It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein's equation (or of the conservation principle that is, itself, a consequence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein's equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed)

Malament, David

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A Remark About the “GeodesicPrinciple” in General Relativity∗Version 3.0David B. MalamentDepartment of Logic and Philosophy of Science3151 Social Science PlazaUniversity of California, IrvineIrvine, CA 92697-5100dmalamen@uci.edu1 IntroductionGeneral relativity incorporates a number of basic principles that correlate space-time structure with physical objects and processes. Among them is theGeodesic Principle: Free massive point particles traverse timelikegeodesics.One can think of it as a relativistic version of Newton’s first law of motion.It is often claimed that the geodesic principle can be recovered as a theoremin general relativity. Indeed, it is claimed that it is a consequence of Einstein’s∗I am grateful to Robert Geroch for giving me the basic idea for the counterexample(proposition 3.2) that is the principal point of interest in this note. Thanks also to HarveyBrown, Erik Curiel, John Earman, David Garfinkle, John Manchak, Wayne Myrvold, JohnNorton, and Jim Weatherall for comments on an earlier draft.1equation (or of the conservation principle ∇aTab = 0 that is, itself, a conse-quence of that equation). These claims are certainly correct, but it may beworth drawing attention to one small qualification. Though the geodesic prin-ciple can be recovered as theorem in general relativity, it is not a consequenceof Einstein’s equation (or the conservation principle) alone. Other assumptionsare needed to drive the theorems in question. One needs to put more in if oneis to get the geodesic principle out. My goal in this short note is to make thisclaim precise (i.e., that other assumptions are needed).All talk about deriving the geodesic principle is a bit delicate because it is notantecedently clear how to formulate it so that it is even a candidate for proof.One way or another, one has to confront the problem of how to associate anenergy-momentum content Tab with a point particle. (Only then can one invokethe conservation principle ∇aTab = 0.) This is a problem even if one is willingto restrict attention to “test particles”, i.e., even if one does not insist that Tabbe recorded on the right side of Einstein’s equation. One might try to workwith energy-momentum “distributions” rather than proper smooth fields, butthere is a natural alternative. In effect, one models a massive point particle as anested sequence of small, but extended, bodies that converges to a point. Oneassociates with each of the bodies a garden variety smooth energy-momentumfield Tab, and requires that, in each case, it satisfy certain constraints. Thenone proves, if one can, that the point to which the bodies converge necessarilytraverses a timelike geodesic.Various theorems in the literature do, in fact, have this form. In all cases,one assumes that the energy-momentum field Tab associated with each smallbody in the sequence satisfies the conservation principle. (This captures theidea that the body is “free” i.e., not exchanging energy-momentum with someexternal field.) That much the theorems have in common. But they differ asto the additional constraints that are imposed. In some cases, very specificassumptions are made about the constitution of the bodies in the sequence. Atheorem in Thomas [6] and Taub [5] is of this type. There one takes each body tobe a blob of perfect fluid, with everywhere non-negative isotropic pressure, thatsatisfies a strong constraint. It is required that the pressure at every point in theblob remains constant over time. Given this assumption (and the conservationprinciple), it is easy to prove that the convergence point of the bodies does, infact, traverse a timelike geodesic.2This result is certainly of interest. But it seems a considerable advance toprove theorems that dispense with special modeling assumptions in favor ofgeneric ones. The result of Geroch and Jang [2] that I’ll formulate in section3 (proposition 3.1) is an example of this latter type. There one only assumesthat the energy-momentum field Tab of each body in the sequence satisfies acertain “energy condition”. It asserts, in effect, that, whatever else is the case,energy propagates within the body at velocities that are timelike. That toois sufficient, together with the conservation principle, to guarantee that theconvergence point of the bodies traverses a timelike geodesic.My point in this note is that the Geroch-Jang theorem fails if one drops theenergy-condition requirement. As we shall see (proposition 3.2), the conserva-tion condition alone imposes no restrictions whatsoever on the wordline of theconvergence point of the bodies. It can be a null or spacelike curve. It can alsobe a timelike curve that exhibits any desired pattern of large and/or changingacceleration.In the Geroch-Jang theorem, one allows oneself to ignore the negligible effecton the background metric made by (the energy-momentum content of) eachbody in the convergent sequence. A stronger result of Ehlers and Geroch [1]relaxes this restriction. There it is not required that the perturbative effectdisappear entirely at each intermediate stage, but only that, in a certain pre-cise sense, it disappear in the limit. In this result too, an energy condition isimposed in lieu of any more specific modeling assumptions about the bodies inthe sequence. And again in this case, the result fails completely without theenergy condition. (The counterexample that we present for the weaker theo-rem (in proposition 3.2) carries over intact to the stronger one.) To keep thepresentation as simple as possible, I will limit my attention to the former.32 The Energy-Momentum Field TabIn this section, we review a few things about the energy-momentum field Tabthat will be important later.1 Some readers may want to skip to section 3.2In what follows, let (M, gab) be a relativistic spacetime, which we here take toconsist of a smooth, connected, four-dimensional differential manifold M , and asmooth metric gab on M of Lorentz signature (1, 3). With this sign convention,a vector ξa at a point counts as timelike if ξaξa > 0, null if ξaξa = 0, causalif ξaξa ≥ 0, and spacelike if ξaξa < 0. We assume that (M, gab) is temporallyorientable, and that some temporal orientation has been specified.Let us start with point particles. It is a basic assumption of relativity theorythat we can associate with every point particle, at every point on its world-line, a four-momentum (or energy-momentum) vector P a that is tangent to itsworldline. We can think of it as encoding several pieces of information. It isstandardly taken for granted that P a is causal. In that case, at least, the lengthof P a gives the mass of the particle:mass = (P aPa)12 .So, in particular, the mass of the particle is strictly positive iff its four-momentumvector field is timelike. Let ξa be a future-directed, unit timelike vector at somepoint on the worldline of the particle. We can think of it as representing the in-stantaneous state of motion of a background observer at that point. Suppose wedecompose P a into two component vectors that are, respectively, proportionalto, and orthogonal to, ξa:P a = (P bξb)︸ ︷︷ ︸energyξa + (P a − (P bξb) ξa)︸ ︷︷ ︸3−momentum. (2.1)The proportionality factor P bξb in the first is standardly understood to give theenergy of the particle relative to ξa; and the second component is understoodto give the three-momentum of the particle relative to ξa.1We will assume familiarity with the basic mathematical formalism of general relativity inwhat follows. For background material, see, e.g., Hawking and Ellis [3], Wald [7], or Malament[4]. The third is a set of unpublished lecture notes that is available online.2All the material in the section is perfectly standard except for one small bit of ad hoc ter-minology. In addition to the weak and dominant energy conditions, we will consider somethingthat we call the “strengthened dominant energy condition”.4Let us now switch from point particles to matter fields, e.g., fluids and elec-tromagnetic fields. Each such field is represented by one or more smooth tensor(or spinor) fields on the spacetime manifold M . Each is assumed to satisfy fieldequations involving the spacetime metric gab.For present purposes, the most important basic assumption about the matterfields is the following.Associated with each matter field F is a symmetric smooth tensorfield Tab characterized by the property that, for all points p in M ,and all future-directed, unit timelike vectors ξa at p, T abξb is thefour-momentum density of F at p as determined relative to ξa.Tab is called the energy-momentum field associated with F . The four-momentumdensity vector T abξb at p can be further decomposed into components propor-tional to, and orthogonal to, ξa (just as with the four-momentum vector P a):T abξb = (Tnb ξnξb)︸ ︷︷ ︸energy densityξa + (T abξb − (Tnb ξnξb) ξa)︸ ︷︷ ︸3−momentum density. (2.2)The coefficient of ξa in the first component, Tabξaξb, is the energy density of Fat p as determined relative to ξa. The second component, Tnb(gan − ξa ξn)ξb,is the three-momentum density of F at p as determined relative to ξa.Various assumptions about matter fields can be captured as constraints onthe energy-momentum tensor fields with which they are associated. The Geroch-Jang theorem makes reference to the third and fourth in the following list.(Suppose Tab is associated with matter field F .)Weak Energy Condition: For all points p inM , and all unit timelike vectorsξa at p, Tab ξaξb ≥ 0.Dominant Energy Condition : For all points p in M , and all unit timelikevectors ξa at p, Tab ξaξb ≥ 0 and T abξb is causal.Strengthened Dominant Energy Condition3: For all points p in M , andall unit timelike vectors ξa at p, Tab ξaξb ≥ 0 and, if Tab 6= 0, then Tabξbis timelike.3This is not a standard name.5Conservation Condition: ∇a Tab = 0 at all points in M.The weak energy condition asserts that the energy density of F (as determinedrelative to any background observer) is everywhere non-negative. The dominantenergy condition adds the requirement that the energy-momentum density of F(as determined relative to a background observer) is causal. It can be under-stood to assert that the energy of F does not propagate at superluminal velocity(relative to any such observer). The strengthened version of the condition justchanges “causal” to “timelike”. Each of the energy conditions is strictly strongerthan the ones that precede it.4The final condition in the list captures the requirement that the energy-momentum carried by F be locally conserved. If two or more matter fields arepresent in the same region of spacetime, it need not be the case that each oneindividually satisfies the condition. Interaction may occur. But presumably inthat case the composite energy-momentum field formed by taking the sum of theindividual ones satisfies the condition. Energy-momentum can be transferredfrom one matter field to another, but it cannot be created or destroyed.Suppose Tab represents the aggregate energy-momentum present in someregion of spacetime. Then, at least if it is understood to arise from “sourcefields” rather than “test fields”, it must satisfy Einstein’s equationRab −12Rgab = 8 pi Tab.The left side is divergence-free: ∇a(Rab − 12Rgab) = 0. (This follows fromBianchi’s identity.) So, in this (source field) case at least, the conservationcondition is a consequence of Einstein’s equation.The dominant energy and conservation conditions have a number of jointconsequences that support the interpretations just given. Here is one. It requiresa preliminary definition.Let (M, gab) be a fixed relativistic spacetime, and let S be an achronal subsetof M (i.e., a subset no two points of which are connected by a smooth timelikecurve). The domain of dependence D(S) of S is the set of all points p in M4If λa is a smooth spacelike field, then Tab = λaλb satisfies the weak, but not the dominant,energy condition. Similarly, if λa is a smooth, non-vanishing null field, then Tab = λaλbsatisfies the dominant, but not the strengthened dominant, energy condition.6with this property: given any smooth causal curve without (past or future)endpoint,5 if (its image) passes through p, then it necessarily intersects S.}D(S)SFigure 2.1: The domain of dependence D(S) of an achronal set S.Proposition 2.1. Let S be an achronal subset ofM . Further let Tab be a smoothsymmetric field on M that satisfies both the dominant energy and conservationconditions. Finally, assume Tab = 0 on S. Then Tab = 0 on all of D(S).The intended interpretation of the proposition is clear. If energy-momentumcannot propagate (locally) outside the null-cone, and if it is conserved, and ifit vanishes on S, then it must vanish throughout D(S). After all, how could it“get to” any point in D(S)? Note that our formulation of the proposition doesnot presuppose any particular physical interpretation of the symmetric field Tab.All that is required is that it satisfy the two stated conditions. (For a proof, seeHawking and Ellis [3, p. 94].)3 A Theorem and A CounterexampleNow we turn to the Geroch-Jang theorem [2] itself.Proposition 3.1. Let (M, gab) be a relativistic spacetime, and let γ : I → Mbe a smooth curve. Suppose that given any open subset O of M containing γ[I],there exists a smooth symmetric field Tab on M such that:5Let γ : I → M be a smooth curve. We say that a point p in M is a future-endpoint of γif, for all open sets O containing p, there exists an s0 in I such that for all s ∈ I, if s ≥ s0,then γ(s) ∈ O, i.e. the image of γ eventually enters and remains in O. (Past-endpoints aredefined similarly.)7(1) Tab satisfies the strengthened dominant energy condition;(2) Tab satisfies the conservation condition;(3) Tab = 0 outside of O;(4) Tab 6= 0 at some point in O.Then γ is a timelike curve, and can be reparametrized so as to be a geodesic.The proposition might be paraphrased this way. Suppose that arbitrarilysmall bodies (with energy-momentum) satisfying conditions (1) and (2) cancontain the image of a curve γ in their worldtubes. Then γ must be a time-like geodesic (up to reparametrization). In effect, as discussed above, we arerepresenting “point particles” as nested convergent sequences of smaller andsmaller extended bodies. Bodies here are understood to be “free” if their in-ternal energy-momentum is conserved (by itself). If a body is acted upon by afield, it is only the composite energy-momentum of the body and field togetherthat is conserved.The proof proceeds by showing that given any worldtube and any energy-momentum field satisfying conditions (1)-(4), the tube must contain the imageof a timelike geodesic. That cannot be true for arbitrarily small tubes containingthe image of the original curve γ unless that curve itself is a timelike geodesic(up to reparametrization).Our formulation of the proposition takes for granted that we can keep thebackground spacetime metric gab fixed while altering the fields Tab that live onM . This is justifiable only to the extent that, once again, we are dealing withtest bodies whose effect on the background spacetime structure is negligible.Though, of course, the proposition has an intended interpretation, it is im-portant that it stands on its own as a well-formed mathematical theorem (asdoes proposition 2.1). It can be proved without any appeal to the interpretationof Tab. It is also noteworthy in the proposition that we do not have to assumethat the initial curve γ is timelike. That is something that we prove.Our main claim, as announced above, is that the proposition fails if condition(1) is dropped. Without it, one cannot prove that the original curve γ mustbe a geodesic (up to a reparametrization), not even if we do assume in advancethat it is timelike. The following proposition gives a counterexample.8Figure 3.1: A non-geodesic timelike curve enclosed in a tube (asconsidered in proposition 3.2).Proposition 3.2. Let (M, gab) be Minkowski spacetime, and let γ : I → Mbe any smooth timelike curve. Then given any open subset O of M containingγ[I], there exists a smooth symmetric field Tab onM that satisfies conditions (2),(3), and (4) in the preceding proposition. (If we want, we can also strengthencondition (4) and require that Tab be non-vanishing throughout some open subsetO1 ⊆ O containing γ[I].)Proof. Let O be an open subset of M containing γ[I], and let f : M → R beany smooth scalar field on M . (Later we will impose further restrictions onf .) Consider the fields Sabcd = f(gadgbc − gacgbd) and T ac = ∇b∇d Sabcd,where∇ is the (flat) derivative operator onM compatible with gab. (So∇agbc =∇agbc = 0.) We haveT ac = (gadgbc − gacgbd)∇b∇df = ∇c∇af − gac (∇b∇bf). (3.1)So T ac is clearly symmetric. It is also divergence-free since∇a Tac = ∇a∇c∇af −∇c∇b∇bf = ∇c∇a∇af −∇c∇b∇bf = 0.(The second equality follows from the fact that ∇ is flat, and so ∇a and ∇ccommute in their action on arbitrary tensor fields.)To complete the proof, we now impose further restrictions on f to insure thatconditions (3) and (4) are satisfied. Let O1 be any open subset of M such that9γ[I] ⊆ O1 and cl(O1) ⊆ O. (Here cl(A) is the closure of A.) Our strategy willbe to choose a particular f on O1, and a particular f on M−cl(O), and thenfill-in the buffer zone cl(O) − O1 any way whatsoever (so long as the resultantfield is smooth). On M−cl(O), we simply take f = 0. This choice guaranteesthat, no matter how we smoothly extend f to all of M , T ac will vanish outsideof O.For the other specification, let p be any point in M , and let χa be the“position field” on M determined relative to p. So ∇a χb = δab everywhere,and χa = 0 at p. (See, for example, proposition 1.7.11 in Malament [4].) OnO1, we take f = −(χnχn). With that choice, Tac is non-vanishing at all pointsin O1. Indeed, we have∇af = −2χn∇aχn = −2χn δan = −2χa,and, therefore,T ac = ∇c∇af − gac (∇b∇bf) = −2∇cχa + 2 gac (∇b χb)= −2 gca + 2 gac δbb = −2 gac + 8 gac = 6 gacthroughout O1.One point about the proof deserves comment. As restricted to O1 and toM−cl(O), the field Tab that we construct does satisfy the strengthened dominantenergy condition. (In the first case, Tab = 6 gab, and in the second case, Tab =0.) But we know – from the Geroch-Jang theorem itself – that it cannot satisfythat condition everywhere. So it must fail to do so in the buffer zone cl(O)−O1.That shows us something. We can certainly choose f in the zone so that itsmoothly joins with our choices for f on O1 and M−cl(O). But, no matter howclever we are, we cannot do so in such a way that T ab (as expressed in (3.1))satisfies the strengthened dominant energy condition.10References[1] J. Ehlers and R. Geroch. Equation of motion of small bodies in relativity.Annals of Physics, 309:232–236, 2004.[2] R. Geroch and P. S. Jang. Motion of a body in general relativity. Journalof Mathematical Physics, 16:65–67, 1975.[3] S.W. Hawking and G. F. R. Ellis. The Large Scale Structure of Space-Time.Cambridge University Press, 1972.[4] D. Malament. Lecture Notes on the Foundations of Relativity Theory.http://www.lps.uci.edu/malament/FndsofGR.html, unpublished.[5] A. H. Taub. On Thomas’ result concerning the geodesic hypothesis. Pro-ceedings of the National Academy of Sciences, 48:1570–1571, 1962.[6] T. Y. Thomas. On the geodesic hypothesis in the theory of gravitation.Proceedings of the National Academy of Sciences, 48:1567–1569, 1962.[7] R. Wald. General Relativity. University of Chicago Press, 1984.11

A Remark About the "Geodesic Principle" in General Relativity

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A Remark About the "Geodesic Principle" in General Relativity

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