Solutions of the Einstein field equations for a bounded and finite discontinuous source, and its generalization: Metric matching conditions and jumping effects

TL;DR

This paper investigates the behavior of solutions to Einstein's equations with bounded, discontinuous sources, deriving conditions for metric jumps and applying the formalism to classical and generalized scenarios in General Relativity.

Contribution

It introduces a formalism for analyzing metric matching conditions with discontinuities and extends the analysis to non-symmetric second derivatives in Einstein's equations.

Findings

Derived general conditions for second derivative jumps across boundaries.

Applied the formalism to the Oppenheimer-Snyder metric, obtaining new results.

Extended the analysis to generalized Einstein equations with non-symmetric second derivatives.

Abstract

We consider the metrics of the General Relativity, whose energy-momentum tensor has a bounded support where it is continuous except for a finite step across the corresponding boundary surface. As a consequence, the first derivative of the metric across this boundary could perhaps present a finite step too. However, we can assume that the metric is ${\cal C}^1$ class everywhere. In such a case, although the partial second derivatives of the metric exhibit finite (no Dirac $\delta$ functions) discontinuities, the Dirac $\delta$ functions will still appear in the conservation equation of the energy-momentum tensor. As a consequence, strictly speaking, the corresponding metric solutions of the Einstein field equations can only exist in the sense of distributions. Then, we assume that the metric considered is ${\cal C}^1$ class everywhere and is a solution of the Einstein field equations in this sense. We explore the consequences of these two assumptions, and in doing so we derive the general conditions that constrain the jumps in the second partial derivatives across the boundary. The example of the Oppenheimer-Snyder metric is considered and some new results are obtained on it. Then, the formalism developed in this exploration is applied to a different situation, i.e., to a given generalization of the Einstein field equations for the case where the partial second derivatives of the metric exist but are not symmetric.