Black hole perturbation in parity violating gravitational theories

TL;DR

This paper analyzes linear perturbations of static, spherically symmetric black holes in parity-violating gravitational theories, revealing potential instabilities and identifying conditions for stable, propagating modes that can be tested observationally.

Contribution

It provides a detailed stability analysis of black hole perturbations in parity-violating theories and derives conditions for stability and mode propagation, highlighting distinctive observational signatures.

Findings

Hamiltonian not bounded from below in general, indicating instability

Stable theories have three propagating modes for a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 modes for a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 a0a0 modes, with distinctive coupling features.

Abstract

We study linear perturbations around the static and spherically symmetric spacetime for the gravitational theories whose Lagrangian depends on Ricci scalar and the parity violating Chern-Simons term. By an explicit construction, we show that Hamiltonian for the perturbation variables is not bounded from below in general, suggesting that such a background spacetime is unstable against perturbations. This gives a strong limit on a phenomenological gravitational model which violates parity. We also provide a necessary and sufficient condition for the theory to belong to a special class in which no such instability occurs. For such theories, the number of propagating modes for $\ell \ge 2$ is three, one from the odd and the other two from the even. Unlike in the case of $f(R)$ theories, those modes are coupled each other, which can be used as a distinctive feature to test the parity violating theories from observations. All the modes propagate at the speed of light. No-ghost condition and no-tachyon condition are the same as those in $f(R)$ theories. For the dipole perturbations, the odd and the even modes completely decouple. The odd mode gives a slowly-rotating BH solution whose metric is linearized in its angular momentum. We provide an integral expression of such a solution. On the other hand, the even mode propagates at the speed of light. For the monopole perturbation, in addition to a mode which just shifts the mass of the background BH, there is also one even mode that propagates at the speed of light.