Linear stability of vector Horndeski black holes

Abstract

Horndeski's vector-tensor (HVT) gravity is described bya Lagrangian in which the field strength fμν = ∂μAν-∂νAμ of a vector field Aμ interacts with a double dualRiemann tensor Lμναβ in the form βLμναβ Fμν Fαβ ,where β is a constant.In Einstein-Maxwell-HVT theory, there are static andspherically symmetric black hole (BH) solutions withelectric or magnetic charges, whose metric componentsare modified from those in the Reissner-Nordström geometry. The electric-magnetic duality of solutions is broken even at the background level by the nonvanishing coupling constant β.We compute a second-order action of BH perturbationscontaining both the odd- and even-parity modes andshow that there are four dynamical perturbationsarising from the gravitational and vector-field sectors.We derive all the linear stability conditions associatedwith the absence of ghosts and radial/angular Laplacianinstabilities for both the electric and magnetic BHs.These conditions exhibit the difference between theelectrically and magnetically charged casesby reflecting the breaking of electric-magnetic dualityat the level of perturbations.In particular, the four angular propagation speedsin the large-multipole limit are different from eachother for both the electric and magnetic BHs.This suggests the breaking of eikonal correspondencebetween the peak position of at least one of the potentials of dynamical perturbations and the radiusof photon sphere.For the electrically and magnetically charged cases,we elucidate parameter spaces of the HVT coupling and the BH charge in which the BHs without naked singularities arelinearly stable.