Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes

TL;DR

<3-5 sentence high-level summary>The paper develops a framework to study perturbations of near-horizon geometries of extreme vacuum black holes by performing a Kaluza-Klein reduction to AdS$_2$ with a homogeneous electric field, yielding a charged scalar equation whose effective mass is determined by eigenvalues on the internal space ${\mathcal{H}}$. It introduces an effective BF bound and a symmetry-based conjecture: if NH perturbations violate this bound under certain symmetry conditions, the full black hole is unstable; it provides substantial evidence from extreme Kerr and cohomogeneity-1 Myers-Perry spacetimes, including analytic results and near-extremal numerical comparisons, and it sketches a scalar-field proof of the conjecture. The work further explores conformal weights and Kerr-CFT implications, showing axisymmetric perturbations have integer weights and predicting instability in $d\ge 7$ MP spacetimes for symmetry-preserving modes. Collectively, the results offer a practical criterion to anticipate instabilities of extreme black holes via NH analysis and deepen the connection between NH dynamics and full spacetime stability.

Abstract

It is shown that the equations governing linearized gravitational (or electromagnetic) perturbations of the near-horizon geometry of any known extreme vacuum black hole (allowing for a cosmological constant) can be Kaluza-Klein reduced to give the equation of motion of a charged scalar field in AdS_2 with an electric field. One can define an effective Breitenlohner-Freedman bound for such a field. We conjecture that if a perturbation preserves certain symmetries then a violation of this bound should imply an instability of the full black hole solution. Evidence in favour of this conjecture is provided by the extreme Kerr solution and extreme cohomogeneity-1 Myers-Perry solution. In the latter case, we predict an instability in seven or more dimensions and, in 5d, we present results for operator conformal weights assuming the existence of a CFT dual. We sketch a proof of our conjecture for scalar field perturbations.

Figures (2)

• Figure 1: Lowest eigenvalues of ${\mathcal{O}^{(2)}}$ plotted against the size of the $AdS$ black hole ($r_+^2/l^2$), in dimensions $d=7$ (left) and $d=9$ (right). The shaded region corresponds to violation of the effective BF bound. The separate curves shown correspond to ${\kappa}=2,3,4,5,6$, moving from left to right as ${\kappa}$ is increased (the curves that are negative for $r_+/l \rightarrow 0$ are ${\kappa}=2$ on the left and ${\kappa}=2,3$ on the right). In both cases, there is some mode that violates the BF bound for any black hole size.
• Figure 2: Eigenvalues of ${\mathcal{O}^{(2)}}$ plotted against the size of the $AdS$ black hole ($r_+^2/l^2$), in dimension $d=5$ (the right hand graph is a zoomed version of the left hand one). The separate curves shown correspond to ${\kappa}=2,3,4,5,6$, moving from left to right as ${\kappa}$ is increased. We find that the generalized BF bound ${\lambda} \geq-1/4$, shown by the horizontal line, is violated by a small amount in various small ranges of black hole size. As ${\kappa}$ is increased, the violation of the BF bound occurs for increasingly large black holes.