LORENTZ-INVARIANT EQUATIONS OF MOTION OF POINT MASSES IN THE GENERAL THEORY OF RELATIVITY

Journal Article · Mon Oct 01 04:00:00 EDT 1962 · Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D

DOI:https://doi.org/10.1103/PhysRev.128.398· OSTI ID:4778432

Havas, P; Goldberg, J N

After a general discussion of the problem of motion in the general theory of relativity a simple derivation of the law of motion is given for single poles of the gravitational field, which is based on a method originally developed by Mathisson. This law follows from the covariant conservation law for the matter energy-momentum tensor alone, without reference to any field equations, and takes the form of a geodesic of the (unknown) metric. Expanding this metric in terms of a power series in a parameter gamma and using the Minkowski proper time to parametrize the world lines of the particles, the (Lorentz-invariant) form of the approximate laws of motion follows. A method is developed to obtain the equations of motion (including the explicit form of the metric in terms of the particle variables) from Einsiein's field equations. A systematic linearization procedure leads to a series of second-order differential equations for the metric; the nth order approximation of the equations of motion, as well as the explicit form of the matter tensor in (n + l)st order, is obtained as an integrability condition on the (n + l)st order opproximation for the metric. No coordinate conditions are required to obtain the general form of the equations of motion; they are needed only to reduce the approximation equations to wave equations and thus to allow their explicit integration in terms of retarded or symmetric potentials. In developing the approximation method it is shown that consistency requires that any set of approximate equations is solved up to'' rather than in'' nth order; this implies that the form of the lower-order metric be maintained, but the motion corresponding to the nth order solutions rather than to lower order ones. In particular, the equations for the first- order metric imply zero-order equations of motion which restrict the particles to zero acceleration; the equations for the second-order metric imply first-order equation of motion involving the first-order metric, but without the previous restriction. In the retarded case the equations of motion contain retarded interactions and radiation terms of the form familiar from electrodynamics; no such terms appear in the symmetric case. The equations of the symmetric case are derivable from a Fokker-type variational principle. The relation of the results obtained to work on Lorentz-invariant equations by other autnors is discussed. In Appendix I a discussion of alternative derivations is prescnted; Appendix II contains remarks on Wheeler-Feynman type considerations for general relativistic equations of motion. (auth)

• Research Organization: Lehigh Univ., Bethlehem, Penna.
• NSA Number: NSA-16-031189
• OSTI ID: 4778432
• Journal Information: Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D, Journal Name: Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D Vol. Vol: 128; ISSN PHRVA
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

CONSERVATION LAWS
DIFFERENTIAL EQUATIONS
EINSTEIN FIELD EQUATIONS
ENERGY
EQUATIONS
FIELD THEORY
GRAVITATION
INTERACTIONS
MOMENTUM
MOTION
PARTICLES
PHYSICS
RADIATIONS
RELATIVITY THEORY
TENSORS
VARIATIONS
VECTORS