Gravitational Perturbations of Higher Dimensional Rotating Black Holes

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 • Gravitational perturbations of D>4 Schwarzschild • Gravitational perturbations of Kerr • Gravitational perturbations of D>4 Myers-Perry

8. Perturbations of D>4 SchwarzschildGibbons & 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: • Equations of motion of metric reduce to single scalar equation • 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, 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

13. Gravitational perturbations • Can decompose into scalar, vector tensor perturbations on CPN. 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!