Black hole perturbations in vector-tensor theories: The odd-mode analysis

TL;DR

This paper analyzes the stability of black hole solutions in generalized Proca theories with vector fields, finding conditions under which these solutions are free from ghost and Laplacian instabilities, especially focusing on odd-parity perturbations.

Contribution

It formulates the odd-parity perturbation analysis in generalized Proca theories and identifies stability conditions for various black hole solutions, including those with vector hair.

Findings

Models with cubic couplings $G_3(X)$ are stable without additional conditions.

Charged stealth Schwarzschild solutions with nonvanishing $A_1$ are unstable near the horizon.

Certain solutions with specific derivative couplings are free from ghost and Laplacian instabilities.

Abstract

In generalized Proca theories with vector-field derivative couplings, a bunch of hairy black hole solutions have been derived on a static and spherically symmetric background. In this paper, we formulate the odd-parity black hole perturbations in generalized Proca theories by expanding the corresponding action up to second order and investigate whether or not black holes with vector hair suffer ghost or Laplacian instabilities. We show that the models with cubic couplings $G_3(X)$, where $X=-A_{\mu}A^{\mu}/2$ with a vector field $A_{\mu}$, do not provide any additional stability condition as in General Relativity. On the other hand, the exact charged stealth Schwarzschild solution with a nonvanishing longitudinal vector component $A_1$, which originates from the coupling to the Einstein tensor $G^{\mu\nu}A_\mu A_\nu$ equivalent to the quartic coupling $G_4(X)$ containing a linear function of $X$, is unstable in the vicinity of the event horizon. The same instability problem also persists for hairy black holes arising from general quartic power-law couplings $G_4(X) \supset \beta_4 X^n$ with the nonvanishing $A_1$, while the other branch with $A_1=0$ can be consistent with conditions for the absence of ghost and Laplacian instabilities. We also discuss the case of other exact and numerical black hole solutions associated with intrinsic vector-field derivative couplings and show that there exists a wide range of parameter spaces in which the solutions suffer neither ghost nor Laplacian instabilities against odd-parity perturbations.