Black holes in the 1/D expansion

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1 Black holes in the 1/D expansion Roberto Emparan ICREA & UBarcelona w/ Tetsuya Shiromizu, Ryotaku Suzuki, Kentaro Tanabe, Takahiro Tanaka

2 R μν = 0 R μν = Λg μν

3 Black holes are very important objects in GR, but they do not appear in the fundamental formulation of the theory They re non-linear, extended field configurations with complicated dynamics

4 Strings are very important in YM theories, but they do not appear in the fundamental formulation of the theory They re non-linear, extended field configurations with complicated dynamics

5 Strings become fundamental objects in the large N limit of SU(N) YM In this limit, they are still extended objects, but their dynamics simplifies drastically

6 Is there a limit of GR in which Black Hole dynamics simplifies a lot? Yes, the limit of large D any other parameter?

7 Is there a limit in which GR can be formulated with black holes as the fundamental (extended) objects? Maybe, the limit of large D

8 BH in D dimensions ds 2 = 1 r 0 r D 3 dt 2 + dr 2 1 r 0 r D 3 + r2 dω D 2

9 Localization of interactions Large potential gradient: Φ r r 0 r D 3 Φ r0 D/r 0 Hierarchy of scales r 0 D r 0 Φ r r 0 r0 D D r

10 Large-D neat separation bh/background r 0 r 0 /D D = 4 D 4

11 r r 0 D r 0 r D 3 0 Flat space Far region

12 r 0 r D 3 = O 1 r r 0 r 0 D r r 0 r 0 D r r 0 : Near-horizon region

13 Near-horizon geometry ds 2 = 1 r 0 r D 3 dt 2 + dr 2 1 r 0 r D 3 + r2 dω D 2 r r 0 D 3 = cosh 2 ρ t near = D 2r 0 t finite as D

14 Near-horizon geometry 2d string bh ds 2 nh 4r 0 2 D 2 tanh2 ρ dt2 near + dρ 2 + r 2 0 (cosh ρ) 4/D 2 dω D 2 2d dilaton string length l s r 0 D Soda 1993 Grumiller et al 2002 RE+Grumiller+Tanabe 2013

15 Near-horizon universality 2d string bh = near-horizon geometry of all neutral non-extremal bhs rotation = local boost (along horizon) cosmo const = 2d bh mass-shift

16 Does this help understand/solve bh dynamics?

17 Quasinormal modes capture interesting perturbative dynamics: -possible instabilities -hydrodynamic behavior but, w/out a small parameter, these modes are not easily distinguished from other more boring quasinormal modes

18 Large D introduces a generic small parameter It isolates the interesting quasinormal modes from the boring modes

19 The distinction comes from whether the modes are normalizable or non-normalizable in the near-horizon region

20 Boring modes Non-normalizable in near-zone Not decoupled from the far zone High frequency: ω D/r 0 Universal spectrum: only sensitive to bh radius Almost featureless oscillations of a hole in flat space

21 Interesting modes Normalizable in near zone Decoupled from the far zone Low frequency: ω D 0 /r 0 Sensitive to bh geometry beyond the leading order Capture instabilities and hydro Efficient calculation to high orders in 1 D

22 Black hole perturbations Quasinormal modes of Schw-(A)dS bhs Gregory-Laflamme instability Ultraspinning instability All solved analytically

23 Fully non-linear large D

24 Replace bh Surface in background What s the dynamics of this surface? D 4 D

25 Large D Effective Theory Solve near-horizon equations Effective theory for the slow decoupling modes

26 Gradient hierarchy Horizon: ρ D Horizon: z 1 ρ z Σ D 3

27 Static geometry ds 2 = N 2 z dρ2 D 2 + g ΩΩ ρ, z dσ D 3 +g tt ρ, z dt 2 + g zz ρ, z dz 2 ρ z Σ D 3

28 Einstein momentum-constraint in ρ: g tt K = const K = mean curvature of horizon surface ds 2 h = g tt z dt 2 + dz 2 + R 2 z dσ D 3 R(z) embedded in background Σ D 3

29 Large D static black holes: Soap-film equation (redshifted) g tt K = const

30 Some applications

31 Soap bubble in Minkowski = Schw BH g tt K = const R 2 + R 2 = 1 R(z) R z = sin z S D 3 z S D 2

32 Black droplets Black hole at boundary of AdS AdS boundary dual to CFT in BH background z AdS bulk Numerical solution: Figueras+Lucietti+Wiseman

33 Our numerical code zmin ; zmax 0.67; r0.5; NDSolve r' z z r z 1 r z 2 z 2 1 r z 2 1 z 2,r zmin r0,r, z,zmin,zmax

34 Black droplets

35 Non-uniform black strings z R z Numerical solution: Wiseman

36 Non-uniform black strings z R z = R(z) D 1/ D K = const R + R 2 + R = const

37 Non-uniform black strings R z R z = R (z) D z

38 At NLO we find a critical dimension D for black strings (from 2nd order to 1st order) at D = 13 Numerical value D 13.5 E Sorkin 2004

39 We ve also solved for stationary black holes Ultraspinning bifurcations of (single-spin) Myers-Perry black holes at a r + = 3, 5, 7, Numerical (D=8): a r + = 1.77, 2.27, 2.72 Dias et al

40 Limitations 1/D expansion breaks down when z D Highly non-uniform black strings AdS black funnels 1/D 1/D

41 Time evolution Long wavelength, slow evolution t,z D 0 can lead to large gradients, fast evolution t,z D if so, breakdown of expansion

42 Conclusions

43 1/D expansion of GR is very efficient at capturing dynamics of horizons Reformulation of a sector of GR: bhs in terms of surfaces (membrane-like) decoupled from bulk (grav waves)

44 1/D: it works (not obvious beforehand!)

45 End

46 Spherical reduction of Einstein-Hilbert ds 2 nh = 4r 0 2 D 2 g 2 μν dx μ dx ν + r 2 0 e 4Φ x D 2 2 dω D 2 g μν 2 (x), Φ x I = d 2 x ge 2Φ R + 4 D 3 D 2 Φ 2 + D 3 D 2 4Φ (D 2) 2 e r 0 2d dilaton gravity

47 Spherical reduction of Einstein-Hilbert ds 2 nh = 4r 0 2 D 2 g 2 μν dx μ dx ν + r 2 0 e 4Φ x D 2 2 dω D 2 D Soda, Grumiller et al I d 2 x ge 2Φ R + 4 Φ 2 + D2 r 0 2 2d string gravity l string 2r 0 D

48 BH perturbations: How accurate? Small expansion parameter: 1 D 3 not quite good for D = 4 But it seems to be 1 2(D 3) not so bad in D = 4, if we can compute to higher order (in AdS: 1 2(D 1) )

49 BH perturbations: How accurate? Small expansion parameter: 1 D 3 not quite good for D = 4 But it seems to be 1 2(D 3) not so bad in D = 4, if we can compute higher orders (in AdS: 1 2(D 1) )

50 Quite accurate Quasinormal frequency in D = 4 (vector-type) Im ωr 0 4D calculation Large D=4 l (angular momentum) Calculation up to 1 D3 yields 6% accuracy in D = 4 6% = 1 2 D 3 4 D=4

51 Quantum effects? Dimensionful scale: L Planck = Għ 1 D 2 Quantum effects governed by r 0 L Planck

52 If r 0 L Planck D 0 the bh is fully quantum: Entropy 0 Temperature Evaporation lifetime 0 But other scalings are possible

53 Scaling r 0 L Planck with D: how large are the black holes, which quantum effects are finite at large D Finite entropy: r 0 L Planck D 1 2 Finite temperature: r 0 L Planck D Finite energy of Hawking radn: r 0 L Planck D 2

54 Black hole perturbations Given the general master equation, it s a straightforward perturbative analysis Leading order is simple and universal (solving in 2D string bh): static modes ω 1 D D r 0 0 Higher order perturbations are not universal, but organized by simple leading order solution