Axially symmetric static sources: A general framework and some analytical solutions

TL;DR

This paper develops a comprehensive framework for axially symmetric static sources in general relativity, deriving key equations, analyzing structure scalars, and presenting new analytical solutions including an incompressible spheroid and anisotropic configurations.

Contribution

It introduces a general formalism for axially symmetric static sources and provides new exact solutions, expanding the understanding of such configurations in Einstein's theory.

Findings

Derived Einstein and conservation equations for anisotropic static fluids with axial symmetry.

Calculated structure scalars and identified inhomogeneity factors.

Presented new analytical solutions, including an incompressible spheroid and anisotropic distributions.

Abstract

We provide all basic equations and concepts required to carry out a general study on axially symmetric static sources. The Einstein equations and the conservation equations are written down for a general anisotropic static fluid endowed with axial symmetry. The structure scalars are calculated and the inhomogeneity factors are identified. Finally some exact analytical solutions were found. One of these solutions describes an incompressible spheroid with isotropic pressure and becomes the well known interior Schwarzschild solution in the spherically symmetric limit, however it cannot be matched smoothly to any Weyl exterior metric. Another family of solutions was found that corresponds to an anisotropic fluid distribution and can in principle be matched to a Weyl exterior.