Structure and evolution of self-gravitating objects and the orthogonal splitting of the Riemann tensor

TL;DR

This paper derives a set of scalar equations from the orthogonal splitting of the Riemann tensor to describe the structure and evolution of self-gravitating, anisotropic, dissipative fluids in general relativity, providing explicit solutions in static cases.

Contribution

It introduces a novel scalar-based formalism for analyzing self-gravitating fluids, linking geometric quantities to physical properties and enabling explicit solution expressions.

Findings

Scalar quantities relate directly to physical properties of the fluid.

Explicit solutions are obtainable in static cases.

The formalism simplifies the Einstein equations for these systems.

Abstract

The full set of equations governing the structure and the evolution of self--gravitating spherically symmetric dissipative fluids with anisotropic stresses, is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity. It is shown that these scalars are directly related to fundamental properties of the fluid distribution, such as: energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux and the active gravitational mass. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through these scalars. Some solutions are exhibited to illustrate this point.