Revisiting linear stability of black hole odd-parity perturbations in Einstein-Aether gravity

TL;DR

This paper investigates the linear stability of black holes against odd-parity perturbations in Einstein-Aether gravity, emphasizing the importance of choosing an Aether-orthogonal frame and analyzing propagation speeds and ghost conditions.

Contribution

It introduces an Aether-orthogonal frame for stability analysis and derives conditions for black hole stability in Einstein-Aether gravity.

Findings

No-ghost conditions match Minkowski perturbations.

Radial and angular propagation speeds are consistent with vector and tensor modes.

Stability depends on Aether derivative coupling coefficients.

Abstract

In Einstein-Aether gravity, we revisit the issue of linear stabilities of black holes against odd-parity perturbations on a static and spherically symmetric background. In this theory, superluminal propagation is allowed and there is a preferred timelike direction along the unit Aether vector field. If we choose the usual spherically symmetric background coordinates with respect to the Killing time $t$ and the areal radius $r$, it may not be appropriate for unambiguously determining the black hole stability because the constant $t$ hypersurfaces are not necessarily always spacelike. Unlike past related works of black hole perturbations, we choose an Aether-orthogonal frame in which the timelike Aether field is orthogonal to spacelike hypersurfaces over the whole background spacetime. In the short wavelength limit, we show that no-ghost conditions as well as radial and angular propagation speeds coincide with those of vector and tensor perturbations on the Minkowski background. Thus, the odd-parity linear stability of black holes for large radial and angular momentum modes is solely determined by constant coefficients of the Aether derivative couplings.