On the Structure of Stationary and Axisymmetric Metrics

TL;DR

This paper analyzes the structure of stationary and axisymmetric solutions to the vacuum Einstein equations, revealing a new formulation that links the metric uniquely to sources in a one-dimensional subspace, with detailed insights into four-dimensional cases like the Kerr black hole.

Contribution

It introduces a novel way to express Einstein's equations for stationary and axisymmetric metrics, identifying sources in a one-dimensional subspace and providing a detailed analysis of four-dimensional solutions.

Findings

New formulation of Einstein equations for these metrics

Unique determination of metrics from sources in a 1D subspace

Detailed analysis of Kerr black hole sources

Abstract

We study the structure of stationary and axisymmetric metrics solving the vacuum Einstein equations of General Relativity in four and higher dimensions, building on recent work in hep-th/0408141. We write the Einstein equations in a new form that naturally identifies the sources for such metrics. The sources live in a one-dimensional subspace and the entire metric is uniquely determined by them. We study in detail the structure of stationary and axisymmetric metrics in four dimensions, and consider as an example the sources of the Kerr black hole.