A variational principle for stationary, axisymmetric solutions of Einstein's equations

TL;DR

This paper introduces a variational principle for stationary, axisymmetric vacuum solutions of Einstein's equations, linking the total mass to critical points and proposing the extreme Kerr initial data as the minimal configuration.

Contribution

It formulates a variational principle expressing the mass as a positive definite integral and relates the minimal mass solution to the extreme Kerr black hole.

Findings

Mass is expressed as a positive definite integral over a hypersurface.

The minimal mass solution, if it exists, corresponds to the extreme Kerr initial data.

Supports the conjecture that the extreme Kerr data minimizes mass among all such configurations.

Abstract

Stationary, axisymmetric, vacuum, solutions of Einstein's equations are obtained as critical points of the total mass among all axisymmetric and $(t,\phi)$ symmetric initial data with fixed angular momentum. In this variational principle the mass is written as a positive definite integral over a spacelike hypersurface. It is also proved that if absolute minimum exists then it is equal to the absolute minimum of the mass among all maximal, axisymmetric, vacuum, initial data with fixed angular momentum. Arguments are given to support the conjecture that this minimum exists and is the extreme Kerr initial data.