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

TL;DR

This work extends perturbation theory of static black holes to Einstein-Maxwell-Dilaton-Axion (EMDA) theories by including an axion coupled via $b\chi\widetilde{F}F$. The authors show that, while perturbation equations can always be separated, diagonalisation is only achievable on a 1-parameter subfamily, notably at $a=1$, $c=-2$, $b^2=1$, where axial and polar sectors are related by a supersymmetric quantum-mechanical structure and the backgrounds connect to ${\cal N}=2$ supergravity (STU truncation with an extra $H_{\mu\nu}$ field). They prove mode stability for the supersymmetric EMDA theory and identify threshold values of $b^2$ above which axial potentials can become negative outside the horizon, implying potential instabilities for larger axion couplings. For the general $b^2$, the analysis reveals a rich structure of axial potentials governed by a cubic for $p$, with precise ranges and positivity properties, highlighting how supergravity embeddings influence perturbative behavior and stability in EMDA black holes.

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 $ε^{μνρσ}\, F_{μν}\, F_{ρσ}$ 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-Nordström 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.

Figures (2)

• Figure 1: Some example plots of the axial and polar potentials $(V_1^-,V_1^+,V_2^-,V_2^+,V_3^-,V_3^+)$ corresponding to the roots $(p_1,p_2,p_3)$ ordered as $p_1<p_2<p_3$, for ${{r_{\!\!{ +}}}}= 1.5\, {{r_{\!\!{ -}}}}$, $\ell=2$, for various values of $b$, as indicated. The dimensionless ratio $r/{{r_{\!\!{ -}}}}$ is plotted from the outer horizon at $r/{{r_{\!\!{ -}}}}=1.5$ to $r/{{r_{\!\!{ -}}}}=6$ along the horizontal axis. The dimensionless quantity $V(r)\, {{r_{\!\!{ -}}}}^2$ is plotted along the vertical axis, for the various potentials $(V_1^-,V_1^+,V_2^-,V_2^+,V_3^-,V_3^+)$. The top pair of almost superimposed potentials in the $b=1$ plots correspond to the middle root $p_2$, with the red line being the axial potential $V_2^-$. The middle pair of almost superimposed $b=1$ plots correspond to the the smallest root $p_1$, with the orange line being the axial potential $V_1^-$. The bottom pair of almost superimposed $b=1$ plots correspond to the largest root $p_3$ (that is, the positive root), with the brown line being the axial potential $V_3^-$. One can see how the potentials change as $b$ is adjusted, and in particular how the axial potential $V_3^-$ becomes negative in a region outside the outer horizon when $b$ is large enough.
• Figure 2: This is the same as the middle-left panel above, showing the potentials $(V_1^-,V_1^+,V_2^-,V_2^+,V_3^-,V_3^+)$ for $b=1$, but now with the potentials $(V^-_{{ H}}, V^+_{{ H}})$ included as well. These are the pair of lines that are just above the previous middle pair $(V_1^-,V_1^+)$. The light blue line is the axial potential $V^-_{{ H}}$.