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Black holes in the 1/D expansion
Roberto Emparan ICREA & U. Barcelona (& YITP Kyoto)
w/ Tetsuya Shiromizu, Ryotaku Suzuki, Kentaro Tanabe, Takahiro Tanaka
Nov 1915 Feb 1917
Black holes in the 1/D expansion Roberto Emparan ICREA & U. - - PowerPoint PPT Presentation
Black holes in the 1/D expansion Roberto Emparan ICREA & U. - - PowerPoint PPT Presentation
Black holes in the 1/D expansion Roberto Emparan ICREA & U. Barcelona (& YITP Kyoto) w/ Tetsuya Shiromizu, Ryotaku Suzuki, Kentaro Tanabe, Takahiro Tanaka Nov 1915 Feb 1917 A dimensionless, adjustable parameter is a good thing to
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A dimensionless, adjustable parameter is a good thing to have for studying a theory
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Quantum ElectroDynamics Perturb around
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Quantum GluoDynamics SU(3) Yang-Mills theory No parameter?
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Quantum GluoDynamics SU(N) Yang-Mills theory parameter!
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What dimensionless parameter in
?
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YM SU(N→∞) Quantum GR SO (D →∞,1)
?
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Quantum GR: SO(D-1,1) local Lorentz group # graviton polarizations grows with D BUT: No topological expansion of Feynman diagrams Even worse: UV behavior infinitely bad
Strominger 1981 Bjerrum-Bohr 2004
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YM SU(N→∞) Quantum GR SO (D →∞,1)
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Classical General Relativity D-diml Einstein’s theory
Well-defined for all D Many problems can be formulated keeping D arbitrary → D = continuous parameter → expand in 1/D
Kol et al RE+Suzuki+Tanabe
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Classical General Relativity D-diml Einstein’s theory
Large D keeps essential physics of D=4
∃ black holes ∃ gravitational waves
simplifies the theory
reformulation in terms of other variables?
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= − 1 −
- +
- 1 −
- + Ω
BH in D dimensions
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Large potential gradient: ⟹ Hierarchy of scales
Localization of interactions
- Φ ∼
- Φ
- ∼ /
Φ
⟷
- ≪
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Fixed
- 1 −
- → 1
→ − + + Ω
Flat, empty space at
- “Far-zone” limit
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“infinitely difficult to catch a line of force”
Black Hole scattering: no deflection
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No absorption of waves with wavelength ∼
Black Hole scattering
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Holes cut out in Minkowski space No interaction
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We are keeping length scales ∼ finite as we send → ∞
“Far-zone” limit
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Now take a limit that does not trivialize the gravitational field
- − ∼
- “Near-horizon” limit
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Near-horizon geometry
- = cosh$
= − 1 −
- +
- 1 −
- + Ω
%&' = 2
- finite
as → ∞
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%)
→ 4
- − tanh $ %&'
- + $
+
- (cosh $)0/ Ω
- Near-horizon geometry
Soda 1993 Grumiller et al 2002
2d string bh
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Physics at
- close to the
horizon is not trivial
Perfect absorption
- f waves with
∼ / 1 ∼ /
“Near-horizon” dynamics
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Not an exact solution Non-trivial interaction
“Near-horizon” dynamics
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2d string bh = near-horizon geometry
- f all neutral non-extremal bhs
rotation = local boost
(along horizon)
cosmo const = 2d bh mass-shift
Near-horizon universality
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Large D Effective Theory
Solve near-horizon equations
integrate-out short-distance dynamics
Boundary conds for far-zone fields
Long-distance effective theory
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Black hole perturbations
Scattering Quasinormal modes Ultraspinning instability Holographic superconductors
Full non-linear GR
General theory of static black holes: Soap-film theory Black droplets Non-uniform black strings
all analytic simple ODE
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Small expansion parameter: 2
- not quite good for = 4 …
But it seems to be
2 ()
not so bad in = 4, if we can compute to higher order
(in AdS:
2 (2))
BH perturbations: How accurate?
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Small expansion parameter: 2
- not quite good for = 4 …
But it seems to be
2 4()
not so bad in = 4, if we can compute higher orders
(in AdS:
2 (2))
BH perturbations: How accurate?
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Quite accurate
−Im 1
− − − − 4D calculation − − − − Large D @ D=4
Calculation up to
2 7 yields 6% accuracy in = 4
Quasinormal frequency in = 4 (vector-type)
ℓ (angular momentum)
6% = 1 2 − 3
<0
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Fully non-linear GR @ large D
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Large-D ⇒ neat separation bh / background Replace bh ⟶ surface in background What eqs determine this surface? ⟶
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Derive them by solving Einstein’s eqs in near-horizon zone ⟶
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Gradient hierarchy
⊥ Horizon: @A ∼ ∥ Horizon: @C ∼ 1
D
$ E
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Einstein ‘momentum-constraint’ in $:
FF
G = mean curvature of ‘horizon surface’
) = HFF D + D + ℛ D Ω
E
D ℛ(D)
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Soap-film equation (redshifted)
FF Valid up to NLO in 1/D (but not at NNLO)
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Some applications
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Soap bubble in Minkowski = Schw BH
= − + D + + Ω = ℛ(D)
−HFFG = const ⇒ ℛJ + ℛ = 1 D ℛ(D)
E
E
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Black droplets
Black hole at boundary of AdS
dual to CFT in BH background
Numerical solution: Figueras+Lucietti+Wiseman
AdS boundary AdS bulk
D
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AdS boundary AdS bulk
D ℛ(D)
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D ℛ(D)
−HFFG = const ⇒ ℛ(D)J = −
D ℛ(D) 1 ± D + ℛ(D) 1 − D 1 − D
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Numerical code
zmin=0.000001; zmax=0.67; r0=.5; NDSolveB:r'@zD−
z r@zD 1− r@zD2+z2 J1−r@zD2N 1−z2
,r@zminDr0>,r,8z,zmin,zmax<F
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Black droplets
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Non-uniform black strings
Numerical solution: Wiseman D
ℛ D
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Non-uniform black strings
G = const
⟹ LJJ(D) + LJ D + L(D) = const
D
ℛ D = 1 + 2L(D)
- ∼ 1/
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Non-uniform black strings
L D
ℛ D = 1 + 2L(D)
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Limitations
1/D expansion breaks down when @C ∼
- Highly non-uniform black strings
- AdS black funnels
1/ 1/
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In progress
Extensions of
FF
Charged black holes Rotating black holes (Time-evolving black holes)
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Conclusions
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1/D: it works
(not obvious beforehand!)
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Static black holes are soap bubbles
at large D (up to NLO)
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Can we reformulate GR around D with black holes as basic (extended) objects?
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