Gaussian coordinate systems for the Kerr metric

TL;DR

This paper develops a comprehensive class of Gaussian coordinate systems for the Kerr metric, linking observer frames to the Hamilton-Jacobi equation, and explores their properties and applications in deriving internal solutions.

Contribution

It introduces a complete set of Gaussian coordinates for the Kerr metric and connects them to the Hamilton-Jacobi equation, facilitating analysis of internal solutions.

Findings

Complete Gaussian coordinate systems for Kerr metric are constructed.

The approach simplifies the quasi-Maxwellian equations in these coordinates.

An example derivation of the Schwarzschild internal metric is provided.

Abstract

We present the whole class of Gaussian coordinate systems for the Kerr metric. This is achieved through the uses of the relationship between Gaussian observers and the relativistic Hamilton-Jacobi equation. We analyze the completeness of this coordinate system. In the appendix we present the equivalent JEK formulation of General Relativity -- the so-called quasi-Maxwellian equations -- which acquires a simpler form in the Gaussian coordinate system. We show how this set of equations can be used to obtain the internal metric of the Schwazschild solution, as a simple example. We suggest that this path can be followed to the search of the internal Kerr metric.