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

𝑺 𝝂𝝃 = 𝟏 𝑺 𝝂𝝃 = −𝚳𝒉 𝝂𝝃

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

BH in D dimensions 𝑒𝑠 2 𝑒𝑡 2 = − 1 − 𝑠 𝐸−3 𝑒𝑢 2 + 0 𝐸−3 + 𝑠 2 𝑒Ω 𝐸−2 1 − 𝑠 𝑠 0 𝑠

Localization of interactions Large potential gradient: Φ 𝑠 Φ 𝑠 ∼ 𝑠 𝐸−3 0 𝑠 𝐸 𝛼Φ ∼ 𝐸/𝑠 0 𝑠 0 𝑠 𝑠 0 ⟹ Hierarchy of scales ⟷ 𝑠 0 𝐸 𝑠 0 𝐸 ≪ 𝑠 0

Large-D ⇒ neat separation bh/background ∼ 𝑠 0 /𝐸 ∼ 𝑠 0 𝐸 = 4 𝐸 ≫ 4

𝑠 ≫ 𝑠 𝑠 0 𝐸−3 0 ⇒ → 0 𝐸 𝑠 Flat space “ Far ” region

𝑠 0 𝐸−3 0 ≲ 𝑠 0 = 𝒫 1 ⟺ 𝑠 − 𝑠 𝑠 𝐸 𝑠 − 𝑠 0 ∼ 𝑠 0 𝐸 𝑠 ≪ 𝑠 0 : “ Near-horizon ” region

Near-horizon geometry 𝑒𝑠 2 𝑒𝑡 2 = − 1 − 𝑠 𝐸−3 𝑒𝑢 2 + 0 𝐸−3 + 𝑠 2 𝑒Ω 𝐸−2 1 − 𝑠 𝑠 0 𝑠 𝐸−3 𝑠 = cosh 2 𝜍 𝑠 0 finite 𝑢 𝑜𝑓𝑏𝑠 = 𝐸 as 𝐸 → ∞ 𝑢 2𝑠 0

Near-horizon geometry 2d string bh 2 2 → 4𝑠 − tanh 2 𝜍 𝑒𝑢 𝑜𝑓𝑏𝑠 0 2 + 𝑒𝜍 2 𝑒𝑡 𝑜ℎ 𝐸 2 2 (cosh 𝜍) 4/𝐸 𝑒Ω 𝐸−2 2 + 𝑠 0 Soda 1993 2d dilaton Grumiller et al 2002 RE+Grumiller+Tanabe 2013 ‘string length’ ℓ 𝑡 ∼ 𝑠0 𝐸

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

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/𝑠 0 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: 𝜕 ∼ 𝐸 0 /𝑠 0 Sensitive to bh geometry beyond the leading order 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

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

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

Quite accurate Quasinormal frequency in 𝐸 = 4 (vector-type) −Im 𝜕𝑠 − 4D calculation 0 − Large D @ D=4 ℓ (angular momentum) 1 𝐸 3 yields 6% accuracy in 𝐸 = 4 Calculation up to 1 6% = 4 𝐸=4 2 𝐸 − 3

Fully non-linear GR @ large D

Replace bh ⟶ Surface in background What’s the dynamics of this surface? 𝐸 → ∞ 𝐸 ≫ 4

Large D Effective Theory Solve near-horizon equations → Effective theory for the ‘slow’ decoupling modes

Gradient hierarchy ⊥ Horizon: 𝜖 𝜍 ∼ 𝐸 ∥ Horizon: 𝜖 𝑨 ∼ 1 𝑨 𝜍 Σ 𝐸−3

Static geometry 𝑒𝑡 2 = 𝑂 2 𝑨 𝑒𝜍 2 𝐸 2 + 𝑕 ΩΩ 𝜍, 𝑨 𝑒Σ 𝐸−3 +𝑕 𝑢𝑢 𝜍, 𝑨 𝑒𝑢 2 + 𝑕 𝑨𝑨 𝜍, 𝑨 𝑒𝑨 2 𝑨 𝜍 Σ 𝐸−3

Einstein ‘momentum - constraint’ in 𝜍 : −𝑕 𝑢𝑢 𝐿 = 2𝜆 𝜆 =surface gravity 𝐿 = mean curvature of ‘horizon surface’ 𝑒𝑡 2 ℎ = 𝑕 𝑢𝑢 𝑨 𝑒𝑢 2 + 𝑒𝑨 2 + ℛ 2 𝑨 𝑒Σ 𝐸−3 embedded in background ℛ(𝑨) Σ 𝐸−3

Large D static black holes: Soap-film equation (redshifted) −𝑕 𝑢𝑢 𝐿 = 2𝜆

Some applications

Soap bubble in Minkowski = Schw BH −𝑕 𝑢𝑢 𝐿 = const ⇒ ℛ ′2 + ℛ 2 = 1 ⇒ ℛ 𝑨 = sin 𝑨 ℛ(𝑨) 𝑨 𝑇 𝐸−3 𝑇 𝐸−2

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

Our numerical code zmin 0.000001; zmax 0.67; r0 .5; r z 2 z2 1 r z 2 1 z NDSolve r' z ,r zmin r0 ,r, z,zmin,zmax 1 z2 r z

Black droplets

Non-uniform black strings 𝑨 ℛ 𝑨 Numerical solution: Wiseman

Non-uniform black strings 𝑨 ℛ 𝑨 = 1 + 2𝒬(𝑨) 𝐸 requires NLO ∼ 1/ 𝐸 𝐿 = const ⟹ 𝒬 ′′ + 𝒬 + 𝒬 ′2 = const

Non-uniform black strings ℛ 𝑨 = 1 + 2𝒬(𝑨) 𝒬 𝑨 𝐸 𝑨

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 = 3, 5, 7, … 𝑠 + 𝑏 𝑠 + = 1.77, 2.27, 2.72 … Numerical (D=8): Dias et al

Extensions Charged black holes RE+Di Dato Time-evolving black holes Minwalla et al

Limitations 1/D expansion breaks down when 𝜖 𝑨 ∼ 𝐸 • Highly non-uniform black strings 1/𝐸 1/𝐸 • AdS black funnels

Long wavelength, slow evolution 𝜖 𝑢,𝑨 ∼ 𝐸 0 can lead to large gradients, fast evolution 𝜖 𝑢,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 2 2 = 4𝑠 2 𝑓 −4Φ 𝑦 2 𝑒𝑦 𝜈 𝑒𝑦 𝜉 + 𝑠 0 𝐸−2 𝑒Ω 𝐸−2 2 𝑒𝑡 𝑜ℎ −𝑕 𝜈𝜉 0 𝐸 2 2 (𝑦) , Φ 𝑦 𝑕 𝜈𝜉 𝐽 = ∫ 𝑒 2 𝑦 −𝑕𝑓 −2Φ 𝑆 + 4 𝐸 − 3 𝐸 − 2 𝛼Φ 2 + 𝐸 − 3 𝐸 − 2 4Φ (𝐸−2) 𝑓 2 𝑠 0 ⟹ 2d dilaton gravity

Spherical reduction of Einstein-Hilbert 2 2 = 4𝑠 2 𝑓 − 4Φ 𝑦 2 𝑒𝑦 𝜈 𝑒𝑦 𝜉 + 𝑠 0 𝐸−2 𝑒Ω 𝐸−2 2 𝑒𝑡 𝑜ℎ −𝑕 𝜈𝜉 0 𝐸 2 𝐸 → ∞ Soda, Grumiller et al 𝐽 → ∫ 𝑒 2 𝑦 −𝑕𝑓 −2Φ 𝑆 + 4 𝛼Φ 2 + 𝐸 2 2 𝑠 0 ⟹ 2d string gravity ℓ 𝑡𝑢𝑠𝑗𝑜𝑕 ∼ 2𝑠 0 𝐸

Quantum effects? Dimensionful scale: 1 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙 = 𝐻ℏ 𝐸−2 𝑠 0 Quantum effects governed by 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙

𝑠 0 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸 0 the bh is fully quantum: If Entropy → 0 Temperature → ∞ Evaporation lifetime → 0 But other scalings are possible

𝑠 0 Scaling 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙 with D: how large are the black holes, which quantum effects are finite at large D 0 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸 1 2 𝑠 Finite entropy: 𝑠 0 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸 Finite temperature: 0 𝑀 𝑄𝑚𝑏𝑜𝑑𝑙 ∼ 𝐸 2 𝑠 Finite energy of Hawking radn:

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