Even-parity black hole perturbations in Minimal Theory of Bigravity

TL;DR

This paper investigates even-parity black hole perturbations within Minimal Theory of Bigravity, analyzing gauge transformations, mode behaviors, and propagating modes for Schwarzschild solutions in a specific theoretical framework.

Contribution

It provides a detailed analysis of even-parity perturbations in MTBG, clarifies gauge transformations, and identifies propagating and instantaneous modes for Schwarzschild black holes.

Findings

$ ext{l}=0$ perturbations can be absorbed by background redefinition.

$ ext{l}=1$ perturbations vanish under regularity conditions.

For $ ext{l} extgreater 1$, two propagating and four instantaneous modes are found.

Abstract

We study even-parity black hole perturbations in Minimal Theory of Bigravity (MTBG). We consider the Schwarzschild solution written in the spatially-flat coordinates in the self-accelerating branch as the background solution. We clarify the gauge transformations for the $\ell=0$, $1$, and $\geq2$ modes with $\ell$ being the angular multipole moments under the joint foliation-preserving diffeomorphism transformation. Requiring that the asymptotic regions in the physical and fiducial sectors share the same Minkowski vacua, the solution to the $\ell=0$ perturbations can be absorbed by a redefinition of the Schwarzschild background. In order to analyze the $\ell=1$ and $\geq 2$ modes, for simplicity we focus on the effectively massless case, where the constant parameter measuring the ratio of the proper times between the two sectors is set to unity and the effective mass terms in the equations of motion vanish. We also find that as a particular solution all the $\ell=1$ perturbations vanish by imposing their regularity at spatial infinity. For each of the $\ell\geq 2$ modes, in the effectively massless case, we highlight the existence of the expected two propagating modes and four instantaneous modes.