Nonmetrical Specification of Space-Time Sources

Abstract

The role played by the energy-momentum conservation law in general relativity is examined. It is noted that this law can be interpreted in two ways. It may be thought of as a condition determining the evolution of the energy-momentum tensor density in time, or it may be thought of as a condition determining the metric. In the present paper, the second of these ways of thinking about the energy-momentum conservation law is explored. Einstein's nonvacuum gravitational field equations (which imply the conservation law) are examined. It is shown that given any analytic symmetric contravariant energy-momentum tensor density as a function of the space-time coordinates, a solution to the gravitational field equations always exists. Furthermore, this solution is such that the law of conservation of energy-momentum is satisfied. The proof uses a coordinate transformation method to exploit the covariance of the energy-momentum conservation law. Riquier's existence theorem enters as an important part of the proof, and a general discussion of Riquier's existence theorem from a physical point of view is given. Both the geodesic nature of the trajectories of free particles and the unit magnitude of the velocity 4-vector are discussed. An interpretation of the above-described results is given.