Stability of Schwarzshild black holes in quadratic gravity with Weyl curvature domination

TL;DR

This paper investigates the linear stability of Schwarzschild black holes in quadratic Weyl curvature gravity, revealing ghost modes and instabilities depending on the Weyl coupling strength, with implications for black hole solutions in modified gravity theories.

Contribution

It provides a detailed analysis of perturbations and stability of Schwarzschild black holes in Weyl-squared gravity, identifying ghost modes and instability regimes across coupling ranges.

Findings

Odd-parity sector has three dynamical degrees of freedom with luminal speeds.

Schwarzschild solutions are unstable for large Weyl coupling due to Laplacian instabilities.

Presence of ghost modes in both parity sectors indicates potential theoretical issues.

Abstract

We study the linear stability of static and spherically symmetric (SSS) black holes (BHs) in the presence of a Weyl-squared curvature besides an Einstein-Hilbert term in the action. In this theory, there is always an exact Schwarzschild BH irrespective of the Weyl coupling constant $\alpha$, with the appearance of a non-Schwarzschild solution for a particular range of the coupling of order $|\alpha| \approx r_h^2$ (where $r_h$ is the horizon radius). On the SSS background, we show that the propagating degrees of freedom (DOFs) are three in the odd-parity sector and four in 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 modes are 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 $|\alpha| \gg r_h^2$. 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 $|\alpha |\lesssim r_h^2$, the WKB approximation does not hold any longer, and a different analysis should be performed if one wants to state the stability of both the Schwarzschild and non-Schwarzschild BH solutions in this range of model parameters.