Spherical Symmetry and Mass‐Energy in General Relativity. I. General Theory

TLDR: In this article, the Bianchi identity is shown to imply that the Misner-Sharp-Hernandez mass function is an integral of two combinations of Einstein's equations for any energymomentum tensor and that mass energy flow is conservative.

Abstract: The mass‐energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner‐Sharp‐Hernandez mass function is an integral of two combinations of Einstein's equations for any energy‐momentum tensor and that mass‐energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass‐function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative‐mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point‐mass solution.