The Dynamics of General Relativity

This article--summarizing the authors' then novel formulation of General Relativity--appeared as Chapter 7 of an often cited compendium edited by L. Witten in 1962, which is now long out of print. Intentionally unretouched, this posting is intended to provide contemporary accessibility to the flavor of the original ideas. Some typographical corrections have been made: footnote and page numbering have changed--but not section nor equation numbering etc. The authors' current institutional affiliations are encoded in: [email protected], [email protected], [email protected] .Comment: 30 pages (LaTeX2e), uses amsfonts, no figure

Coordinate Invariance and Energy Expressions in General Relativity

The invariance of various definitions proposed for the energy and momentum of the gravitational field is examined. We use the boundary conditions that g_(μν) approaches the Lorentz metric as 1/r, but allow g_(μν,α) to vanish as slowly as 1/r. This permits "coordinate waves." It is found that none of the expressions giving the energy as a two-dimensional surface integral are invariant within this class of frames. In a frame containing coordinate waves they are ambiguous, since their value depends on the location of the surface at infinity (unlike the case where g_(μν,α) vanishes faster than 1/r). If one introduces the prescription of space-time averaging of the integrals, one finds that the definitions of Landau-Lifshitz and Papapetrou-Gupta yield (equal) coordinate-invariant results. However, the definitions of Einstein, Møller, and Dirac become unambiguous, but not invariant. The averaged Landau-Lifshitz and Papapetrou-Gupta expressions are then shown to give the correct physical energy-momentum as determined by the canonical formulations Hamiltonian involving only the two degrees of freedom of the field. It is shown that this latter definition yields that inertial energy for a gravitational system which would be measured by a nongravitational apparatus interacting with it. The canonical formalism's definition also agrees with measurements of gravitational mass by Newtonian means at spacial infinity. It is further shown that the energy-momentum may be invariantly calculated from the asymptotic form of the metric field at a fixed time

Consistency of the Canonical Reduction of General Relativity

The question of consistency of the canonical reduction of general relativity (obtained by eliminating constraints and also imposing coordinate conditions in the action or generator) is examined. It is shown that the equations of motion obtained from this "reduced" formalism agree with the original Einstein equations. Agreement is also established for the generators of space‐time translations. In order to establish consistency, it is necessary to discard certain well‐defined divergence terms in the original Lagrangian. These would otherwise appear as nondivergences in the reduced Lagrangian, incorrectly altering the equations

Wave Zone in General Relativity

It is shown that in general relativity a "wave zone" may be defined for systems which are asymptotically flat. In this region, gravitational radiation propagates freely, independent of its interior sources, and obeys the superposition principle. The independent dynamical variables of the full theory which describe the radiation are shown to be coordinate invariant in the wave zone and to satisfy the linearized theory's equations there. Thus, the basic properties of free waves in linear field theories (e.g., electrodynamics) are reproduced for the gravitational case. True waves are also clearly distinguished from so-called "coordinate waves." Reduction to asymptotic form (taking leading powers of 1/r ), is not identical to linearization, since, for example, the Newtonian-like 1/r part of the metric begins quadratically in the linear theory's variables. The Poynting vector of the full theory, which measures energy flux in the wave zone, is correspondingly shown to be given by the linearized theory's formula. This Poynting vector is also shown to be coordinate-invariant in the wave zone. All the physical quantities may therefore be evaluated in any frame becoming rectangular sufficiently rapidly. A brief discussion of measurements of the canonical variables in the wave zone is given. The relation between the present work and other treatments of gravitational radiation is examined

Dynamical Structure and Definition of Energy in General Relativity

The problem of the dynamical structure and definition of energy for the classical general theory of relativity is considered on a formal level. As in a previous paper, the technique used is the Schwinger action principle. Starting with the full Einstein Lagrangian in first order Palatini form, an action integral is derived in which the algebraic constraint variables have been eliminated. This action possesses a "Hamiltonian" density which, however, vanishes due to the differential constraints. If the differential constraints are then substituted into the action, the true, nonvanishing Hamiltonian of the theory emerges. From an analysis of the equations of motion and the constraint equations, the two pairs of dynamical variables which represent the two independent degrees of freedom of the gravitational field are explicitly exhibited. Four other variables remain in theory; these may be arbitrarily specified, any such specification representing a choice of coordinate frame. It is shown that it is possible to obtain truly canonical pairs of variables in terms of the dynamical and arbitrary variables. Thus a statement of the dynamics is meaningful only after a set of coordinate conditions have been chosen. In general, the true Hamiltonian will be time dependent even for an isolated gravitational field. There thus arises the notion of a preferred coordinate frame, i.e., that frame in which the Hamiltonian is conserved. In this special frame, on physical grounds, the Hamiltonian may be taken to define the energy of the field. In these respects the situation in general relativity is analogous to the parametric form of Hamilton's principle in particle mechanics

Interior Schwarzschild Solutions and Interpretation of Source Terms

The solutions of the Einstein field equations, previously used in deriving the self-energy of a point charge, are shown to be nonsingular in a canonical frame, except at the position of the particle. A distribution of "dust" of finite extension is examined as the model whose limit is the point particle. The standard "proper rest-mass density" is related to the bare rest-mass density. The lack of singularity of the initial metric The solutions of the Einstein field equations, previously used in deriving the self-energy of a point charge, are shown to be nonsingular in a canonical frame, except at the position of the particle. A distribution of "dust" of finite extension is examined as the model whose limit is the point particle. The standard "proper rest-mass density" is related to the bare rest-mass density. The lack of singularity of the initial metric g_(µν) is in contrast to the Schwarzschild type singularity of standard coordinate systems. Our solutions for the extended source are nonstatic in general, corresponding to the fact that a charged dust is not generally in equilibrium. However, the solutions become static in the point limit for all values of the bare-source parameters. Similarly, the self-stresses vanish for the point particle. Thus, a classical point electron is stable, the gravitational interaction cancelling the electrostatic self-force, without the need for any extraneous "cohesive" forces

Coordinate Invariance and Energy Expressions in General Relativity

The invariance of various definitions proposed for the energy and momentum of the gravitational field is examined. We use the boundary conditions that g_(μν) approaches the Lorentz metric as 1/r, but allow g_(μν,α) to vanish as slowly as 1/r. This permits "coordinate waves." It is found that none of the expressions giving the energy as a two-dimensional surface integral are invariant within this class of frames. In a frame containing coordinate waves they are ambiguous, since their value depends on the location of the surface at infinity (unlike the case where g_(μν,α) vanishes faster than 1/r). If one introduces the prescription of space-time averaging of the integrals, one finds that the definitions of Landau-Lifshitz and Papapetrou-Gupta yield (equal) coordinate-invariant results. However, the definitions of Einstein, Møller, and Dirac become unambiguous, but not invariant. The averaged Landau-Lifshitz and Papapetrou-Gupta expressions are then shown to give the correct physical energy-momentum as determined by the canonical formulations Hamiltonian involving only the two degrees of freedom of the field. It is shown that this latter definition yields that inertial energy for a gravitational system which would be measured by a nongravitational apparatus interacting with it. The canonical formalism's definition also agrees with measurements of gravitational mass by Newtonian means at spacial infinity. It is further shown that the energy-momentum may be invariantly calculated from the asymptotic form of the metric field at a fixed time

Interior Schwarzschild Solutions and Interpretation of Source Terms

The solutions of the Einstein field equations, previously used in deriving the self-energy of a point charge, are shown to be nonsingular in a canonical frame, except at the position of the particle. A distribution of "dust" of finite extension is examined as the model whose limit is the point particle. The standard "proper rest-mass density" is related to the bare rest-mass density. The lack of singularity of the initial metric The solutions of the Einstein field equations, previously used in deriving the self-energy of a point charge, are shown to be nonsingular in a canonical frame, except at the position of the particle. A distribution of "dust" of finite extension is examined as the model whose limit is the point particle. The standard "proper rest-mass density" is related to the bare rest-mass density. The lack of singularity of the initial metric g_(µν) is in contrast to the Schwarzschild type singularity of standard coordinate systems. Our solutions for the extended source are nonstatic in general, corresponding to the fact that a charged dust is not generally in equilibrium. However, the solutions become static in the point limit for all values of the bare-source parameters. Similarly, the self-stresses vanish for the point particle. Thus, a classical point electron is stable, the gravitational interaction cancelling the electrostatic self-force, without the need for any extraneous "cohesive" forces

Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem

The gravitational effect on the classical Coulomb self-energy of a point charge is calculated rigorously. It is shown that the total mass then becomes finite (although still quite large), and that it depends only on the charge and not on the bare mechanical mass. Thus, a particle acquires mass only when it has nongravitational interactions with fields of nonzero range. In order to treat this problem, it is necessary to extend the canonical formalism, previously obtained for the free gravitational field, to include coupling with the Maxwell field and the point charge system. It is shown that the canonical variables of the gravitational field are unaltered while those of the matter system are natural generalizations of their flat space forms. The determination of the total energy of a state can still be made from knowledge of the spatial metric at a given time. The self-mass of a particle is then the total energy of a pure one-particle state, i.e., a state containing no excitations of the canonical variables of the Maxwell or Einstein fields. Solutions corresponding to pure particle states of two like charges are also obtained, and their energy is shown consistent with the one-particle results