Stability of the Schwarzschild-de Sitter black hole in the dRGT massive gravity theory

Abstract

The Schwarzschild-de Sitter solution in the Einstein theory with a positive cosmological constant $\Lambda=m^2/\alpha$ becomes an exact solution to the dRGT non-linear massive gravity theory with the mass parameter $m$ when the theory parameters $\alpha$ and $\beta$ satisfy the relation $\beta=\alpha^2$. We study the perturbative behaviour of this black hole solution in the non-linear dRGT theory with $\beta=\alpha^2$. We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of the gauge-invariant variables. This implies that this black hole is stable in the dRGT theory as far as the spacetime structure is concerned in contrast to the case of the bi-Schwarzschild solution in the bi-metric theory. However, we have also found a pathological feature that the general solution to the perturbation equations contain a single arbitrary function of spacetime coordinates. This implies a degeneracy of dynamics in the St\"uckelberg field sector at the linear perturbation level in this background. Physical significance of this degenercy depends on how the St\"uckelberg fields couple observable fields.