Black holes in the 1/D expansion Roberto Emparan ICREA & - - PowerPoint PPT Presentation

Black holes in the 1/D expansion

Roberto Emparan ICREA & UBarcelona

w/ Tetsuya Shiromizu, Ryotaku Suzuki, Kentaro Tanabe, Takahiro Tanaka

𝑺𝝂𝝃 = 𝟏 𝑺𝝂𝝃 = −𝚳𝒉𝝂𝝃

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

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

Strings become fundamental objects in the large N limit of SU(N) YM In this limit, YM can be reformulated using worldsheet variables Strings are still extended objects, but their dynamics simplifies drastically

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

any other parameter?

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

𝑒𝑡2 = − 1 − 𝑠 𝑠

𝐸−3

𝑒𝑢2 + 𝑒𝑠2 1 − 𝑠 𝑠

𝐸−3 + 𝑠2𝑒Ω𝐸−2

BH in D dimensions

Large potential gradient: ⟹ Hierarchy of scales

Localization of interactions

𝑠

𝑠 𝐸

Φ 𝑠 ∼ 𝑠 𝑠

𝐸−3

𝛼Φ

𝑠0

∼ 𝐸/𝑠0

Φ 𝑠

⟷ 𝑠0 𝐸

𝑠0 𝐸 ≪ 𝑠0

Large-D ⇒ neat separation bh/background

𝐸 = 4 𝐸 ≫ 4

∼ 𝑠0 ∼ 𝑠0/𝐸

𝑠 ≫ 𝑠 𝐸 ⇒ 𝑠0 𝑠

𝐸−3

→ 0 “Far” region Flat space

𝑠0 𝑠

𝐸−3

= 𝒫 1 ⟺ 𝑠 − 𝑠

0 ≲ 𝑠

𝐸

𝑠 − 𝑠0 ∼ 𝑠0 𝐸

𝑠 ≪ 𝑠

0: “Near-horizon” region

Near-horizon geometry

𝑠 𝑠0

𝐸−3

= cosh2𝜍

𝑒𝑡2 = − 1 − 𝑠 𝑠

𝐸−3

𝑒𝑢2 + 𝑒𝑠2 1 − 𝑠 𝑠

𝐸−3 + 𝑠2𝑒Ω𝐸−2

𝑢𝑜𝑓𝑏𝑠 = 𝐸 2𝑠 𝑢

finite as 𝐸 → ∞

𝑒𝑡𝑜ℎ

2 → 4𝑠 2

𝐸2 − tanh2 𝜍 𝑒𝑢𝑜𝑓𝑏𝑠

2

+ 𝑒𝜍2 + 𝑠

2 (cosh 𝜍)4/𝐸 𝑒Ω𝐸−2 2

Near-horizon geometry

Soda 1993 Grumiller et al 2002 RE+Grumiller+Tanabe 2013

2d string bh

‘string length’ ℓ𝑡 ∼ 𝑠0

𝐸

2d dilaton

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

Does this help understand/solve bh dynamics?

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

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

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

‘Boring’ modes

Non-normalizable in near-zone Not decoupled from the far zone High frequency: 𝜕 ∼ 𝐸/𝑠0 Universal spectrum: only sensitive to bh radius Almost featureless oscillations of a hole in flat space

‘Interesting’ modes

Normalizable in near zone Decoupled from the far zone Low frequency: 𝜕 ∼ 𝐸0/𝑠0 Sensitive to bh geometry beyond the leading

• rder

Capture instabilities and hydro Efficient calculation to high orders in 1

𝐸

Black hole perturbations

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

Small expansion parameter:

1 𝐸−3

not quite good for 𝐸 = 4 … But it seems to be

1 2(𝐸−3)

not so bad in 𝐸 = 4, if we can compute to higher order

(in AdS:

1 2(𝐸−1))

How accurate?

Small expansion parameter:

1 𝐸−3

not quite good for 𝐸 = 4 … But it seems to be

1 𝟑(𝐸−3)

not so bad in 𝐸 = 4, if we can compute higher orders

(in AdS:

1 2(𝐸−1))

How accurate?

Quite accurate

−Im 𝜕𝑠

− 4D calculation − Large D @ D=4

Calculation up to

1 𝐸3 yields 6% accuracy in 𝐸 = 4

Quasinormal frequency in 𝐸 = 4 (vector-type)

ℓ (angular momentum)

6% = 1 2 𝐸 − 3

4 𝐸=4

Fully non-linear GR @ large D

Replace bh ⟶ Surface in background

What’s the dynamics of this surface?

𝐸 ≫ 4 𝐸 → ∞

Large D Effective Theory

Solve near-horizon equations → Effective theory

for the ‘slow’ decoupling modes

Gradient hierarchy

⊥ Horizon: 𝜖𝜍 ∼ 𝐸 ∥ Horizon: 𝜖𝑨 ∼ 1

𝑨

𝜍

Σ𝐸−3

Static geometry

𝑒𝑡2 = 𝑂2 𝑨 𝑒𝜍2 𝐸2 + 𝑕ΩΩ 𝜍, 𝑨 𝑒Σ𝐸−3 +𝑕𝑢𝑢 𝜍, 𝑨 𝑒𝑢2 + 𝑕𝑨𝑨 𝜍, 𝑨 𝑒𝑨2 𝑨

𝜍

Σ𝐸−3

Einstein ‘momentum-constraint’ in 𝜍:

−𝑕𝑢𝑢𝐿 = 2𝜆

𝐿 = mean curvature of ‘horizon surface’

𝑒𝑡2 ℎ = 𝑕𝑢𝑢 𝑨 𝑒𝑢2 + 𝑒𝑨2 + ℛ2 𝑨 𝑒Σ𝐸−3 embedded in background Σ𝐸−3

ℛ(𝑨)

𝜆=surface gravity

Large D static black holes: Soap-film equation (redshifted)

−𝑕𝑢𝑢𝐿 = 2𝜆

Some applications

Soap bubble in Minkowski = Schw BH

−𝑕𝑢𝑢𝐿 = const ⇒ ℛ′2 + ℛ2 = 1

𝑨 ℛ(𝑨)

⇒ ℛ 𝑨 = sin 𝑨

𝑇𝐸−3

𝑇𝐸−2

Black droplets

Black hole at boundary of AdS

dual to CFT in BH background

Numerical solution: Figueras+Lucietti+Wiseman

AdS boundary AdS bulk

𝑨

Our numerical code

zmin 0.000001; zmax 0.67; r0 .5; NDSolve r' z

z r z 1 r z 2 z2 1 r z 2 1 z2

,r zmin r0 ,r, z,zmin,zmax

Black droplets

Non-uniform black strings

Numerical solution: Wiseman 𝑨

ℛ 𝑨

Non-uniform black strings

𝐿 = const

⟹ 𝒬′′ + 𝒬 + 𝒬′2 = const

𝑨

ℛ 𝑨 = 1 + 2𝒬(𝑨) 𝐸

∼ 1/ 𝐸

requires NLO

Non-uniform black strings

𝒬 𝑨

𝑨

ℛ 𝑨 = 1 + 2𝒬(𝑨) 𝐸

At NLO there appears a critical dimension 𝐸∗ for black strings

(from 2nd order to 1st order)

at 𝐸∗ = 13

Suzuki+Tanabe

Numerical value 𝐸∗ ≃ 13.5

E Sorkin 2004

Formulation for stationary black holes Ultraspinning bifurcations of (single-spin) Myers-Perry black holes at 𝑏 𝑠+ = 3, 5, 7, … Numerical (D=8):

𝑏 𝑠+ = 1.77, 2.27, 2.72 …

Dias et al

Extensions

Charged black holes

RE+Di Dato

Time-evolving black holes Minwalla et al

Limitations

1/D expansion breaks down when 𝜖𝑨 ∼ 𝐸

• Highly non-uniform black strings
• AdS black funnels

1/𝐸 1/𝐸

Long wavelength, slow evolution 𝜖𝑢,𝑨 ∼ 𝐸0 can lead to large gradients, fast evolution 𝜖𝑢,𝑨 ∼ 𝐸 if so, breakdown of expansion

Conclusions

1/D expansion of GR is very efficient at capturing dynamics of horizons Reformulation of a sector of GR: bh’s in terms of (membrane-like) surfaces

decoupled from bulk (grav waves)

1/D: it works

(not obvious beforehand!)

End

Spherical reduction of Einstein-Hilbert

𝑒𝑡𝑜ℎ

2 = 4𝑠 2

𝐸2 −𝑕𝜈𝜉

2 𝑒𝑦𝜈𝑒𝑦𝜉 + 𝑠 2𝑓−4Φ 𝑦 𝐸−2 𝑒Ω𝐸−2 2

𝑕𝜈𝜉

2 (𝑦) , Φ 𝑦 𝐽 = ∫ 𝑒2𝑦 −𝑕𝑓−2Φ 𝑆 + 4 𝐸 − 3 𝐸 − 2 𝛼Φ 2 + 𝐸 − 3 𝐸 − 2 𝑠

2

𝑓

4Φ (𝐸−2)

⟹ 2d dilaton gravity

Spherical reduction of Einstein-Hilbert

𝑒𝑡𝑜ℎ

2 = 4𝑠 2

𝐸2 −𝑕𝜈𝜉

2 𝑒𝑦𝜈𝑒𝑦𝜉 + 𝑠 2𝑓−4Φ 𝑦 𝐸−2 𝑒Ω𝐸−2 2

𝐸 → ∞

𝐽 → ∫ 𝑒2𝑦 −𝑕𝑓−2Φ 𝑆 + 4 𝛼Φ 2 + 𝐸2 𝑠

2

⟹ 2d string gravity

ℓ𝑡𝑢𝑠𝑗𝑜𝑕 ∼ 2𝑠 𝐸

Soda, Grumiller et al

Dimensionful scale: 𝑀𝑄𝑚𝑏𝑜𝑑𝑙 = 𝐻ℏ

1 𝐸−2

Quantum effects governed by

𝑠0 𝑀𝑄𝑚𝑏𝑜𝑑𝑙

Quantum effects?

If

𝑠0 𝑀𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸0 the bh is fully quantum:

Entropy → 0 Temperature → ∞ Evaporation lifetime → 0 But other scalings are possible

Scaling

𝑠0 𝑀𝑄𝑚𝑏𝑜𝑑𝑙 with D:

how large are the black holes, which quantum effects are finite at large D Finite entropy: 𝑠

0 𝑀𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸 1 2

Finite temperature: 𝑠

0 𝑀𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸

Finite energy of Hawking radn: 𝑠

0 𝑀𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸2

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 𝜕 ∼

1 𝐸 𝐸 𝑠0 → 0

Higher order perturbations are not universal, but

• rganized by simple leading order solution