A Proof for a Theorem of Wald in Arbitrary Dimensions

TL;DR

This paper provides an elementary proof of Wald's theorem in arbitrary dimensions, establishing the integrability of certain subspaces crucial for deriving the most general black hole solutions in higher-dimensional general relativity.

Contribution

It offers a simplified, elementary proof of Wald's theorem applicable to any number of dimensions, enabling the derivation of the general metric ansatz for D-dimensional vacuum Einstein solutions with multiple symmetries.

Findings

Elementary proof of Wald's theorem in arbitrary dimensions

Derivation of the most general metric ansatz for D-dimensional black holes

Extension of integrability results to higher-dimensional spacetimes

Abstract

Static, axisymmetric solutions form a large class of important black holes in classical GR. In four dimensions, the existence of their most general metric ansatz relies on the fact that two-dimensional subspaces of the tangent space at each point spanned by vectors orthogonal to the time-translation and rotation Killing fields are integrable. This was first proved by Wald via an application of Frobenius theorem. In this note, we furnish an elementary proof for this theorem by Wald in arbitrary dimensions which yields the metric ansatz for the most general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal and commuting Killing vector fields.