Transcript 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
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Compare D=5 black ring
with Myers-Perry black
hole with single angular
momentum.
“Extremal” MP solution is
nakedly singular.
A
Black ring has greater
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 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
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”)
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 Schrödinger 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 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 N≥2 (D=2N+3≥7) then
• Einstein equations reduce to effective
Schrödinger 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!