Gravitational Perturbations of Higher Dimensional Rotating ...

Gravitational Perturbations of Higher Dimensional Rotating ...

Gravitational Perturbations of Higher Dimensional Rotating Black Holes Harvey Reall University of Nottingham Collaborators: Hari Kunduri, James Lucietti Motivation 1

Compare D=5 black ring with Myers-Perry black hole with single angular momentum. Extremal MP solution is nakedly singular. Black ring has greater A entropy than near-extremal

MP. Is MP black hole unstable near extremality? J Motivation 2 D>5 MP black hole with some angular momenta vanishing and others large looks locally like black brane. Emparan & Myers

Expect Gregory-Laflamme instability. Motivation 3 Stationary implies axisymmetric Hollands, Ishibashi, Wald But: all known D>4 black holes have multiple rotational symmetries! Do there exist less symmetric solutions? Could look for such solutions as stationary

axisymmetric gravitational perturbations of Myers-Perry HSR Motivation 4 Rotating black hole in AdS: Small perturbations with amplified by superradiant scattering Reflected back towards hole by AdS potential barrier Process repeats: instability!

Cant happen for Hawking & HSR (superradiant modes dont fit into AdS box) Motivation 4 Superradiant instability shown to occur for scalar field perturbations of small Kerr-AdS holes in D=4 Cardoso & Dias What about large Kerr-AdS, gravitational perturbations, or D>4?

What is critical value for ? What is end point of instability? Outline of talk 1. Gravitational perturbations of D>4 Schwarzschild 2. Gravitational perturbations of Kerr 3. Gravitational perturbations of D>4 Myers-Perry

Perturbations of D>4 Schwarzschild Gibbons & Hartnoll, Ishibashi & Kodama Spherical symmetry: classify gravitational perturbations as scalar, vector, tensor e.g. Eqs of motion for each type reduce to single scalar equation of Schrdinger form (x=tortoise coordinate):

Form of potential implies : stable! Gravitational Perturbations of Kerr Teukolsky Two miracles make problem tractable:

1. Equations of motion of metric pert. reduce to single scalar equation 2. This equation admits separation of variables (related to existence of Killing tensor) Perturbations of Myers-Perry Miracle 2 occurs for some MP black holes Frolov & Stojkovic, Ida, Uchida & Morisawa: can study scalar field perturbations

Miracle 1 (apparently) does not occur: gravitational perturbations hard! Can make progress in special case Equal angular momenta MP black hole exhibits symmetry enhancement when some angular momenta are equal Maximal enhancement for D=2N+3 dimensions, all angular momenta equal

No a priori reason to expect instability in AF case Equal angular momenta D=2N+3, Ji=J implies cohomogeneity-1 (metric depends only on radial coord) Horizon is homogeneously squashed S2N+1=S1 bundle over CPN:

Rotation is in direction Gravitational perturbations Can decompose into scalar, vector tensor perturbations on CPN. Focus on tensors: need N2 (D=2N+37) then Einstein equations reduce to effective Schrdinger equation:

Results Asymptotically flat: no sign of instabilty (proof?) No evidence for existence of new AF solutions with less symmetry than MP Asymptotically AdS: superradiant instability when , for both large and small black holes

Endpoint of instability? For given m, unstable MP-AdS separated from stable MP-AdS by critical solution admitting stationary nonaxisymmetric zero mode Is there a corresponding branch of stationary nonaxisymmetric black holes? Could this be the endpoint of the superradiant instability?

Future directions Quasinormal modes Vector, scalar perturbations, D=5 Less symmetric black holes: some, but not all, angular momenta equal, or all angular momenta equal for even D. No longer cohomogeneity-1, but neither is Kerr!

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