Static Solutions of Einstein's Equations with Spherical Symmetry

TL;DR

This paper compares various static spherically symmetric solutions of Einstein's equations, including Schwarzschild and solutions with matter, highlighting differences in coordinate definitions and special cases with altered symmetry.

Contribution

It systematically analyzes and compares different static spherically symmetric solutions, including those with matter and altered symmetry, clarifying their relationships.

Findings

Different coordinate definitions lead to varied solutions.

Solutions with non-zero energy density and pressure are examined.

Special cases with changed spherical symmetry are calculated.

Abstract

The Schwarzschild solution is a complete solution of Einstein's field equations for a static spherically symmetric field. The Einstein's field equations solutions appear in the literature, but in different ways corresponding to different definitions of the radial coordinate. We attempt to compare them to the solutions with nonvanishing energy density and pressure. We also calculate some special cases with changes in spherical symmetry.