Perturbations of Black Holes in Einstein-Maxwell-Dilaton-Axion (EMDA) Theories

TL;DR

This paper investigates linear perturbations of static black holes within Einstein-Maxwell-Dilaton-Axion (EMDA) theories, revealing special cases with supersymmetric properties and analyzing stability across different parameter regimes.

Contribution

It extends previous work to include axion fields, identifies a parameter-dependent decoupling of perturbation equations, and uncovers a supersymmetry-inspired transformation linking axial and polar perturbations.

Findings

Perturbation equations can be separated and decoupled in a 1-parameter family of EMDA theories.

A remarkable transformation relates axial and polar perturbations when the theory parameter b=1, linked to supersymmetry.

Mode stability is proven for the supersymmetric case; potential instabilities may occur for large parameter b.

Abstract

We extend our earlier work on the linearised perturbations of static black holes in Einstein-Maxwell-Dilaton (EMD) theories to the case where the black holes are solutions in an enlarged theory including also an axion. We study the perturbations in a 3-parameter family of such EMDA theories. The systems of equations describing the linearised perturbations can always be separated, but they can only be decoupled when the three parameters are restricted to a 1-parameter family of EMDA theories, characterised by a parameter $b$ that determines the coupling of the axion to the $\epsilon^{\mu\nu\rho\sigma}\, F_{\mu\nu}\, F_{\rho\sigma}$ term. In the specific case when $b=1$, the theory is related to an ${\cal N}=2$ supergravity. In this one case we find that the perturbations in the axial and the polar sectors are related by a remarkable transformation, which generalises one found by Chandrasekhar for the perturbations of Reissner-Nordstr\"om in Einstein-Maxwell theory. This transformation is of a form found in supersymmetric quantum mechanical models. The existence of such mappings between the axial and polar perturbations appears to correlate with those cases where there is an underlying supergravity supporting the solution, even though the black hole backgrounds are non-extremal and therefore not supersymmetric. We prove the mode stability of the static black hole solutions in the supersymmetric EMDA theory. For other values of the parameter $b$ in the EMDA theories that allow decoupling of the modes, we find that one of the radial potentials can be negative outside the horizon if $b$ is sufficiently large, raising the possibility of there being perturbative mode instabilities in such a case.