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

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 G₃(X), where X=−A_μA^μ/2 with a vector field A_μ, 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 A₁, which originates from the coupling to the Einstein tensor G^μνA_μ A_ν equivalent to the quartic coupling G₄(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 G₄(X) ⊃ β₄ Xⁿ with the nonvanishing A₁, while the other branch with A₁=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.