Reflection Symmetry in Higher Dimensional Black Hole Spacetimes

TL;DR

This paper proves that higher-dimensional, asymptotically flat, stationary, and axisymmetric vacuum black hole spacetimes possess a reflection symmetry reversing time and angular coordinates, extending known 4D results.

Contribution

It generalizes the existence of $t$-$$ reflection isometries to higher dimensions under specific symmetry and trivial action conditions.

Findings

Existence of $t$-$$ reflection isometry in higher dimensions.

The proof uses the first law of black hole mechanics.

Applicable to vacuum black hole spacetimes with trivial isometry group action.

Abstract

In 4 spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein's equation there exists a "$t$-$\phi$" reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in $d \geq 4$ spacetime dimensions such that the action of the isometry group is trivial. (Here "trivial" means that if the "axes"---i.e., the points where the axial Killing fields are linearly dependent---are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a "$t$-$\phi$" reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.