Universal features of black holes in the large D limit Roberto - PowerPoint PPT Presentation

Universal features of black holes in the large D limit Roberto Emparan ICREA & U. Barcelona w/ Kentaro Tanabe, Ryotaku Suzuki, Daniel Grumiller

Why black hole dynamics is hard Non-decoupling : BH is an extended object whose dynamics mixes strongly with background BH’s own dynamics not well-localized, not decoupled

Why black hole dynamics is hard BHs, like other extended objects, have (quasi-) normal modes but typically localized at some distance from the horizon ∼ photon orbit in AF in AdS backgrounds may be further away → hard to disentangle bh dynamics from background dynamics

Why black hole dynamics is hard BH dynamics lacks a generically small parameter Decoupling requires a small parameter Near-extremality does it: AdS/CFT-type decoupling Develop a throat effective radial potential

Large D limit Kol et al RE+Suzuki+Tanabe 1/D as small parameter Separates bh’s own dynamics from background spacetime – sharp localization of bh dynamics BH near-horizon well defined – a very special 2𝐸 bh Somewhat similar to decoupling limit in ads/cft

Large D limit Far-region : background spacetime w/ holes only knows bh size and shape → far-zone trivial dynamics Near-region : – non-trivial geometry – large universality classes eg neutral bhs (rotating, AdS etc)

Large D expansion may help for – calculations: new perturbative expansion – deeper understanding of the theory (reformulation?) Universality (due to strong localization) is good for both

Large D black holes Basic solution 𝑒𝑠 2 𝑒𝑡 2 = − 1 − 𝑠 𝐸−3 0 𝑒𝑢 2 + 𝐸−3 + 𝑠 2 𝑒Ω 𝐸−2 1 − 𝑠 𝑠 0 𝑠 length scale 𝑠 0

Large D black holes 𝑠 0 not the only scale Small parameter 1 𝐸 ⟹ scale hierarchy 𝑠 0 𝐸 ≪ 𝑠 0 This is the main feature of large-D GR

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

Far zone Fixed 𝑠 > 𝑠 0 𝐸 → ∞ 𝐸−3 𝑔 𝑠 = 1 − 𝑠 0 → 1 𝑠 𝑒𝑡 2 → −𝑒𝑢 2 + 𝑒𝑠 2 + 𝑠 2 𝑒Ω 𝐸−2 Flat, empty space at 𝑠 > 𝑠 0 no gravitational field

0 𝐸 0 scale 𝒫 𝑠 Far zone geometry Holes cut out in Minkowski space

0 𝐸 0 scale 𝒫 𝑠 Far zone Holes cut out in Minkowski space No wave absorption (perfect reflection) for 𝐸 → ∞

Near zone Gravitational field appreciable only in thin near-horizon region 𝐸−3 𝑠 0 = 𝒫 1 ⟺ 𝑠 − 𝑠 0 < 𝑠 0 𝐸 𝑠 𝑠 − 𝑠 0 ∼ 𝑠 0 𝐸

Near zone Keep non-trivial gravitational field: Length scales ∼ 𝑠 0 /𝐸 away from horizon Surface gravity 𝜆 ∼ 𝐸/𝑠 0 finite 𝐸−3 Near-horizon coordinate: 𝑆 = 𝑠 𝑠 0 All remain 𝒫(1) where grav field is non-trivial

Near zone 𝑒𝑠 2 𝐸−3 𝑒𝑡 2 = − 1 − 𝑠 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 zone 2 2 → 4𝑠 − tanh 2 𝜍 𝑒𝑢 𝑜𝑓𝑏𝑠 + 𝑒𝜍 2 + 𝑠 0 2 𝑒Ω 𝐸−2 2 2 𝑒𝑡 𝑜ℎ 0 𝐸 2 Elitzur et al 2d string black hole Mandal et al Witten Soda ℓ 𝑡𝑢𝑠𝑗𝑜𝑕 ∼ 𝑠 𝛽′ ~ 𝑠 0 2 Grumiller et al 0 𝐸 , 𝐸

Near zone universality: neutral bhs 2d string bh is near-horizon geometry of all neutral non-extremal bhs - rotation appears as a local boost (in a third direction) - cosmo const shifts 2d bh mass More near-horizon structure than just Rindler limit

Near zone universality Charge modifies near-horizon geom some are ‘ stringy ’ bhs eg, 3d black string Horne+Horowitz but many different solutions possess same near-horizon universality classes

Large D expansion: 1. BH quasinormal modes 2. Instability of rotating bhs

Massless scalar field Φ = 𝑠 − 𝐸−2 □Φ = 0 2 𝜚 𝑠 𝑓 −𝑗𝜕𝑢 𝑍 ℓ (Ω) 𝑒 2 𝜚 2 + 𝜕 2 − 𝑊 𝑠 𝜚 = 0 ∗ 𝑒𝑠 ∗ 𝑠 ∗ ) ∗ : tortoise coord 𝑊(𝑠 𝑠 ∗ 𝑠 horizon 0 infty

Massless scalar field ∗ → 𝐸 2 𝐸 → ∞ 𝑊 𝑠 2 Θ(𝑠 ∗ − 𝑠 0 ) 4𝑠 ∗ Truncated flat-space barrier 2 𝐸 2𝑠 ∗ ) 𝑊(𝑠 0 𝑠 ∗ 𝑠 horizon 0 infty

Massless scalar field ∗ → 𝐸 2 𝑊 𝑠 2 Θ(𝑠 ∗ − 𝑠 0 ) 4𝑠 ∗ 𝐸 𝜕 > 2𝑠 0 : perfectly absorbed 2 𝐸 2𝑠 0 𝜕 = 𝒫(𝐸 0 )/𝑠 0 : perfectly reflected 𝑠 ∗ 𝑠 horizon 0 infty

Schwarzschild bh grav perturbations Kodama+Ishibashi Gravitational scalar, vector, tensor modes 𝑇𝑃(𝐸 − 1) reps ∗ ) 𝑊(𝑠 𝐸 = 7 ℓ = 2 𝑠 ∗

Schwarzschild bh grav perturbations scalar vector tensor 𝐸 = 500 Potential seen by ℓ = 500 𝜕𝑠 0 = 𝒫(𝐸)

Schwarzschild bh grav perturbations scalar vector tensor Potential seen by 𝜕𝑠 0 = 𝒫(1) ℓ = 𝒫 1 𝐸 = 1000 ℓ = 2

Quasinormal modes Free, damped oscillations of black hole 𝑊 outgoing ingoing 𝑠 ∗ horizon infty

Quasinormal modes QNMs as bound states in inverted potential 𝑊 −𝑊 analytic continuation 𝑠 ∗ horizon infty

𝜕𝑠 0 = 𝒫(𝐸) QNMs ∗ → 𝐸 2 𝑊 𝑠 2 Θ(𝑠 ∗ − 𝑠 0 ) 4𝑠 ∗ 𝜕𝑠 0 = 𝒫 𝐸 high-frequency (‘scaling’ modes) 𝑠 𝑠 ∗ 0

𝜕𝑠 0 = 𝒫(𝐸) QNMs ∗ → 𝐸 2 Holes in flat space 𝑊 𝑠 2 Θ(𝑠 ∗ − 𝑠 0 ) 4𝑠 ∗ Universal structure ∀ static, AF bhs 𝑠 𝑠 ∗ 0

𝜕𝑠 0 = 𝒫(𝐸) QNMs −𝑊 𝑊 Triangular well → Airy wavefns 𝑠 𝑠 ∗ 0

𝜕𝑠 0 = 𝒫(𝐸) QNMs −𝑊 𝑊 𝑙 = 2 𝑙 = 1 𝑠 𝑠 ∗ 0 Airy zeroes 1 3 𝑓 𝑗𝜌 ⇒ 𝜕 (ℓ,𝑙) 𝑠 0 = 𝐸 𝐸 2 + ℓ − 2 + ℓ 𝑏 𝑙 2

Universal spectrum @ large D 1 3 𝑓 𝑗𝜌 𝜕 (ℓ,𝑙) 𝑠 0 = 𝐸 𝐸 2 + ℓ − 2 + ℓ 𝑏 𝑙 2 Depends only on bh radius 𝒔 𝟏 Same spectrum for: • any charges, dilaton coupling etc • scalar, vector, tensor perturbations

Universal spectrum @ large D 1 3 𝑓 𝑗𝜌 0 = 𝐸 𝐸 𝜕 (ℓ,𝑙) 𝑠 2 + ℓ − 2 + ℓ 𝑏 𝑙 2 spectrum of scalar oscillations of a hole in space Im𝜕 → 0 : Re𝜕 ∼ 𝐸 −2 3 sharp resonances ‘normal modes’ of bh

𝜕𝑠 0 = 𝒫(1) QNMs More complicated wave eqn but we’ve solved it up to 𝐸 −3 for vectors 𝐸 −2 for scalars (no tensors)

Quantitative accuracy 𝝏𝒔 𝟏 = 𝓟(𝟐) modes Vector mode (purely imaginary) • At 𝐸 = 100 : ℓ = 2 mode Im 𝜕𝑠 0 = -1.01044742 (analytical) -1.01044741 (numerical Dias et al )

Quantitative accuracy 𝝏𝒔 𝟏 = 𝓟(𝟐) modes Vector mode (purely imaginary) • At 𝐸 = 100 : ℓ = 2 mode Im 𝜕𝑠 0 = -1.01044742 (analytical) -1.01044741 (numerical Dias et al ) • At 𝐸 = 4 : − 4D exact ‘algebraically special’ mode −Im 𝜕𝑠 0 − Large D ℓ

Quantitative accuracy 𝝏𝒔 𝟏 = 𝓟(𝑬) modes Re 𝜕𝑠 0 = 𝐸 2 + ℓ : good at moderate 𝐸 Re 𝜕𝑠 ℓ = 2 0 𝐸 : only good at very high 𝐸 Im 𝜕𝑠 0 ∼ 𝐸 1 3 𝐸 ≳ 300 (!)

Instability of rotating bhs

Hi-D bhs have ultra-spinning regimes Expect instabilities: – axisymmetric – non-axisymmetric (at lower rotation) Confirmed by numerical studies Dias et al Hartnett+Santos Shibata+Yoshino Analytically solvable in 1 𝐸 expansion thanks to universality features – also in AdS

Equal-spin, odd-D, Myers-Perry black holes → only radial dependence → ODEs But equations are coupled – analytically hopeless

Dias, Figueras, Monteiro, Reall, Santos 2010

Equations do decouple for rotation=0 Large D expansion: Leading large D near-horizon: rotating bh is just a boost of Schw → rotating eqns decouple can be solved analytically Beyond leading order, MP metric is not boosted Schw, but LO boost allows to decouple eqns

Analytical computation of QNMs • Axisymmetric instability for 3 𝑏 > 2 𝑠 + • Non-axisymmetric instability for 1 𝑏 > 2 𝑠 + = .71 𝑠 + Comparison to numerical: D =5: 𝑏 > .81𝑠 D =15: 𝑏 > .73𝑠 + , + Hartnett+Santos

Outlook

Any problem that can be formulated in arbitrary D is amenable to large D expansion simpler, even analytically solvable

Universal features Far: empty space ∀ bhs Near: 2D string bh ∀ neutral bhs

BH dynamics splits into: 𝜕𝑠 0 = 𝒫(𝐸) : non-decoupled modes – scalar field oscillations of a hole in space – universal normal modes 𝜕𝑠 0 = 𝒫(𝐸 0 ) : decoupled modes – localized in near-horizon region