Stability of Schwarzshild black holes in quadratic gravitywith Weyl curvature domination

Abstract

We study the linear stability of static and spherically symmetric (SSS)black holes (BHs) in the presence of a Weyl-squared curvaturebesides an Einstein-Hilbert term in the action.In this theory, there is always an exact Schwarzschild BHirrespective of the Weyl coupling constant α,with the appearance of a non-Schwarzschild solutionfor a particular range of the coupling oforder |α| ≈ r h 2 (where r h isthe horizon radius).On the SSS background, we show that the propagating degreesof freedom (DOFs) are three in the odd-parity sector and fourin the even-parity sector. Since the number of total seven DOFs coincides with those on the Minkowski and isotropic cosmological backgrounds, the Weyl gravity does not pose a strong coupling problem associated with the vanishing kinetic term of dynamical perturbations.The odd-parity perturbations possess at least one ghost mode,but the propagation speeds of all three dynamical modesare luminal.In the even-parity sector, our analysis, based on the WKB approximation, shows that, besides the appearance of at least one ghost mode, the Schwarzschild solution is prone to both radial and angular Laplacian instabilities of several dynamical perturbations for the Weyl coupling in the range |α| ≫ r h 2.For large radial and angular momentum modes, the time scales of such instabilities are much shorter than the horizon distance r h divided by the speed of light.In the coupling regime |α|≲ r h 2, the WKB approximation does not hold any longer, and a different analysis should be performedif one wants to state the stability of both the Schwarzschild andnon-Schwarzschild BH solutions in this range of model parameters.