On Hawking's Local Rigidity Theorems for Charged Black Holes

TL;DR

This paper proves the existence of a Hawking vector field near bifurcate horizons in Einstein-Maxwell spacetimes without requiring analyticity, and shows axial symmetry arises under certain conditions, aiding black hole classification.

Contribution

It extends Hawking's local rigidity theorems to non-analytic Einstein-Maxwell spacetimes and establishes conditions for axial symmetry without analyticity assumptions.

Findings

Existence of a Hawking vector field near bifurcate horizons.

Conditions under which spacetime is locally axially symmetric.

Extension of rigidity results beyond analytic spacetimes.

Abstract

We show the existence of a Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein-Maxwell space-time without assuming the underlying space-time is analytic. It extends one result of Friedrich, R\'{a}cz and Wald, which was limited to the interior of the black hole region. Moreover, we also show, in the presence of an additional Killing vector field $T$ which tangent to the horizon and not vanishing on the bifurcate sphere, then space-time must be locally axially symmetric without the analyticity assumption. This axial symmetry plays a fundamental role in the classification theory of stationary black holes.