Perturbations of black holes in Einstein-Maxwell-dilaton-axion theories

We extend our earlier work on the linearized 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 three-parameter family of such (Einstein-Maxwell-dilaton-axion) EMDA theories. The systems of equations describing the linearized perturbations can always be separated, but they can only be decoupled when the three parameters are restricted to a one-parameter family of EMDA theories, characterized by a parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>b</a:mi> </a:math> that determines the coupling of the axion to the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>ε</c:mi> <c:mrow> <c:mi>μ</c:mi> <c:mi>ν</c:mi> <c:mi>ρ</c:mi> <c:mi>σ</c:mi> </c:mrow> </c:msup> <c:msub> <c:mi>F</c:mi> <c:mrow> <c:mi>μ</c:mi> <c:mi>ν</c:mi> </c:mrow> </c:msub> <c:msub> <c:mi>F</c:mi> <c:mrow> <c:mi>ρ</c:mi> <c:mi>σ</c:mi> </c:mrow> </c:msub> </c:math> term. In the specific case when <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>b</e:mi> <e:mo>=</e:mo> <e:mn>1</e:mn> </e:math> , the theory is related to an <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi mathvariant="script">N</g:mi> <g:mo>=</g:mo> <g:mn>2</g:mn> </g:math> supergravity. In this one case we find that the perturbations in the axial and the polar sectors are related by a remarkable transformation, which generalizes 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 nonextremal 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>b</j:mi> </j:math> 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 <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>b</l:mi> </l:math> is sufficiently large, raising the possibility of there being perturbative mode instabilities in such a case.