General solutions of Einstein's spherically symmetric gravitational equations with junction conditions

TL;DR

This paper derives the most general solutions to Einstein's spherically symmetric equations with junction conditions, including anisotropic fluids and exotic configurations, providing a comprehensive framework for modeling spherical gravitational systems.

Contribution

It presents a unified method to solve Einstein's equations with junction conditions for spherically symmetric bodies, including anisotropic and exotic solutions, extending previous work.

Findings

Existence of a total mass function $M(r,t)$ is rigorously proved.

General solutions satisfying junction conditions are obtained for anisotropic fluids.

Various exotic solutions, including gravitational instantons and higher-dimensional models, are explored.

Abstract

Einstein's spherically symmetric interior gravitational equations are investigated. Following Synge's procedure, the most general solution of the equations is furnished in case $T^{1}_{1}$ and $T^{4}_{4}$ are prescribed. The existence of a total mass function, $M(r,t)$, is rigorously proved. Under suitable restrictions on the total mass function, the Schwarzschild mass $M(r,t)=m$, implicitly defines the boundary of the spherical body as $r=B(t)$. Both Synge's junction conditions as well as the continuity of the second fundamental form are examined and solved in a general manner. The weak energy conditions for an \emph{arbitrary boost} are also considered. The most general solution of the spherically symmetric anisotropic fluid model satisfying both junction conditions is furnished. In the final section, various exotic solutions are explored using the developed scheme including gravitational instantons, interior $T$-domains and $D$-dimensional generalizations.