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

2. 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 entropy than near-extremal MP. Is MP black hole unstable near extremality? A J

3. 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.

4. Motivation 3 Stationary implies axisymmetric Hollands, Ishibashi, Wald But: all known D>4 black holes all have more than 2 symmetries! Do there exist less symmetric solutions? Could look for such solutions as stationary axisymmetric gravitational perturbations of Myers-Perry HSR

5. 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! Can’t happen for Hawking & HSR (superradiant modes don’t fit into AdS “box”)

6. 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?

7. Outline of talk 1.Gravitational perturbations of D>4 Schwarzschild 2.Gravitational perturbations of Kerr 3.Gravitational perturbations of D>4 Myers-Perry

8. 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 Schrödinger form (x=tortoise coordinate): Form of potential implies : stable!

9. Gravitational Perturbations of Kerr Teukolsky Two miracles make problem tractable: 1.Equations of motion of metric reduce to single scalar equation 2.This equation admits separation of variables (related to existence of Killing tensor)

10. 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…

11. 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

12. Equal angular momenta D=2N+3, J i =J implies cohomogeneity-1 (metric depends only on radial coord) Horizon is homogeneously squashed S 2N+1 =S 1 bundle over CP N : Rotation is in direction

13. Gravitational perturbations Can decompose into scalar, vector tensor perturbations on CP N. Focus on tensors: need N≥2 (D=2N+3≥7) then Einstein equations reduce to effective Schrödinger equation:

14. 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

15. 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?

16. 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!