A Survey of Spherically Symmetric Spacetimes

TL;DR

This survey comprehensively reviews the properties and descriptions of spherically symmetric spacetimes, emphasizing the polar-areal coordinate form and its applications in solving Einstein equations.

Contribution

It provides a detailed overview of various descriptions of spherically symmetric spacetimes and highlights the utility of polar-areal coordinates in theoretical and computational contexts.

Findings

Polar-areal coordinates align well with Newtonian notions.

Monotonicity of Hawking mass in these coordinates.

Application to Einstein-Klein-Gordon equations.

Abstract

We survey many of the important properties of spherically symmetric spacetimes as follows. We present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-areal coordinates and its properties. In particular, we show how the metric component functions chosen are extremely compatible with notions in Newtonian mechanics. We also show the monotonicity of the Hawking mass in these coordinates. As an example, we discuss how these coordinates and the metric can be used to solve the spherically symmetric Einstein-Klein-Gordon equations. We conclude with a brief mention of some applications of these properties.