Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations

TL;DR

This paper proves that all vacuum stationary-axisymmetric black holes in general relativity are stable against axisymmetric electromagnetic perturbations, including those with a positive cosmological constant, under certain conditions.

Contribution

It establishes the positive definiteness of the canonical energy for axisymmetric electromagnetic perturbations, demonstrating stability of a broad class of black holes in vacuum general relativity.

Findings

Vacuum black holes are stable to axisymmetric electromagnetic perturbations.

Results extend to asymptotically deSitter black holes with positive cosmological constant.

Stability holds for extremal black holes if initial perturbations vanish near the horizon.

Abstract

We consider arbitrary stationary and axisymmetric black holes in general relativity in $(d +1)$ dimensions (with $d \geq 3$) that satisfy the vacuum Einstein equation and have a non-degenerate horizon. We prove that the canonical energy of axisymmetric electromagnetic perturbations is positive definite. This establishes that all vacuum black holes are stable to axisymmetric electromagnetic perturbations. Our results also hold for asymptotically deSitter black holes that satisfy the vacuum Einstein equation with a positive cosmological constant. Our results also apply to extremal black holes provided that the initial perturbation vanishes in a neighborhood of the horizon.