On the effective metric of axial black hole perturbations in DHOST gravity

TL;DR

This paper investigates axial black hole perturbations in shift-symmetric DHOST theories, revealing that their dynamics can be described by an effective metric, often related to the background metric via disformal transformations, with implications for black hole properties.

Contribution

The study extends the first-order formulation to DHOST theories and shows that axial perturbations are governed by an effective metric, generalizing previous results from Horndeski theories.

Findings

Axial perturbations in DHOST are equivalent to GR perturbations in an effective metric.

In quadratic DHOST, the effective metric is obtained via a disformal transformation of the background metric.

Examples include stealth solutions with shifted horizons and solutions with naked singularities.

Abstract

We study axial (or odd-parity) perturbations about static and spherically symmetric hairy black hole (BH) solutions in shift-symmetric DHOST (Degenerate Higher-Order Scalar-Tensor) theories. We first extend to the family of DHOST theories the first-order formulation that we recently developed for Horndeski theories. Remarkably, we find that the dynamics of DHOST axial perturbations is equivalent to that of axial perturbations in general relativity (GR) evolving in a, distinct, effective metric. In the particular case of quadratic DHOST theories, this effective metric is derived from the background BH metric via a disformal transformation. We illustrate our general study with three examples of BH solutions. In some so-called stealth solutions, the effective metric is Schwarzschild with a shifted horizon. We also give an example of BH solution for which the effective metric is associated with a naked singularity.