in the slice yields the “synchronization defect” in the terminology of Bazański [1988] which is in turn closely related to the Sagnac effect [Post 1967, Ashtekar and Magnon 1975, Van Bladel 1984, Henrickson and Nelson 1985] involving null rather than spacelike curves. This phenomenon in the stationary case is best described by interpreting the spacetime region (4)M together with the parametrized nonlinear reference frame as a (local) line bundle with group Flow(e0) and (4) base space equal to the quotient space M/ Flow(e0), for which e0 is a fundamental vector field. This line bundle has a natural connection whose horizontal subspaces are just the local rest spaces of m. The = associated connection 1-form is just θ0 = ω0 − M and the condition that it transform under the (trivial) 0 a adjoint representation of the group is just £e0 θ = −(e0Ma)ω = 0 which is satisfied by the condition of stationarity. A compatible slicing is just a particular local trivialization. The spacetime metric is just a fiber invariant Kaluza-Klein metric on this bundle, with the dual lapse describing the metric along the fiber direction or vertical subspace and the spatial metric the metric on the (4) (4) horizontal subspace. If s : M/ Flow(e0) → M is a section, i.e., a valid compatible slice hypersurface, then pulling down the connection 1-form s∗θ0to the quotient space yields a 1-form which is the projection of the sign reversed dual shift 1-form associated with this hypersurface. For the section st which maps up = to the slice Σt, this is just the projection of −M. Given any curve in a slice starting at a particular point, one can project the curve down to the quotient space and then lift it back up to the spacetime horizontally to obtain a new curve emanating from the same point as the original curve. Such a curve describes a synchronization of clocks carried by the observers whose worldlines intersect the original (and final) curve. The integrability of this synchronization operation depends on the vanishing of the curvature of the connection which is just the projected exterior derivative of the connection 1-form = = Dθ0 = P (m)dθ0 = −d(m)M = −dM (5.5) which is proportional to the rotation tensor of m, whose spatial dual leads to the gravitomagnetic vector field in the threading point of view. (The proportionality factor of the dual lapse function leads to a field

130 Figure 5.1: TO DO

Figure 5.2: TO DO which is invariant under reparametrization of the threading congruence.) A flat connection leads to a family of integral submanifolds of the horizontal distribution which correspond to the elements of the new slicing orthogonal to the threading congruence. This is the static subcase. The parallel transport on the quotient space associated with this bundle connection transports coordinate clock time around in “space” in a way which represents local synchronization. Parallel transport around a closed curve leads to a translation along the fiber by an amount which represents the synchronization defect of the curve. One can always introduce another uniquely defined translation along the fiber above the given curve in the quotient space by instead following the future-directed null curve, leading to a “future-null lift” of the curve. This represents a null synchronization rather than a spacelike synchronization of time parameters along the fibers.

Suppose one has a parametrized curve segment {C(λ) λ ∈ [0, λ0]} in the quotient space. This represents a sheaf of threading curves in the spacetime or a “Sagnac tube” [Ashtekar and Magnon 1975]. From any point c(0) lying above the initial point C(0) in the quotient space (i.e., lying on the threading curve C(0)), one can construct three parametrized curves through c(0) in this sheaf which project down to C at corresponding parameter values. The first curve c is just the representative of C lying in the slice containing c(0). The second cS is the horizontal lift of C through c(0) consisting of those points on the sheaf which are spatially synchronized relative to the threading. The third curve cN is is the future-directed null representative of C, consisting of those points of the sheaf related by a null synchronization of the threading time parameters. (See Figure 5.1.) SD Define t (λ) to be the amount of translation of cS(λ) along the fiber from c(λ), i.e., the difference in SE value of the time function from c(λ) to cS(λ). Similarly let t (λ) be the corresponding quantity for CN (λ). When C is a simple closed curve with C(λ0) = C(0), then c is also a simple closed curve. For notational convenience, let a minus sign subscript denote denote the same curves with the reverse orientation and a plus sign subscript the original curves. This leads to new parametrized curves cS− and cN− defined in the SD SE same way relative to the curve C−, and corresponding values t± and t± for the translation intervals for these curves and the originals after one revolution in the quotient space. The synchronization defect for the simple closed curve C, or equivalently for c, is just the quantity

SD SD SD ∆t = t+ = −t− (5.6)

and the Sagnac effect is SE SE SE SD ∆t = t+ − t− = 2∆t . (5.7) The Sagnac effect is just a slightly different geometric realization of the synchronization effect for a stationary spacetime in a stationary nonlinear reference frame. Figure 5.2 illustrates this situation. To derive this result, let q be the unit spatial vector field relative to the threading at each point of the 0 above sheaf which is obtained by horizontally lifting the tangent C (λ) and normalizing it. Then k± = m ± q specify the two null directions tangent to the sheaf at each point. The spatial synchronization curve cS is the integral curve of q through c(0), while the future-null synchronization curve cN is the integral curve of k+ through c(0), each suitably reparametrized. > 0 The first curve cS is an integral curve of the distribution determined by θ or equivalently by θ = = −1 [ −1 [ −M m = dt − M; the second curve cN is an integral curve of the distribution determined by −M k = = dt−(M +M −1q[). (Here integral curve is used in the sense of a partial integral submanifold of the distribution 0 0 0 determined by a 1-form.) Apart from Lie translation along e0, the three tangents c , cS, and cN all have the

131 = same spatial projection so the integral of M along the three curves is the same

Z = Z = Z = M = M = M c cS cN Z (5.8) SD = dt = t (λ0) cS = 0 and equals the change in the time coordinate along the spatial synchronization curve cS since (dt−M)(cS) = 0. When c is a simple closed curve, this is just the synchronization defect

I = SD SD ∆t = t+ = M, (5.9) c

while the integral on cS− gives the opposite result due to the reverse orientation. On the other hand the null circuit times are Z = SE −1 [ t± = (M ± M q ) . (5.10) cN± 0 0 0 0 Again, since apart from Lie translation along e0 the tangents cN+, −cN−, cS+, and c all have the same spatial projection by construction, the integrals Z Z Z Z M −1q[ = − M −1q[ = M −1q[ = M −1ds (5.11) cN+ cN− cS+ cS+ are all equal and give the integral of the reciprocal of the dual lapse with respect to the arclength of the spacelike spatial synchronization curve cS+. For the same reason the integrals

Z = Z = Z = M = − M = M cN+ cN− cS+ (5.12) I = = M = ∆tSD c are all equal to the Sagnac defect, so the average null circuit time is Z SE 1 SE SE −1 tavg = 2 (t+ + t− ) = M ds , (5.13) cS+ while the diﬀerence is the Sagnac eﬀect

SE SE SE SD ∆t = t+ − t− = 2∆t . (5.14) Using Stoke’s theorem the Sagnac eﬀect can be re-expressed as a surface integral over any simple 2-surface Ξ whose boundary is the closed curve c

I = Z = Z = Z 1 SE −1 ∗(m) [ 2 ∆t = M = dM = d(m)M = M [ H(m) ] , c Ξ Ξ Ξ Z Z (5.15) −1 −1 = M H~ (m) η(m) = M H~ (m) ·m dS~m . Ξ Ξ This formula is the threading point of view expression for the four-dimensional integral of Ashtekar and Magnon [1975]. The final form of the integral uses the classical notation dS~m for the projection orthogonal = to m of the vector differential of surface area on the 2-surface Ξ. Note that the spacetime differential of M equals its spatial differential since it is a stationary field. Thus one can interpret the Sagnac effect as the quantity of projected flux contained in the closed curve of the quotient of the dual gravitomagnetic vector field by the dual lapse function. The term “projected flux” or equivalently “projected surface integral” is necessary to distinguish the classical notation dS~m from

132 the diﬀerential of surface area dS~ of the surface Ξ. For example, in the case of a spacelike slicing, if Ξ is an interior surface of c lying in the slice, then

1 a b c H~ (m) ·m dS~m|Ξ = H(m) η(m)|Ξ = H(m) η(m)abc ω ∧ ω |Ξ 2 (5.16) 1 a b c ~ ~ = 2 γLH(m) ηabc θ ∧ θ |Ξ = [PH(m)] η|Ξ = H(m) ·n dSn|Ξ shows that an additional gamma factor becomes explicit when rewritten in terms of the differential of surface area dS~n on the surface Ξ. The notation |Ξ indicates the restriction of a differential form to the surface Ξ. This same integral may be expressed by projecting everything down to the quotient space, where the projection of the threading spatial metric P (m)[ provides a Riemannian metric. One obtains an ordinary surface integral of the projected dual gravitomagnetic vector field divided by the projected dual lapse function over the projection of the surface Ξ into the quotient space, or equivalently as a line integral of the projected dual shift over the closed curve C which bounds Ξ, in each case using the projected threading spatial metric on the quotient space I = Z 1 SE −1 ~ ~ 2 ∆t = M = (Ξ)M H(m) · dS. (5.17) C π In the case that the slicing is spacelike and Ξ lies entirely in the slice of the closed curve c, this may also be transformed into an integral in terms of the slicing spatial gravitational fields Z SE −1 2 h i ∆t = N γL PH~ (m) η Ξ Z (5.18) −1 4 h ↔ i = N γL H~ + (H~ ·n ~v)~v + 2~v ×n (~g + (SYM H) ·n ~v) η , Ξ since the restriction of η(m) to Ξ is the same as that of γLη. It is also useful to introduce the relative Sagnac effect in order to put the magnitude of the Sagnac effect into perspective 1 SE R −1 ~ ~ 2 ∆t Ξ M H(m) ·m dSm null = R −1 . (5.19) ∆t c M ds This quantity is just the “corrected” projected surface integral of the gravitomagnetic field divided by the “corrected” arclength of the bounding curve, where “corrected” refers to the additional factor of M −1 which converts the proper time along the congruence to coordinate time. Each of the time intervals discussed above is an interval of coordinate time. However, as noted above, along a given threading curve, the dual lapse M is constant so the proper time interval along a given threading curve C which corresponds to the Sagnac effect or the null transit times can be obtained just by multiplication of the latter interval by the value of the dual lapse on that threading curve.

5.3 Rotating spatial Cartesian coordinates in ﬂat spacetime

No discussion of spatial gravitational forces would be complete without relating them to the familiar notions of centrifugal and Coriolis forces which arise in everyday experience. It is these forces which have been generalized by the splitting discussion of the acceleration equation, so it is important to understand exactly how they fit into the general picture as a special case. The point of view in which they are discussed in classical mechanics is neither the slicing or threading point of view but rather the reference point of view. Consider adapted coordinates {t, xa} for a nonlinear reference frame in an arbitrary spacetime. The reference decomposition of the geodesic equation considered as a (contravariant) vector field equation is just its coordinate decomposition. The calculation of the spatial reference projection of that equation may be found in Forward’s discussion [1961] of linearized general relativity based on Møller’s first edition discussion [1952] of spatial gravitational forces using the coordinate time as the parameter for the geodesic, but modified to use the spatial coordinate decomposition of the geodesic equation rather than the threading decomposition. Linearization allows one to sidestep the some of the complications which arise in the general case, but the logarithmic time derivative of the coordinate gamma factor Γ defined by Eq.(??) leads to an additional

133 acceleration-dependent term proportional to the velocity. (Compare with Eq.(6.25) of Misner, Thorne and Wheeler [1973].) However, in ﬂat spacetime with a coordinate system {t, xa} consisting of a rigidly rotating system of orthonormal Cartesian spatial coordinates and the usual time coordinate t of the related nonrotating inertial coordinates, the complications wash out and lead to the usual formulas

t¨= 0 , (5.20) ¨ ˙ ~ ~x = ~g(ref) + ~x × H(ref) .

Here the dot is the usual time coordinate derivative and the usual vector notation is employed. The spatial forces are ~ ~ ~g(ref) = −Ω × (Ω × ~x) , (5.21) ~ ~ H(ref) = 2Ω , where Ω~ is the constant vector describing the angular velocity of the rotating system, identified with a constant vector field. The reference gravitoelectric field is exactly the centrifugal force (per unit mass) and the reference gravitomagnetic field vector is just twice the global rotation of the coordinate system, leading to the Coriolis force (per unit mass). To see how these relate to the slicing and threading fields, one must evaluate them for the nonlinear reference frame for which these coordinates are adapted. The slicing is a flat slicing of Minkowski spacetime but the threading is an inhomogeneous tilted threading relative to that slicing, and it is timelike only within a certain cylinder in space called the light cylinder inside of which the velocity of rotation is less than the speed of light. Thus the slicing point of view holds everywhere, while the threading point of view is limited to the interior of the light cylinder. Since the spatial metric is Euclidean in the slicing point of view, one can use the customary vector notation unambiguously in expressing the line element (instead of metric to allow the dot product notation)

ds2 = −dt2 + (d~x + Ω~ × ~xdt) · (d~x + Ω~ × ~xdt) , (5.22) N = 1 , N~ = Ω~ × ~x, gab = δab .

The differentials d~x + Ω~ × ~xdt are the rotated differentials of the nonrotating Cartesian coordinates. The slicing spatial gravitational fields in this notation are ↔ ~ ~ ~ ~g(n) = 0 = SYM H(n) , H(n) = 2Ω = H(ref) (5.23) and the expansion tensor vanishes θ(n) = 0. The Lorentz boost parameters are

2 −1/2 ~v = N~ , γL = (1 − v ) , (5.24) leading to the threading quantities

= −1 2 a b −2 M = γL , M = γL δabv dx , γab = δab + γL MaMb , (5.25) 2 b a 2 a g(m)a = γL δabg(m)(ref) ,H(m) = γL H(ref) .

Thus it is the gravitoelectric force in the threading point of view, apart from a time reparametrization and a projection, which corresponds to the centrifugal force. The slicing gravitomagnetic vector field directly equals the reference gravitomagnetic vector field, which in turn is twice the global angular velocity of the coordinate system, and leads to the Coriolis force. The threading gravitomagnetic vector field differs from twice the constant global rotation vector by the time reparametrization and a projection in such a way that it equals the inhomogeneous local rotation of the threading congruence. Since the slicing spatial velocity differs from the reference spatial velocity by the velocity of the threading curves ν(n)a = dxa/dt + N a , (5.26)

134 the slicing second-order acceleration equation takes the form

D(U, P, n)2xa/dt2 = d2xa/dt2 ~ a a b = [~ν(n) ×n H(n)] + N |bN (5.27) ~ a a b a b = [d~x/dt ×n H(n)] + 2N[b| ]N + N |bN ~ a a b = [d~x/dt ×n H(n)] + Nb| N . The resultant shift derivative term corresponds to the reference gravitoelectric ﬁeld, i.e., the centrifugal force term. For the threading point of view where there is no relative velocity term

ν(m)a = M −1dxa/dt , (5.28) the second-order acceleration equation is instead

2 a 2 2 a 2 a b c D(U, m) x /dt = d x /dt + Γ(m) bcdx /dt dx /dt (5.29) 2 a = M [−2~g(m) ·m ~ν(m) ~ν(m) + ~g(m) + ~ν(m) ×m H~ (m)] . Here the quadratic term in the spatial velocity involving the gravitoelectric ﬁeld cancels the connection term, leaving a direct correspondence between the threading and reference spatial gravitational forces. The spatial curvature in the threading point of view is easily evaluated from the splitting of the vanishing spacetime curvature tensor. Since the expansion tensor is zero, one has in the projected computational frame

(4) ab R cd = 0 → (5.30) ab ab a b ab R(lie)(m) cd = R(sym)(m) cd = −2ω(m) [cω(m) d] − 2ω(m) ω(m)cd . The double-sided spatial dual gives the equivalent Einstein tensor

ab ab a b 4 a b G(lie)(m) = G(sym)(m) = 3ω(m) ω(m) = 3γL Ω Ω . (5.31)

5.4 Stationary axially-symmetric case: rotating Minkowski, G¨odel and Kerr spacetimes

A very interesting class of stationary spacetimes are the stationary axisymmetric spacetimes, which are characterized by the existence of a pair of commuting Killing vector fields ξ(t) and ξ(φ), respectively timelike and spacelike in some region of the spacetime, associated with the isometric action in this region of the abelian group R×SO(2,R) of time translations and rotations around an axis of symmetry. The axial Killing vector field ξ(φ) is uniquely defined by the condition that its one-parameter subgroup have closed spacelike orbits diffeomorphic to S1 with periodicity of 2π. When spatial infinity exists, at least in the direction orthogonal to the axis of symmetry, one may define ξ(t) as the unique linearly independent Killing vector field which at spatial infinity is timelike, has unit length and is orthogonal to ξ(φ). In this same region of spacetime there exists a privileged class of observers, the so called “stationary observers”, which are characterized by timelike worldlines contained in a single orbit of the group and invariant under the action of the group. A family of stationary observers which covers the stationary region of the spacetime obviously is a preferred candidate for a threading congruence. Such a family has a timelike unit four-velocity u which is tangent to the orbits of the symmetry group and commutes with its generators. Such a four-velocity u is characterized by a single angular velocity function Ω which is invariant under the group (a stationary function)

α −1/2 u = |ξ ξ(Ω)α| ξ(Ω) , (Ω) (5.32) ξ(Ω) = ξ(t) + Ωξ(φ) , ξ(t)Ω = ξ(φ)Ω = 0 .

The condition that u be timelike restricts the angular velocity to lie between two values Ωmin and Ωmax.

135 The one-parameter family of families of stationary observers for which Ω is a constant have worldlines which are orbits of a one-parameter subgroup of the symmetry group generated by the Killing vector field ξ(Ω). Such observers are called Killing observers and are associated with a stationary parametrized nonlinear reference frame with threading tangent e0 = ξ(Ω). The components of the metric will be stationary functions in the associated computational frame, leading to stationary spatial fields in the slicing and threading splittings. However, just as in flat spacetime in rotating cylindrical coordinates, there will always be an orbit of some maximum circumference beyond which u will be spacelike and the threading point of view breaks down, unless Ω = 0. These special observers have zero rotation at spatial infinity and are commonly referred to as the static observers. This is the usual threading associated with the Boyer-Lindquist [1967] coordinates for a black hole, and the associated threading point of view is valid only outside the ergosphere, where the value Ω = 0 lies in the allowed range [Ωmin, Ωmax]. As one moves inward into the ergosphere, the threading switches from being timelike to spacelike through the outer boundary (the so-called static limit) where it is null. The non-Killing stationary observers have differential rotation relative to the Killing observers. If one takes a threading point of view with e0 = ξ(Ω) in this case, the metric becomes explicitly time-dependent, i.e., one obtains nonstationary spatial fields from the associated splittings of the spacetime metric. Of all families of stationary observers, a rotation-free congruence, if it exists, provides a preferred slicing of the stationary region by the family of orthogonal hypersurfaces it defines. Such a slicing is stationary, i.e., taken into itself under the isometry group. The most convenient parametrized nonlinear reference frame is therefore the stationary one with a Killing threading associated with the choice e0 = ξ(t) and such a preferred slicing, leading to stationary spatial fields in both points of view. For the orthogonally transitive case, where the group orbits admit a family of orthogonal 2-surfaces, Carter [1969] has shown that there exists a unique rotation-free stationary family of observers with u orthogonal to the axial Killing vector field ξ(φ). These are called the “locally nonrotating observers” [Bardeen 1970, Misner, Thorne and Wheeler 1973] or the “zero angular momentum observers” (ZAMO’s) [Thorne et al 1986, Thorne and Macdonald 1982]. This provides such a spacetime with a preferred slicing, while the static observers (the Killing observers which are nonrotating at spatial infinity) provide a preferred threading. Black hole spacetimes have an additional preferred threading which is adapted to the geometric structure of the spacetime in a more subtle way. This is the Carter threading associated with the canonical orthonormal frame defined by the separability properties of various geometric equations on the spacetime [Carter 1968, 1973]. This orthonormal frame is the one in which the Boyer-Lindquist representation of the metric is usually given [Misner, Thorne and Wheeler 1973], although it is not usually explicitly referred to. For the charged black hole, this family of observers (the “Carter observers” with unit velocity u(carter)) measures parallel electric and magnetic fields, for example, as was used by Damour and Ruffini [1975] in quantum electrodynamical considerations for black holes. In any case, the electric and magnetic parts of the Weyl curvature associated with this splitting are also aligned, i.e., proportional, a simple statement that is difficult to find in the literature in spite of all the articles which have used an orthonormal frame adapted to the Carter observers. This is easily derived using the more transparent formulas of Marck [1983] which also corrects some sign errors in the Carter expressions. All of these threadings and the single slicing are well behaved and have the appropriate causal behavior for both the slicing and threading points of view outside of the ergosphere. The Boyer-Lindquist [1967] coordinates {t, r, θ, φ} are adapted coordinates for the nonlinear reference frame consisting of this preferred slicing and the static threading associated with ξ(t). The field m associated with the static observers is boosted with respect to n in the direction of the axial orbits opposing the corotating direction (negative angular momentum), while the field u(carter) is boosted in the corotating direction (positive angular momentum). This latter direction becomes null leaving the ergosphere on its inner boundary, the event horizon, where the radial direction also becomes null, switching the causal nature of the slicing. Thus the slicing point of view for this nonlinear reference frame is valid only outside the horizon, while the threading point of view is valid only outside the ergosphere. The two standard choices of metric variables for an orthogonally transitive stationary axisymmetric spacetime, as given by Demiansky [1985] for example, correspond to the slicing and threading decompositions

136 respectively

(4)g = −fdt˜ ⊗ dt + f˜−1ρ2(dφ − ωdt˜ ) ⊗ (dφ − ωdt˜ ) + e2µ(dρ ⊗ dρ + dz ⊗ dz) , = −f(dt − ωdφ) ⊗ (dt − ωdφ) + f −1[ρ2(dφ ⊗ dφ) + e2γ (dρ ⊗ dρ + dz ⊗ dz) , (5.33) 2 ˜ ˜−1 2 2µ 2µ φ N = f , (gab) = diag(f ρ , e , e ) ,N = −ω˜ , 2 −1 2 2γ 2γ M = f , (γab) = f diag(ρ , e , e ) ,Mφ = ω .

The threading variables prove more useful in obtaining exact solutions for these models. The simplest example of this class is flat spacetime with a nonlinear reference frame associated with rotating cylindrical coordinates (t, ρ, φ, z) = (t, ρ, φ − Ωt, z), where the barred coordinates are standard cylindrical coordinates, and the unbarred angular coordinate plane rotates about the z axis with constant angular velocity Ω ≥ 0 in the counterclockwise direction. The Killing vector fields described above are ξ(t) = ∂/∂t, ξ(φ) = ∂/∂φ and ξ(Ω) = ∂/∂t. In addition, the coordinates are adapted to the Killing vector field ∂/∂z which extends the axial symmetry of the coordinate computational frame to cylindrical symmetry. In the slicing point of view one has the decomposition of the metric

[ 2 N = 1 , N~ = Ω∂φ ,P (n) = dρ ⊗ dρ + ρ dφ ⊗ dφ + dz ⊗ dz , (5.34) where 2 −1/2 v = ρΩ , γL = (1 − v ) , dφ = dφ + Ωdt . (5.35) The spatial metric on the time slices is flat, the gravitoelectric field is zero and the gravitomagnetic field is spacetime-homogeneous 1 ~ ↔ ~g = 0 , 2 H = Ω∂z , SYM H = 0 . (5.36) In the threading point of view valid up to the the radial coordinate value ρ = Ω−1, one has instead a spatially inhomogeneous splitting

= −1 2 [ 2 M = γL , M = (γLρ) Ωdφ , P (m) = dρ ⊗ dρ + (γLρ) dφ ⊗ dφ + dz ⊗ dz , (5.37) and the spatially inhomogeneous spatial gravitational ﬁelds

[ 2 1 ~ 2 ~g(m) = (γLΩ) ρdρ , 2 H(m) = γL Ω∂z . (5.38) The Sagnac eﬀect for a φ-coordinate circle traced out in the co-rotating direction of increasing φ is

I = 1 SE 2 2 2 ∆t = M = 2(πρ )γL Ω , (5.39) S1 where the integral over S1 is shorthand for the integral over φ in the positive direction. This relativistic result was given implicitly by Landau and Lifshitz [1975]. The average time for a null circuit of the Sagnac tube is I avg −1 2 ∆t = M ds = 2πργL Ω (5.40) S1 so the relative Sagnac effect is 1 ∆tSE 2 = ρΩ = v , (5.41) ∆tavg which is the relative speed at a given radius. In this example, the slicing gravitomagnetic vector field is just the constant vector field of the global angular velocity of the associated rotating cartesian coordinate system, while the threading gravitomagnetic vector field is the inhomogeneous local rotation of the inhomogeneous congruence of time coordinate lines which becomes singular at the radius at which the rotational velocity equals the speed of light, beyond which the time lines are spacelike and the threading point of view breaks down. On the other hand the spacetime homogeneous cosmological models [Ryan and Shepley 1975] expressed in a stationary reference frame have spacetime homogeneous threading congruences and set up spacetime homogeneous spatial gravitational fields

137 in the threading point of view. The Gödelsolution [Gödel1949] is the most famous such cosmological model, a solution of the Einstein equations with dust and nonzero cosmological constant, but the stationary reference frame in which it is usually presented has a timelike homogeneous slicing, so only the threading point of view can be considered. The comoving fluid nonlinear reference frame is geodesic leading to zero gravitoelectric field, leaving only a spacetime homogeneous gravitomagnetic field associated with the local rotation of the fluid flow lines. Jacobs and Wilkins [1988] discuss the Gödelsolution in this light using a linear analogy with electromagnetism. When the stationary nonlinear reference frame of a spacetime homogeneous cosmological model has a spatially homogeneous spacelike slicing, the same is true of the slicing spatial gravitational fields, namely they are also spacetime homogeneous. Ozsváthand Schücking [1969] describe a class of such dust models with S3 spatial slices, using not only a spatially homogeneous stationary reference frame but introducing also the normal and comoving perfect fluid reference frames for the same slicing. The latter two reference frames are spatially homogeneous but nonstationary and set up time-dependent spatially homogeneous spatial gravitational fields, just like the nonstationary spatially homogeneous cosmological models. This is analogous to the nonstationary reference frames associated with non-Killing stationary observers in the stationary axially symmetric case.

138 Chapter 6

Perturbation problems

6.1 Linearization about an orthogonal nonlinear reference frame 6.2 Post-Newtonian approximation 6.3 The Newtonian limit 6.4 Friedmann-Robertson-Walker Perturbations

139 Appendix A

Formulas from diﬀerential geometry

This summary of formulas from differential geometry continues the remarks of the introduction. It is assumed that the reader has some familiarity with many of these concepts, at least in coordinate notation, or at the level of the text Gravitation by Misner, Thorne and Wheeler [1973]. The purpose of this appendix is to have a reference guide for the particular choice of conventions used here and the specific formulas for the frame components of many geometric objects whose coordinate frame representation is more widely known. More details may be found by consulting Spivak [1965], Hicks [1971], Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick [1982], Schouten [1954] or other well known texts on differential geometry.

A.1 Manifold α Let M be an n-dimensional manifold, with local coordinates {x }α=1,...,n when needed. These local coordinates deﬁne a local coordinate frame {∂/∂xα} with dual frame {dxα}.

A.2 Frame and dual frame α It is often more convenient to work with a (local) noncoordinate frame {eα} with dual frame {ω } (sometimes called co-frame) satisfying the duality condition

α α ω (eβ) = δ β . (A.1) If one expresses these ﬁelds in local coordinates

β β α α β eα = e α∂/∂x , ω = ω βdx , (A.2)

α α then the matrices of components (e β) and (ω β) are inverse matrices. The differential properties of the frame are characterized by the structure functions of the frame, which define the Lie brackets of the frame vector fields and the exterior derivatives of the dual frame 1-forms

γ α 1 α β γ [eα, eβ] = C αβeγ , dω = − 2 C βγ ω ⊗ ω . (A.3) p A q -tensor ﬁeld can be expressed in terms of the frame as

α... β α... α S = S β...eα ⊗ · · · ⊗ ω ⊗ · · · ,S β... = S(ω , . . . , eβ,...) . (A.4) It is convenient to adopt the notation

ωα1...αp = ωα1 ∧ · · · ∧ ωαp (A.5)

0 to more compactly represent a p-form, or antisymmetric p -tensor 1 S = S ωα1...αp = S ωα1...αp , (A.6) p! α1...αp |α1...αp|

1 where the vertical bar notation indicates a sum only over strictly increasing sequences of indices. A similar p notation eα1...αp will be used to express p-vector fields, namely antisymmetric 0 -tensor fields. A tensor field whose components are antisymmetric in a subset of p covariant indices may be interpreted as a tensor-valued p-form

S = Sα... e ⊗ · · · ⊗ ωβ ⊗ · · · ⊗ ωγ1 ⊗ · · · ⊗ ωγp β...γ1...γp α 1 α... β γ1...γp = S eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ ω (A.7) p! β...γ1...γp β α... = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ S β... . where 1 Sα... = Sα... ωγ1...γp (A.8) β... p! β...γ1...γp are the components of the tensor-valued arguments of the tensor-valued p-form. In most of the literature this collection of p-forms is often used in place of the tensor-valued p-form it represents. One may take the exterior derivative of this set of p-forms, which in turn deﬁnes the exterior derivative of the tensor-valued p-form β α... dS = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ dS β... . (A.9) This clearly depends on the choice of frame.

A.3 Linear transformations A function whose value at each point is a linear transformation of the tangent space may be identified with 1 a 1 -tensor field in a natural way, acting on vector fields by right contraction

α β B = B βeα ⊗ ω , (A.10) α β B X = B βX eβ .

α n α The matrix of components (B β) then acts on the R -vector of components (X ) by matrix multiplication. 1 Similarly right contraction between 1 -tensor ﬁelds corresponds to matrix multiplication of their square matrices of components α α γ [A B] β = A γ B β . (A.11) It is convenient to use the abbreviation A2 = A A. (A.12) The identity transformation is represented by the unit tensor or Kronecker delta tensor δ

α β δ = δ βeα ⊗ ω . (A.13)

1 The trace of a 1 -tensor corresponds to the trace of the linear transformation

α Tr A = A α . (A.14)

The abbreviation Tr2 A = (Tr A)2 is convenient. Any such tensor can be decomposed into its pure trace and tracefree parts 1 A = [Tr A]δ + A(TF) , n (A.15) 1 A(TF) = A − [Tr A]δ . n

2 A.4 Change of frame 1 α Given a nondegenerate 1 -tensor field A, i.e., such that the determinant det(A β) is everywhere nonvanishing, it has a corresponding inverse tensor field A−1 whose component matrix is the inverse matrix of the component matrix of A −1 −1 α γ α A A = δ , A γ A β = δ β . (A.16) These may be used to define a change of frame by defining new frame vector fields and dual 1-forms by

−1 −1 β α α α β eα = A eα = A αeβ , ω = ω A = A βω . (A.17) p Under such a change of frame, the components of a q -tensor ﬁeld “transform” under the representation ρp,q of the general linear group GL(n, R)

α... α −1 δ γ... S β... = A γ ··· A β ··· S δ... p,q α... (A.18) ≡ [ρ (A)S] β... . The derivative of a 1-parameter family of such transformations leads to the associated representation σp,q of the Lie algebra of linear transformations gl(n, R)

p,q α... α γ... γ α... [σ (A)S] β... ≡ A γ S β... + ... − A βS γ... − .... (A.19) This linear operator σp,q, which is convenient to abbreviate by just σ when the context is clear, appears in the component formula for any diﬀerential operator which obeys a product rule with respect to the tensor product. It is convenient to allow the argument of σ either to be a matrix-valued function (depending on 1 the choice of frame) or the corresponding 1 -tensor, whose component matrix is then understood to be substituted in its place. Occasionally tensor densities and oriented tensor densities prove useful. These transform with an additional factor which is a power (called the weight) of the inverse of the determinant of the linear transformation matrix α... −W α −1 δ γ... S β... = (det A) A γ ··· A β ··· S δ... p,q α... (A.20) ≡ [ρW (A)S] β... . p,q The derivative of a 1-parameter family of such transformations leads to the associated representation σW of the Lie algebra of linear transformations gl(n, R)

p,q α... α γ... γ α... γ α... [σW (A)S] β... ≡ A γ S β... + ... − A βS γ... − ... − WA γ S β... . (A.21) It is diﬃcult to ﬁnd clear discussions of these objects in modern textbooks. Oriented tensor densities instead transform with an additional sign factor which is the sign of the determinant of the transformation matrix, thus undergoing an additional sign change for an orientation changing transformation with negative determinant.

A.5 Metric

If (gαβ) is a symmetric nondegenerate matrix-valued function of signature s, i.e., can be reduced to a diagonal matrix with N negative diagonal values and n − N positive values, with s = (n − N) − N = n − 2N, and if (gαβ) is its inverse, then α β −1 αβ g = gαβ ω ⊗ ω , g = g eα ⊗ eβ (A.22) locally deﬁnes a pseudo-Riemannian (Riemannian if N = 0) metric and its inverse on M, satisfying

−1 αγ α g g = δ , g gγβ = δ β . (A.23)

Let g = | det(gαβ)| be the absolute value of the determinant of the matrix of components of the metric in this frame, the sign being (−1)N . For a Lorentz metric, one has N = 1 or N = n − 1. The metric determinant derivative satisﬁes αβ −1 d ln g = g dgαβ = Tr[g dg] . (A.24)

3 Given a choice of orientation on M, namely an everywhere nonzero n-form O, then {eα} is an oriented frame if O(e1, . . . , en) > 0 everywhere. In an oriented frame, the unit volume-form associated with the metric is deﬁned by η = g1/2ω1 ∧ · · · ∧ ωn . (A.25) For a Lorentz metric it is often convenient to use instead the indices 0, . . . , n − 1 for certain choices of frame, choosing the orientation so that in an oriented frame of this type one has

η = g1/2ω0 ∧ · · · ∧ ωn−1 . (A.26)

Since the metric is nondegenerate, it determines an isomorphism between the tangent and cotangent spaces at each point which in index-notation corresponds to “raising” and “lowering” indices. For a vector ﬁeld X and a 1-form θ one has [ β X = g X,Xα = gαβX , (A.27) ] −1 α αβ θ = g θ , θ = g θβ . As indicated in the introduction, the sharp and ﬂat notation for an arbitrary tensor will be understood to mean the tensor obtained by raising or lowering respectively all of the indices which are not already of the appropriate type.

A.6 Unit volume n-form

α1...αn The Levita-Civita permutation symbols α1...αn and are totally antisymmetric with

1...n = 1 = 1...n , (A.28) so that it vanishes unless α1 . . . αn is a permutation of 1 . . . n, in which case its value is the sign of the permutation. The components of the unit volume n-form are related to these objects by

1/2 α1...αn N −1/2 ηα1...αn = g α1...αn , η = (−1) g α1...αn . (A.29)

For a Lorentz 4-dimensional spacetime, it is convenient to use the ordered index labels 0,1,2,3 so that one has the Misner, Thorne and Wheeler convention

0123 0123 = 1 = (A.30) and therefore the unit-oriented 4-form has components

1/2 0123 −1/2 η0123 = g , η = −g . (A.31)

Others prefer the Ellis convention using the index labels 1,2,3,4, but one ﬁnds both sign conventions η1234 = 1/2 1/2 g and η1234 = −g in use, making comparisons diﬃcult.

A.7 Connection A general linear connection or covariant derivative ∇ on M is (locally) determined by its components in a frame. Adopting the “del” convention for the order of the covariant indices, these components are deﬁned by γ β β γ ∇eα eβ = Γ αβeγ ↔ ∇eα ω = −Γ αγ ω . (A.32) p p The covariant derivative ∇S of an arbitrary q -tensor S is a q+1 -tensor with components

α... α... α... [∇S] β...γ ≡ ∇γ S β... ≡ S β...;γ α... α δ... δ α... = S β...,γ + Γ γδS β... + · · · − Γ γβS δ... − · · · (A.33) α... p,q α... ≡ S β...,γ + [σ (ωωωγ )S] β... .

4 p p,q p,q For a q -tensor density of weight W , one need only replace σ (ωωωγ ) by σW (ωωωγ ), adding the additional term δ α... −W Γ γδS β.... 1 The 1 -tensor-valued connection 1-form for the frame {eα} is deﬁned by α β α γ β ωωω = eα ⊗ ωωω β ⊗ ω = Γ γβeα ⊗ ω ⊗ ω , (A.34) but is often treated as a matrix-valued 1-form whose entries are

α α γ ωωω β = Γ γβω . (A.35) The γ component of this matrix of 1-forms is understood to be the argument of the the linear operator σ in the deﬁnition of the covariant derivative. The components of the torsion tensor of the connection are

α α α T βγ = 2Γ [βγ] − C βγ (A.36) and deﬁne a vector-valued torsion 2-form

α 1 α βγ α ΘΘΘ = 2 T βγ ω , ΘΘΘ = eα ⊗ ΘΘΘ . (A.37) Of course as tensor fields ΘΘΘ = T , but the different notation helps distinguish the tensor-valued differential form interpretation from the tensor, and provides a notation for evaluating only the differential form arguments of the tensor. In most of the literature ΘΘΘ is treated as an Rn-valued 2-form (ΘΘΘα), as are all other vector-valued forms. The frame independent definition of the torsion 2-form, with its 2-form arguments evaluated on a pair of vector fields X and Y is ∇XY − ∇Y X − [X,Y ] = ΘΘΘ(X,Y ) . (A.38) Evaluation of this formula using the frame vectors themselves leads to the above component formula. This may also be expressed equivalently in terms of the exterior derivative dωα + ωωωα ∧ ωβ = ΘΘΘα , β (A.39) dϑϑϑ + ωωω ∧ϑϑϑ = ΘΘΘ , where the right contraction is between the tensor-valued indices and the wedge is between the differential form indices, and α α β ϑϑϑ = eα ⊗ ϑϑϑ = δ βeα ⊗ ω (A.40) is the identity tensor thought of as a vector-valued 1-form with component 1-forms ϑϑϑα = ωα equal to the frame 1-forms. A connection for which the torsion is zero is called torsion-free or symmetric, since in a coordinate frame the connection components are symmetric in the pair of covariant indices

α α α T βγ = 0 = C βγ → Γ [βγ] = 0 . (A.41) For a general connection, one can always introduce a 1-parameter family of related connections whose torsion is any real multiple of the original torsion by using a multiple of the torsion tensor itself as a diﬀerence tensor between the two connections. In particular there is a symmetric connection SYM ∇ with zero torsion and a transposed connection ∇˜ with opposite torsion α α α α 1 α 1 α Γ βγ = Γ (βγ) + Γ [βγ] = Γ (βγ) + 2 C βγ + 2 T βγ , α α 1 α α 1 α [SYM Γ] βγ = Γ βγ − T βγ = Γ (βγ) + C βγ , 2 2 (A.42) ˜α α α α 1 α 1 α Γ βγ = Γ βγ − T βγ = Γ (βγ) + 2 C βγ − 2 T βγ α α = Γ γβ + C βγ . In a coordinate frame these reduce to simple relationships α α ˜α α [SYM Γ] βγ = Γ (βγ) , Γ βγ = Γ γβ . (A.43) Such triplets of connections play an important role in the geometry of Lie groups.

5 A.8 Metric connection A connection is said to be metric with respect to a given metric g if that metric is covariant constant

0 = g = g − g Γδ − g Γδ αβ;γ αβ,γ δβ γα αδ γβ (A.44) = gαβ,γ − Γαγβ − Γβγα → gαβ,γ = 2Γ(α|γ|β) ,

extending the index-shifting convention to the components of the connection. This relation may be inverted using the deﬁnition of the torsion to express the components of the connection entirely in terms of the torsion, the structure functions of the frame and the ordinary derivatives of the metric components. Using the notation

A{δβγ}− = Aδβγ − Aβγδ + Aγδβ . (A.45) and shifting indices on the structure functions with the frame component matrix of the spacetime metric, the connection components can be written

α 1 αδ Γ βγ = g (g{δβ,γ} + C{δβγ} + T{δβγ} ) , 2 − − − (A.46) α α = Γ(g) βγ + K βγ ,

where α 1 αδ Γ(g) βγ = g (g{δβ,γ} + C{δβγ} ) , 2 − − (A.47) α 1 αδ K βγ = 2 g T{δβγ}− . α Γ(g) βγ are the components of the unique metric connection for which the torsion vanishes, often just called “the metric connection.” Its connection components in a coordinate frame reduce to the Christoffel α symbols of the metric. K βγ are the components of the difference tensor between the general metric connection and the unique one associated with the metric. This difference tensor is sometimes referred to as the contorsion tensor.

A.9 Curvature The components of the curvature tensor of a connection in a frame are deﬁned by

α α α α α α R βγδ = Γ δβ,γ − Γ γβ,δ − C γδΓ β + Γ γΓ δβ − Γ δΓ γβ . (A.48)

1 This tensor is explicitly antisymmetric in its last pair of indices and so deﬁnes a 1 -tensor-valued 2-form ΩΩΩ = R β α 1 α β γδ ΩΩΩ = eα ⊗ ω ⊗ ΩΩΩ β = 2 R βγδeα ⊗ ω ⊗ ω , (A.49) whose tensor-valued components are often thought of as a matrix-valued 2-form whose entries are

α 1 α γδ ΩΩΩ β = 2 R βγδω . (A.50)

1 The 1 -tensor values of this 2-form define linear transformations of the tangent and cotangent spaces. The invariant definition of the curvature tensor of a connection with the 2-form arguments evaluated on a pair of vector fields X and Y , and acting as a linear transformation on the vector field Z, is

{[∇X, ∇Y ] − ∇[X,Y ]}Z = ΩΩΩ(X,Y ) Z. (A.51)

When evaluated on the frame vectors themselves, one obtains the above component formula. This may also be represented alternatively in terms of the connection and curvature forms

α α γ α dωωω β + ωωω γ ∧ ωωω β = ΩΩΩ β , (A.52) dωωω + ωωω ∧ωωω = ΩΩΩ

where again the right contraction refers to the adjacent tensor-valued indices and the wedge product to the diﬀerential form indices.

6 The totally covariant Riemann tensor obtained by lowering its ﬁrst index can be re-expressed in terms of the covariant connection component derivatives using (A.44) to re-express the metric derivatives in terms of the symmetric part of the connection. For a torsion-free metric connection one ﬁnds

Rαβγδ = Γαδβ,γ − Γαγβ,δ − C γδΓαβ + ΓδαΓ γβ − ΓγαΓ δβ . (A.53)

This in turn may be used to make the second derivatives of the metric coeﬃcients explicit in a coordinate α frame (C βγ = 0) leading to the Landau-Lifshitz formula (92.4) from their Classical Theory of Fields

1 Rαβγδ = 2 (gαδ,βγ + gβγ,αδ − gαγ,βδ − gβδ,αγ ) + ΓδαΓ γβ − ΓγαΓ δβ . (A.54)

A.10 Total covariant derivative The total covariant derivative of a tensor field along a parametrized curve generalizes the ordinary derivative of a function along such a curve. It is called the intrinsic or absolute derivative in the older literature [Synge and Schild 1949, Møller 1952] and occurs as a special case of an induced connection over a map in the modern literature [Sachs and Wu 1977, Bishop and Goldberg 1968]. The terminology “total covariant derivative” as used in this text may be a result of teaching too many calculus classes while thinking about relativity. Given a parametrized curve c(λ), with corresponding tangent vector c0(λ), one can covariantly differentiate any tensor S(λ) defined along the curve by correcting the ordinary derivative by the representation of the connection 1-form matrix evaluated on the tangent vector

α... α... 0 α... DS β.../dλ = dS β.../dλ + [σ(ωωω(c (λ)))S(λ)] β... . (A.55)

The correction terms come from the product rule and the total covariant derivatives of the frame vectors and dual 1-forms along the curve

0 β α 0 α β D[eα ◦ c(λ)]/dλ = ωωω(c (λ)) αeβ ,D[ω ◦ c(λ)]/dλ = −ωωω(c (λ)) βω . (A.56)

For a tensor ﬁeld S, the total covariant derivative of its restriction to the curve S(λ) = S ◦ c(λ) agrees with the covariant derivative of S along the tangent vector at each point of the curve

[∇c0(λ)S] ◦ c(λ) = DS(λ)/dλ . (A.57)

It is convenient to follow the sloppy convention of not distinguishing between the two types of derivatives and suppressing the parametrization variable, writing simply

DS/dλ = ∇c0 S. (A.58) Given a local coordinate system, a parametrized curve is represented by the composed functions cα(λ) = xα ◦ c(λ), the derivatives of which provide the coordinate frame components of its tangent vector [c0(λ)]α = dcα(λ)/dλ = cα0(λ), often sloppily represented by the symbols xα(λ) and dxα(λ)/dλ respectively. For a vector ﬁeld X(λ) deﬁned along the curve, the coordinate frame expression for its total covariant derivative then takes the form α α α β0 γ DX (λ)/dλ = dX (λ)/dλ + Γ βγ ◦ c(λ)c (λ)X (λ) . (A.59) For the tangent vector itself, this becomes

0 α 2 α α β γ [Dc (λ)] = d c (λ)/dλ + Γ βγ ◦ c(λ)dc (λ)/dλdc (λ)/dλ . (A.60)

Under a change of parametrization λ = f(λ) of the curve, the total covariant derivative changes according to the familiar chain rule DS/dλ = (dλ/dλ) DS/dλ . (A.61)

7 A.11 Parallel transport and geodesics A tensor ﬁeld S(λ) deﬁned along a parametrized curve c(λ) is said to be parallel transported (sometimes “propagated”) along the curve if its total covariant derivative is identically zero

DS(λ)/dλ = 0 . (A.62)

A parametrized curve whose tangent vector is parallel transported along the curve is called a geodesic

Dc0(λ)/dλ = 0 . (A.63)

When expressed in a coordinate system, these are second order ordinary diﬀerential equations for the values of the coordinate functions along the parametrized curve.

A.12 Generalized Kronecker deltas The following identities deﬁne the generalized Kronecker deltas and relate them to the Levi-Civita epsilon and to the unit volume n-form η when a metric is available

δα1...αn = α1...αn = (−1)N ηα1...αn η , β1...βn β1...βn β1...βn 1 δα1...αp = δα1...αpγp+1...γn β1...βp (n − p)! β1...βpγp+1...γn 1 (A.64) = (−1)N ηα1...αpγp+1...γn η (n − p)! β1...βpγp+1...γn 1 = (−1)N+p(n−p) ηα1...αpγp+1...γn η . (n − p)! β1...βpγp+1...γn

Alternative deﬁnitions are instead

α1 α1 δ β1 ··· δ βp

α1...αp . . α1 αp δ = . . = p!δ [β ··· δ β ] β1...βp . . 1 p αp αp δ β1 ··· δ βp α2...αp (A.65) = pδα1 δ [β1 β2...βp] p X α ...αˆ ...α = (−1)i−1δαi δ 1 i p , β1 β2... βp i=1 where here the hatted index notation implies the removal of the index from the indicated position.

A.13 Symmetrization/antisymmetrization Given an object with only covariant or contravariant indices, or a subset of only covariant or contravariant indices from those of a mixed object, only can always project out the purely symmetric and purely 0 antisymmetric parts. For example for a p -tensor ﬁeld one has 1 β1...βp [ALT S]α ...α = S = δ Sβ ...β , 1 p [α1...αp] p! α1...αp 1 p 1 X [SYM S] = S = S . (A.66) α1...αp (α1...αp) p! σ(α1)...σ(αp) σ

8 A.14 Exterior product The exterior or wedge product of p 1-forms like the frame 1-forms is deﬁned antisymmetrizing without the factorial factor

α ...α ωα1...αp = ωα1 ∧ · · · ∧ ωαp = p! ω[α1 ⊗ · · · ⊗ ωαp] = δ 1 p ωβ1 ⊗ · · · ⊗ ωβp . (A.67) β1...βp With this convention the components of an antisymmetric tensor are themselves the expansion coeﬃcients in the basis {ω|α1...αp|}. This leads to a correction factor for the exterior product of p-forms with p > 1. For a p-form S and a q-form T , one has (p + q)! [S ∧ T ]α1...αp+q = S[α1...αp Tαp+1...αp+q ] p!q! (A.68) = δβ1...βpγ1...γq S T . α1 ... αp+q |β1...βp| |γ1...γp| One has the identity S ∧ T = (−1)pq T ∧ S. (A.69)

A.15 Hodge star duality operation The Hodge star duality operation is deﬁned for a p-form S to be the (n − p) form ∗S with components 1 [ ∗S] = S ηα1...αp (A.70) αp+1...αn p! α1...αp αp+1...αn following the right dual convention of Misner, Thorne and Wheeler in which the contraction of S] with η has η on the right. By introducing the (n − p)-forms 1 ηα1...αp = ∗ωα1...αp = ηα1...αp ωαp+1...αn , (A.71) (n − p)! αp+1...αn one can re-express this in the following way 1 ∗S = S ηα1...αp p! α1...αp (A.72) 1 = S] p η , p! where the generalized left contraction of p indices indicates the contraction of the last p contravariant indices of the left factor with the ﬁrst p covariant indices of the right factor in order. Note the extreme cases

∗1 = η , ∗η = (−1)N . (A.73)

Using the above deﬁnitions and identities one establishes the identity

∗ ∗S = (−1)N+p(n−p)S (A.74)

for a p-form S, where the term N in the sign arises from the signature and the second from the interchange of the group of indices. For a 4-dimensional Lorentzian manifold and a 3-dimensional Riemannian manifold respectively, this reduces to

n = 4,N = 1 : ∗ ∗S = (−1)p−1S (spacetime) (A.75) n = 3,N = 0 : ∗ ∗S = S (space) .

This is a special case of the more general identity for a p-form S and a q-form T with q ≥ p 1 ∗(S ∧ ∗T ) = (−1)N+(n−q)(q−p) S] p T. (A.76) p!

9 With p = 0 and S = 1 this reduces to the previous identity. With p = q one instead has 1 ∗(S ∧ ∗T ) = (−1)N S] p T, (A.77) p!

or since ∗η = (−1)N , removing the duality operation from each side leads to 1 S ∧ ∗T = S] p T η = hS, T i η , (A.78) p! where 1 hS, T i = Sα1...αp T (A.79) p! α1...αp deﬁnes a natural local (pointwise) inner product between p-forms.

A.16 Complex Duality Operation For a 4-dimensional Lorentz spacetime, 2-forms (“bivectors”) and 4-forms which are symmetric under ex- change of the ﬁrst and second index pairs (like the Riemann and Weyl tensors) play an important role. With N = 1 and p = 2, one has ∗ ∗S = −S for all 2-forms. The minus sign comes from the fact that raising all the indices of the unit volume 4-form in an orthonormal frame changes the sign of the ordered index component as in (A.29). By using the square root of the negative metric determinant rather than of its absolute value in the deﬁnition of the volume form, the extra sign in (A.29) is eliminated

1/2 α1...α4 −1/2 α1...α4 α1...α4 Eα1...α4 = (−g) α1...α4 = iηα1...αn ,E = (−g) = −iη . (A.80) Using this purely imaginary 4-form instead of η for the duality operation on 2-form index pairs deﬁnes the “hook” duality operation with symbol ˘, apparently used by Veblen and von Neumann, and later Taub. This leads to ˘ S = i ∗S on 2-forms which therefore satisfy ˘ ˘ S = S, so that one can ﬁnd eigentensors of the operation with eigenvalues ±1, called self-dual (+) and anti-self-dual (−) respectively. Self-duality then translates back to i ∗S = S or ∗S = −iS.

A.17 Exterior derivative The exterior derivative is deﬁned to obey the following identity for a p-form S and a q-form T

d(S ∧ T ) = dS ∧ T + (−1)pS ∧ dT , (A.81)

and to reduce to the ordinary diﬀerential of a 0-form or function

α df = ∂αf ω . (A.82) For a p-form 1 S = S ωα1...αp , (A.83) p! α1...αp this identity yields 1 1 1 dS = dS ∧ ωα1...αp + S dωα1...αp = [dS] ωα1...αp+1 , (A.84) p! α1...αp p! α1...αp (p + 1)! α1...αp+1 while for the basis p-forms it yields

p α1...αp 1 X i−1 αi α1...βγ...αp dω = − 2 (−1) C βγ ω . (A.85) i=1 Putting these together yields the formula

1 β [dS]α1...αp+1 = (p + 1)(∂[α1 Sα2...αp+1] − 2 pC [α1α2 S|β|α3...αˆi...αp+1]) , (A.86)

10 where the vertical bar delimiters exclude the indicated index from the antisymmetrization. Only the first term remains in a coordinate frame. This result may be re-expressed in terms of a general connection as 1 γ [dS]α1...αp+1 = (p + 1)(∇[α1 Sα2...αp+1] + 2 pT [α1α2 S|γ|α3...αî...αp]) . (A.87) For a symmetric connection only the first term remains yielding a formula which arises by substituting a covariant derivative for the ordinary derivative in the coordinate frame formula, i.e., the “comma to semicolon” rule. The exterior derivative may be expressed in a frame-independent way by evaluating its vector field arguments

p+1 X i−1 X i+j dS(X1,...,Xp+1) = (−1) XiS(X1,..., Xˆj,...,Xp+1)+ (−1) S([Xi,Xj],X1,..., Xˆi,..., Xˆj,...,Xp+1) i=1 i

A.18 Differential form divergence operator For a (p − 1)form S and a p-form T , the identities d(S ∧ ∗T ) = dS ∧ ∗T + (−1)p−1S ∧ d ∗T = hdS, T iη − (−1)N+p+(n−p+1)(p−1)hS, ∗d ∗T i η (A.90) = hdS, T iη − hS, δT i η define the adjoint operator δ for a p-form δT = (−1)N+1+n(p−1) ∗d ∗T. (A.91) For an appropriate space of p-forms for which the integral of the above exact n-form vanishes, this is the adjoint operator with respect to the global inner product defined by the integral of the local inner product over the manifold Z hhdS, T ii = hdS, T iη = hhS, δT ii . (A.92)

This has the interesting special cases n = 4,N = 1 : δT = ∗d ∗T (spacetime) (A.93) n = 3,N = 0 : δT = (−1)p ∗d ∗T (space) for a 4-dimensional Lorentzian manifold and a 3-dimensional Riemannian manifold respectively. For a metric connection ∇, it is easy to obtain a more familiar expression using the covariant derivative expression for the exterior derivative

α1...αp α1...αp α1...αp [∇α1 Sα2...αp ]T η = ∇α1 [Sα2...αp T ]η − Sα2...αp [∇α1 T ]η . (A.94) Identifying this equation with the above identity, the divergence term corresponding to the exact term above, leads one to the simple result that δ is just the negative of the covariant divergence operator α [δT ]α2...αp = −∇ Tαα2...αp = −[div T ]α2...αp . (A.95) The covariant divergence term and exact term are related by the following identity for a 1-form X δX η = ∗δX = (−1)N+1 ∗ ∗d ∗X = −d ∗X (A.96) which can also be written in the form d(X] η) = [div X] η . (A.97)

11 A.19 De Rham Laplacian

Again for the appropriate space of diﬀerential forms, one has the self-adjoint operator ∆(dR) deﬁned by

∆(dR) = δd + dδ (A.98)

which is called the de Rham Laplacian, diﬀering from the covariant Laplacian ∆

α [∆S]α1...αp = −∇α∇ Sα1...αp (A.99)

by curvature terms which arise from the Ricci identity. For the linear connection associated with a metric it is given by p X β [∆(dR)S]α1...αp = [∆S]α1...αp + R αi Sα1...β...αp i=1 p X β γ − R αi αj Sα1...β...γ...αp i6=j=1 p (A.100) X i β = [∆S]α1...αp + (−1) R αi Sβα2...αî...αp i=1 p X i+j β γ + (−1) R αi αj Sβγα1...αî...αˆj ...αp , i6=j=1 γ where Rαβ = R αγβ is the symmetric Ricci tensor defined below. For an n = 4 Lorentzian spacetime (signature − + ++) and a 1-form A with F = dA, and dF = 0, δF = −4πJ, one immediately finds ∆ A − dδA = −4πJ , (dR) (A.101) ∆(dR)F = −4πdJ .

A.20 Covariant exterior derivative The covariant exterior derivative operator D is defined for tensor-valued differential forms [Schouten 1954, Misner, Thorne and Wheeler 1973] as a derivative operator which acts as the exterior derivative on the form indices and as the covariant derivative on the tensor-valued indices (wedging the extra covariant index into the form indices). This operator may be defined by the following formula

β α... S = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ S , β... (A.102) β α... β α... DS = eα ⊗ · · · ⊗ ω ⊗ · · · ⊗ dS β... + [∇∧][eα ⊗ · · · ⊗ ω ⊗ · · ·] ⊗ S β... , where the notation ∇∧ in the final formula indicates a wedge product between the additional covariant index of the covariant derivative and the coefficient differential forms. For an ordinary differential form, D reduces to d, while for an ordinary tensor, it reduces to ∇. With the sloppy conventional notation

α... α... [DS] β... → DS β... , (A.103) the formula can be written in the more conventional form

α... α... α... DS β... = dS β... + [σ(ωωω) ∧ S] β... (A.104) or suppressing indices entirely as DS = dS + σ(ωωω) ∧ S. (A.105) With this compact notation one easily derives the identity

D2S = σ(ΩΩΩ) ∧ S (A.106)

12 which generalizes the identity d2 = 0 for ordinary diﬀerential forms. The component formula for the covariant exterior derivative includes torsion terms to compensate for the noncovariant derivatives in the exterior derivative part

α... α... [DS β...]α1...αp+1 = (p + 1)(∇[α1 S |β...|α2...αp+1] (A.107) 1 γ α... + 2 pT [α1α2 S |β...γ|[α2...αî...αp+1 ) . where the vertical bar delimiters exclude the indicated indices from the antisymmetrization. For a symmetric connection only the first term remains, yielding a formula which can be obtained by the “comma-to- semicolon” transformation of the familiar coordinate frame formula for the exterior derivative. Note that the definition of the torsion 2-form is very simple in terms of this notation. It is just the covariant exterior derivative of the identity tensor

ΘΘΘ = Dϑϑϑ . (A.108)

A metric may be considered as a tensor-valued 0-form whose covariant exterior derivative is

γ γ Dgαβ = dgαβ − gγβωωω α − gαγωωω β = dgαβ − 2ωωω(αβ) . (A.109) For a metric connection and a frame in which the metric components are constants, one has

0 = Dgαβ = dgαβ → ωωω(αβ) = 0 , (A.110) i.e., the connection 1-form is antisymmetric in its tensor-valued indices. This is true for orthonormal frames.

A.21 Ricci identities p The Ricci identity for a q -tensor ﬁeld S is

α... α... α... α... [∇γ , ∇δ]S β... = 2S β...;[δγ] = [σ(ΩΩΩγδ)S] β... − T γδ∇S β... . (A.111)

p 2 Interpreting S as a q -tensor valued 0-form, this becomes a special case of the identity D S = σ(ΩΩΩ)S. For a metric and a metric connection one has

2 γδ γ γ D gαβ = gαβ;[δγ]ω = −gγβΩΩΩ α − gαγΩΩΩ β = −2ΩΩΩ(αβ) . (A.112) Thus for a metric connection, the curvature 2-form is also antisymmetric in its tensor-valued indices

2 D gαβ = 0 → ΩΩΩ(αβ) = 0 or R(αβ)γδ = 0 . (A.113)

A.22 Bianchi identities of the ﬁrst and second kind The Bianchi identity of the ﬁrst kind re-expresses the covariant exterior derivative of the torsion 2-form and is a direct consequence of the previous identity for the vector-valued 1-form ϑϑϑ

α 2 α α β 1 α βγδ DΘΘΘ = D ϑϑϑ = ΩΩΩ β ∧ ϑϑϑ = 2 R [βγδ]ω . (A.114) For a symmetric connection this reduces to the familiar identity

α α α α 3R [βγδ] = R βγδ + R γδβ + R δβγ = 0 . (A.115) The Bianchi identity of the second kind re-expresses the covariant exterior derivative of the curvature 2-form DΩΩΩ = 0 or α α α α (A.116) 3R β[γδ;] = R βγδ; + R βδ;γ + R βγ;δ = 0 . For a symmetric metric connection, namely the unique such connection associated with a given metric, the cyclic Bianchi identity of the ﬁrst kind (for zero torsion) together with the antisymmetry of the ﬁrst

13 index pair (covariant constant metric) leads to the pair interchange symmetry for the Riemann curvature tensor of the metric. Summing the four cyclic permutations of the Bianchi identity of the ﬁrst kind 0 = 3(R + R + R + R ) = ... α[βγδ] β[γδα] γ[δαβ] δ[αβγ] (A.117) = 2(Rαγβδ − Rβδαγ ) , leads to twelve terms, eight of which cancel in pairs due to the antisymmetry of the ﬁrst index pair while the remaining four collapse to two terms. Relabeling the result yields the pair interchange symmetry

Rαβγδ = Rγδαβ . (A.118)

A.23 Ricci Tensor, Scalar Curvature and Einstein tensor For a metric connection the natural contraction of the Riemann curvature tensor deﬁnes the Ricci curvature

γ Rαβ = R αγβ (A.119) which is a symmetric tensor due to the Bianchi identities of the first kind. Its trace defines the scalar curvature α γα R = R α = R γα . (A.120) The following combination is called the Einstein tensor, whose divergence-free property is shown in the next section 1 G = R − Rg . (A.121) αβ αβ 2 αβ In n = 2 dimensions, there is only one independent component of the Riemann curvature tensor which up to a sign depending on the signature of the metric, is the Gaussian curvature, and the Einstein tensor vanishes identically. In n = 3 dimensions, the Weyl tensor defined in the next section vanishes identically, so that the Riemann and Ricci tensors contain the same information and the Einstein tensor is just the double dual of the curvature tensor up to sign.

A.24 Contracted Bianchi Identities of the Second Kind and the Weyl Tensor The Riemann curvature tensor may be expressed in terms of the Weyl curvature tensor, the Ricci tensor, and the curvature scalar as follows [Eisenhart 1966] (which may be taken as a deﬁnition of the Weyl curvature tensor) αβ αβ [α β] αβ R γδ = C γδ + 4/(n − 2)R [γ δ δ] − R/[(n − 1)(n − 2)]δ γδ (A.122) αβ [α β] αβ = C γδ + 4/(n − 2)Q [γ δ δ] + R/[n(n − 1)]δ γδ where the second equality expresses it instead using the tracefree part of the Ricci tensor

α α α Q β = R β − 1/n δ β , (A.123)

α while the Weyl tensor is itself tracefree C βαδ = 0. For n = 2 one has instead αβ αβ R γδ = R δ γδ (A.124) where R = K is the Gaussian curvature in the Euclidean signature case. For n = 3 in the Euclidean signature case the Riemann curvature tensor has the same number of independent components as the Ricci tensor and Einstein tensor, and can be represented as the double dual of the Einstein tensor

αβ αβµ ν R γδ = η γδν G µ . (A.125) The once contracted Bianchi identity of the second kind (using the tracefree property of the Weyl tensor) reduces to αβ γ 0 = 3R [γδ;]δ α αβ β = R δ;α + 2R [δ;] (A.126) αβ β = C δ;α + (n − 3)/(n − 2) R δ ,

14 where β β β R δ = 2(R [δ − R/[2(n − 1)]δ [δ);] , (A.127) thus expressing the divergence of the Weyl tensor for n > 3 in terms of a vector-valued 2-form, the Cotton tensor [Cotton 1899], which is the covariant exterior derivative of a linear combination of the Ricci tensor and the scalar curvature seen as a vector valued 1-form. The twice contracted Bianchi identity of the second kind in any dimension

αβ γ δ 0 = R [γδ;]δ αδ β 1 = −2(Rδ − Rδδ ) (A.128) 2 ;δ δ = −2G ;δ , which states that the Einstein tensor is divergence-free, immediately leads to the tracefree property of the Cotton tensor β β R αβ = −G α;β = 0 , (A.129) whose totally antisymmetric part is easily seen to be zero using the symmetry of the Ricci and metric tensors

3R[αβγ] = Rαβγ + Rβγα + Rγαβ = 0 . (A.130)

The divergence of the Cotton-tensor is also zero

β β R δ;β = 2G [δ;|β|;] = 0 . (A.131)

For n = 3 dimensions where the Weyl tensor vanishes identically, one can take the dual of the covariant index pair of the Cotton tensor to obtain a two-index tensor, the Cotton-York tensor [York 1971]

αβ α βγδ α α βγδ C = R γδη = (R γ − 1/4Rδ γ );δη , (A.132) which can then be decomposed into its symmetric and antisymmetric parts, the latter of which is the spatial dual of a vector αβ (αβ) [αβ] (αβ) αβγ C = C + C = Y + η Aγ (A.133) α αδ αβ δα R βγ = C ηδβγ = ηβγδY + Aδδβγ . The explicitly symmetric York tensor and the vector (which vanishes by the twice contracted Bianchi identity) are

αβ γδ(α β) αβ (TF) αβ αβ y = η R γ;δ = −[Scurl(Ricci)] = −[Scurl(Ricci )] = −[Scurl(Einstein)] , α α (A.134) Aβ = R βα = G β;α = 0 ,

The subsequent equalities for the York tensor y hold since the symmetrized covariant exterior derivative “Scurl”, see section (2.6)) of a symmetric 2-tensor interpreted as a vector-valued 1-form annihilates the pure trace part (and automatically yields a tracefree result, making the York tensor tracefree since the Ricci tensor is symmetric). The York tensor is automatically divergence-free as a consequence of the same property for the Cotton tensor.

A.25 Conformal Transformations in 3 Dimensions

Introduce the notation Ua = ∇aU for a scalar U, which satisﬁes

∇aUb = ∇a∇bU = ∇b∇aU = ∇bUa (A.135)

15 abc or equivalently η ∇bUc = 0.ˆ Then the standard conformal transformation laws for the metric and the curvature tensors are

2U ab −2U ab 1/6 U 1/6 gãb = e gab , g˜ = e g , g˜ = e g , (A.136) 3U abc −3U abc ηãbc = e ηabc , η˜ = e η , (A.137) b b b e e b ∇˜ dX c = ∇dX c + C deX c − C dcX e , (A.138) c c c c C ab = 2δ (aUb) − gabU ,C [ab] = 0ˆ , (A.139) e e Rãc = Rac − ∇aUc + UaUc + gac(−∇ Ue − U Ue) , (A.140) −2U e e R˜ = e [R − 4∇ Ue − 2U Ue], (A.141) e Gãc = Gac − ∇aUc + UaUc + gac∇ Ue , (A.142) a −2U a −2U R˜ c = e Rãc , ∇˜ dR c = e (∇d − 2Ud)Rãc . (A.143)

The York tensor then satisﬁes

−5/6 5/6 ea −5/6 5/6 edc a edc˜ a g g˜ y˜ = g g˜ η˜ ∇˜ dR˜ c = η (∇d − 2Ud)R˜ c = yea + [Scurl(−∇U + U ⊗ U) + U × (∇U − R)]ea = yea − [Scurl(∇U) + U × R]ea + [Scurl(U ⊗ U) + U × ∇U]ea . (A.144)

The ﬁrst pair of terms in square brackets is identically zero for a gradient Ua = ∇aU by the relevant Ricci identity, while the second is identically zero for a gradient just by symmetry of the second derivatives alone, leading to the conformal invariance of the Cotton-York contravariant tensor-density Y ea = g5/6yea (see exercise 21.22 of Misner, Thorne and Wheeler [1973]). It must therefore be zero for any metric which is conformally ﬂat.

A.26 n = 3 Structure Functions and Orthonormal Frame Connection Compo- nents In any frame in n = 3 dimensions, the structure constant functions are antisymmetric in the lower indices, so a 2-index object can be deﬁned by taking the spatial dual on those indices, then decomposing the result into its symmetric and antisymmetric parts, the latter of which can be expressed as the dual of a 1-index object αδ α δβγ C = C βγ η = C(αδ) + C[αδ] (A.145) (αδ) αδγ = m + η aγ , α βδ δα C βγ = ηβγδm + aδδβγ , where αδ (α δ)βγ 1 α m = C βγ η , aγ = 2 C γα . (A.146) α 3 The mixed object C β may be interpreted as a “vector”-valued 1-form in the sense of an R -valued 1-form. This notation was introduced by Ellis and MacCallum (1969) after the decomposition was described by Behr (1968, 2002). In any frame for which the metric components are constants (an orthonormal frame in general or an invariant metric on a Lie group, for example), the antisymmetry of the matrix-valued connection 1-form

γ 0 = ω(αβ) = Γ(α|γ|β)ω (A.147) means that the connection components are antisymmetric in their outer indices, so one may take their spatial dual to deﬁne a matrix-valued scalar which may then be interpreted as a “vector”-valued 1-form

β αβγ Γ δ = Γ ηαγδ , (A.148)

16 which in turn can be decomposed into its symmetric and antisymmetric parts and pure trace and tracefree parts Γβδ = Γ(βδ) + Γ[βδ] . (A.149) The relationship between these two “vector”-valued 1-forms in an orthonormal frame follows from the specialization ∂αgβγ = 0 of the general formula for the metric connection to the following result

α 1 α α α 1 α α 1 α α Γ βγ = 2 (C βγ + Cβ γ + Cγ β) = 2 C βγ + C(β γ) ≡ 2 C βγ + K βγ , (A.150) where 1 Kαβγ = − 2 £eα gβγ , (A.151) namely β β 1 γ β Γ δ = C δ − C γ δ δ 2 (A.152) β 1 γ β β γ = (m δ − 2 m γ δ δ) + η δ aγ . The double dual of the Riemann tensor yields a symmetric tensor, namely the sign-reversed Einstein tensor ∗ ∗ α 1 αγδ µν 1 αγδ µν α [ R ] β = η Rγδ ηµνβ = δ Rγδ = −G β , 4 4 βµν (A.153) αβ αβµ ν R γδ = −η ηγδν G µ . For a metric with constant frame components, the Einstein tensor then reduces to

α γδ α α 1 α µ ν γδ G = (G β) = (−ηβ ∂γ Γ δ + Γ C β − η µν Γ γ Γ δη β) 2 (A.154) 2 1 2 1 2 f T f = 2m − m Tr m − 2 (Tr m − 2 Tr m − 6af a )I − 2aa − 2a Kf ,

α α T β α using a more eﬃcient matrix notation m = (m β), a = (a ), a = (aα), Kα = (Kα γ ), I = (δ β). Similarly

α γδ α α 1 α µ ν γδ R = (R β) = (−ηβ ∂γ Γ δ + Γ C β − 2 η µν Γ γ Γ δη β) 2 2 1 2 T f = 2m − m Tr m − (Tr m − 2 Tr m)I − 2aa − 2a Kf , (A.155) 2 1 2 f Tr R = −(Tr m − 2 Tr m) − 6af a .

A.27 Lie derivative The Lie derivative along a vector field X is defined so that it is the ordinary derivative along X of a function f and the Lie bracket by X of another vector field Y ,

£Xf = Xf , £XY = [X,Y ] , (A.156) and it is then extended to all other tensor ﬁelds so that it obeys the obvious product rule for the evaluation of a tensor ﬁeld on all of its arguments (reducing it to a function). Solving this product rule for the Lie derivative of the tensor alone yields

(1) (q) [£XS](θ(1), . . . , θ(p),X ,...X ) (1) (q) = £ [S(θ(1), . . . , θ(p),X ,...X )] X (A.157) (1) (q) − S(£Xθ(1), . . . , θ(p),X ,...X ) − ... (1) (q) − S(θ(1), . . . , θ(p), £XX ,...X ) − .... The Lie derivative is a linear operator which also acts on diﬀerential forms (sends p-forms into p-forms) and obeys the usual product rules with respect to the tensor and exterior products (no extra signs). p This last equation immediately leads to a formula for the components of the Lie derivative of a q -tensor ﬁeld in a frame by evaluating it on the frame vectors and 1-forms

α... α... £XS β... ≡ [£XS] β... α... γ p,q α... (A.158) = S β...,γ X + [σ (L(X, e))S] β... ,

17 where the component matrix which is the argument of σp,q is given by

α α α L(X, e) β = [£Xeβ] = −[£Xω ]β α α γ = −X ,β + C γβX (A.159) α α γ = −X ;β + Γ˜ γβX ,

and σp,q is the associated representation σp,q of the Lie algebra of linear transformations gl(n, R) introduced in section A.4. The last equality holds when a linear connection ∇ is available and enables the Lie derivative to be expressed in terms of the covariant derivative ∇X and the transposed covariant derivative ∇˜ S

α... ˜ α... p,q α... £XS β... = ∇XS β... − [σ (∇X)S] β... . (A.160) For a symmetric connection, this reduces to

α... α... γ p,q α... £XS β... = S β...;γ X − [σ (∇X)S] β... , (A.161) which is the formula which results from the substitution of a semicolon for the comma in the formula for the coordinate frame components of the Lie derivative. For a differential form S, the Lie derivative can be expressed in terms of the exterior derivative and the contraction operation £XS = X dS + d(X S) . (A.162) The geometric significance of the Lie derivative comes from its relation to the generator of a 1-parameter family of diffeomorphisms. If Xλ denotes the flow of a vector field, then each tensor field S may be dragged along by the flow for each value of λ to define a 1-parameter family of tensor fields, indicated by the abbreviation Sλ = XλS,S0 = S. (A.163) Then the parameter derivative of this family at λ = 0 defines the Lie derivative reversed in sign

d/dλ Sλ|λ=0 = −£XS. (A.164) p For a q -tensor density ﬁeld of weight W the formula for the components of the Lie derivative has one p extra term compared to a q -tensor ﬁeld

α... α... £XS β... ≡ [£XS] β... α... γ p,q α... = S β...,γ X + [σW (L(X, e))S] β... (A.165) α... γ p,q α... γ α... = S β...,γ X + [σ (L(X, e))S] β... − W L(X, e) γ S β... .

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