Black hole perturbation in the most general scalar-tensor theory with second-order field equations I: The odd-parity sector

TL;DR

This paper analyzes odd-parity linear perturbations in the most general scalar-tensor theories with second-order equations, deriving stability conditions applicable to various gravity models like Brans-Dicke, $f(R)$, and Galileon theories.

Contribution

It provides a unified relativistic framework for analyzing odd-parity perturbations in general scalar-tensor theories, deriving concise stability criteria.

Findings

Quadratic action reduces to a single dynamical variable.

Derived explicit no-ghost and no-gradient instability conditions.

Applicable to multiple gravity theories including Brans-Dicke, $f(R)$, and Galileon.

Abstract

We perform a fully relativistic analysis of odd-type linear perturbations around a static and spherically symmetric solution in the most general scalar-tensor theory with second-order field equations in four-dimensional spacetime. It is shown that, as in the case of general relativity, the quadratic action for the perturbations reduces to the one having only a single dynamical variable, from which concise formulas for no-ghost and no-gradient instability conditions are derived. Our result is applicable to all the theories of gravity with an extra scalar degree of freedom. We demonstrate how the generic formulas can be applied to some particular examples such as the Brans-Dicke theory, $f(R)$ models, and Galileon gravity.