Lorentz-Invariant Equations of Motion of Point Masses in the General Theory of Relativity

Havas, Péter; Goldberg, Joshua N.

doi:10.1103/physrev.128.398

Cited by 163 publications

(73 citation statements)

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“…(38) Note that the right-hand side of this equation is evaluated on the worldline, and once evaluated, it contains a term proportional to −ȧ µ , corresponding to the antidamping phenomenon discovered by Havas [47] (as corrected by Havas and Goldberg [48]). However, my assumed expansion of the acceleration has forced the righthand side to be evaluated for a = a (0) = 0, which serves to automatically yield an "order-reduced" equation with no higher-order derivatives (assuming, of course, that a time-derivative does not change the order of a term in the expansion).…”

Section: Singular Expansionmentioning

confidence: 99%

Self-consistent gravitational self-force

2010

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I review the problem of motion for small bodies in General Relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed; I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body's past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long timescales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.

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Section: Singular Expansionmentioning

confidence: 99%

Self-consistent gravitational self-force

2010

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“…Fokker actions for point-particles in post-Minkowskian gravity Havas and Goldberg [19] derived equations of motion for point particles in general relativity by expanding the metric and demanding that the covariant conservation law for the stress-energy tensor be satisfied to first order in the perturbation, with a time-symmetric, half-advanced + half-retarded field for the first-order metric. They found a Fokker action, an action integral I for which lim δI = 0 (in the sense of Eq.…”

Section: Action At a Distance Theory For Post-minkowskian Gravitymentioning

confidence: 99%

Post-Minkowski action for point particles and a helically symmetric binary solution

Friedman

Uryū

2006

Phys. Rev. D

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Two Fokker actions and corresponding equations of motion are obtained for two point particles in a post-Minkowski framework, in which the field of each particle is given by the half-retarded + half-advanced solution to the linearized Einstein equations. The first action is parametrization invariant, the second a generalization of the affinely parametrized quadratic action for a relativistic particle. Expressions for a conserved 4-momentum and angular momentum tensor are obtained in terms of the particles' trajectories in this post-Minkowski approximation. A formal solution to the equations of motion is found for a binary system with circular orbits. For a bound system of this kind, the post-Minkowski solution is a toy model that omits nonlinear terms of relevant postNewtonian order; and we also obtain a Fokker action that is accurate to first post-Newtonian order, by adding to the post-Minkowski action a term cubic in the particle masses. Curiously, the conserved energy and angular momentum associated with the Fokker action are each finite and well-defined for this bound 2-particle system despite the fact that the total energy and angular momentum of the radiation field diverge. Corresponding solutions and conserved quantities are found for two scalar charges (for electromagnetic charges we exhibit the solution found by Schild). For a broad class of parametrization-invariant Fokker actions and for the affinely parametrized action, binary systems with circular orbits satisfy the relation dE = ΩdL (a form of the first law of thermodynamics), relating the energy, angular velocity and angular momentum of nearby equilibrium configurations.

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“…In particular, in (1937,1940) Mathisson casually shows that applying the principle only to the lowest order term in the expansion (because such a tensor eld might be representative of a spinless point particle ) entails the geodesic equation. Though Mathisson does not make explicit use of distributions in these 0 th -order derivations, such an appeal can naturally be read into this technique as was done later by Havas and Goldberg (1962). See also (Tulczyjew, 1959) for a distributional reconstruction of Mathisson's work.…”

Section: Pressure Treating the Wood: Distributional Energy-momentummentioning

confidence: 99%

Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

Tamir

2012

Studies in History and Philosophy of Science Part B: Studies in

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In his initial formulation of the general theory of relativity, Einstein's proposal that freely falling gravitating massive bodies follow geodesic paths was submitted as an independent fundamental principle. By adopting this geodesic principle to supply the theory's law of motion, Einstein was immediately able to recover both the free-fall motion of bodies in non-relativistic regimes and the previously anomalous precession of the perihelion of Mercury. Over the last century numerous ostensible proofs claiming to have derived the geodesic principle from Einstein's eld equations have been developed. As a result physicists and philosophers of science alike frequently herald Einstein's theory for having the unique distinction of being able to derive its dynamical law of motion from its own eld equations.In this paper I critically survey the multiple attempts to derive the geodesic principle in the context of Einstein's theory. Grouping these results into three major families, which I refer to as (1) limit operation proofs, (2) 0 th -order proofs, and (3) singularity proofs, I argue that none of these strategies successfully demonstrates the geodesic principle, canonically interpreted as a dynamical law that massive bodies must actually follow geodesic paths in Einstein's theory.Speci cally, I argue for the following three claims: First, limit operation proofs fail to demonstrate that massive bodies are ever guaranteed to follow geodesic paths. Second, on the contrary 0 thorder proofs demonstrate that extended massive bodies generically deviate from uniformly geodesic paths. Moreover, the only potentially extended distributions of matter and energy that fail to avoid a uniform geodesic evolution are highly unstable, deviating from such motion under arbitrary perturbations of their angular momentum (or higher order moments). Third, thanks to certain mathematical theorems concerning distribution theory, alternative representations of massive bodies as unextended point particles must result either in precluding the possibility of coupling the particle to the spacetime metric in a way that is coherent with Einstein's eld equations or in having to excise the particle (and its would-be path) from spacetime entirely. This three pronged argument reveals that not only does the geodesic law of motion fail to be a deductive consequence of the eld equations, but also any attempt to canonically interpret the geodesic principle in such a way requires * In the following, M will be taken to be a smooth, orientable, four-dimensional manifold, and (M, g ab ) will be referred to as a Lorentzian spacetime if g ab is a smooth metric of signature (+, −, −, −) de ned on M. Excepting quoted material all further notational conventions follow that of (Wald, 1984).1 Forthcoming: Studies in History and Philosophy of Modern Physics that either the gravitating body is not massive, its existence violates Einstein's eld equations, or it does not exist within the spacetime manifold at all (let alone along a geodesic).Having rejected the canonical in...

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Lorentz-Invariant Equations of Motion of Point Masses in the General Theory of Relativity

Cited by 163 publications

References 50 publications

Self-consistent gravitational self-force

Self-consistent gravitational self-force

Post-Minkowski action for point particles and a helically symmetric binary solution

Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

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