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

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

Kol et al RE+Suzuki+Tanabe

Large D limit

Far-region: background spacetime w/ holes

• nly 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

Basic solution

𝑒𝑡2 = − 1 − 𝑠 𝑠

𝐸−3

𝑒𝑢2 + 𝑒𝑠2 1 − 𝑠 𝑠

𝐸−3 + 𝑠2𝑒Ω𝐸−2

length scale 𝑠

Large D black holes

𝑠

0 not the only scale

Small parameter 1 𝐸 ⟹ scale hierarchy

𝑠0 𝐸 ≪ 𝑠0

This is the main feature of large-D GR

Large D black holes

Large potential gradient: ⟹ Hierarchy of scales

𝑠0 𝐸 ≪ 𝑠0

Localization of interactions

𝑠

𝑠 𝐸

Φ 𝑠 ∼ 𝑠 𝑠

𝐸−3

𝛼Φ

𝑠0

∼ 𝐸/𝑠0

Φ 𝑠

⟷ 𝑠0 𝐸

Fixed 𝑠 > 𝑠0 𝐸 → ∞

𝑔 𝑠 = 1 − 𝑠0 𝑠

𝐸−3

→ 1 𝑒𝑡2 → −𝑒𝑢2 + 𝑒𝑠2 + 𝑠2𝑒Ω𝐸−2

Flat, empty space at 𝑠 > 𝑠

no gravitational field

Far zone

Far zone geometry

Holes cut out in Minkowski space

scale 𝒫 𝑠

0𝐸0

Holes cut out in Minkowski space No wave absorption (perfect reflection) for 𝐸 → ∞

scale 𝒫 𝑠

0𝐸0

Far zone

Gravitational field appreciable only in thin near-horizon region

𝑠0 𝑠

𝐸−3

= 𝒫 1 ⟺ 𝑠 − 𝑠0 < 𝑠0 𝐸

Near zone

𝑠 − 𝑠0 ∼ 𝑠0 𝐸

Keep non-trivial gravitational field: Length scales ∼ 𝑠0/𝐸 away from horizon Surface gravity 𝜆 ∼ 𝐸/𝑠0 finite Near-horizon coordinate: 𝑆 = 𝑠 𝑠0

𝐸−3

All remain 𝒫(1) where grav field is non-trivial

Near zone

Near zone

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 → ∞

2d string black hole

Elitzur et al Mandal et al Witten

Near zone

𝑒𝑡𝑜ℎ

2 → 4𝑠 2

𝐸2 − tanh2 𝜍 𝑒𝑢𝑜𝑓𝑏𝑠

2

+ 𝑒𝜍2 + 𝑠

2𝑒Ω𝐸−2 2

ℓ𝑡𝑢𝑠𝑗𝑜𝑕 ∼ 𝑠 𝐸 , 𝛽′ ~ 𝑠0 𝐸

2 Soda Grumiller et al

2d string bh is near-horizon geometry

• f 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: neutral bhs

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

Near zone universality

Large D expansion:

• 1. BH quasinormal modes
• 2. Instability of rotating bhs

Massless scalar field

□Φ = 0

Φ = 𝑠−𝐸−2

2 𝜚 𝑠 𝑓−𝑗𝜕𝑢 𝑍

ℓ(Ω)

𝑒2𝜚 𝑒𝑠

∗ 2 + 𝜕2 − 𝑊 𝑠 ∗

𝜚 = 0

𝑊(𝑠

∗)

infty horizon

𝑠

𝑠

∗

𝑠

∗: tortoise coord

Massless scalar field

𝑊(𝑠

∗)

infty horizon

𝐸 2𝑠

2

Truncated flat-space barrier

𝑊 𝑠

∗ → 𝐸2

4𝑠

∗ 2 Θ(𝑠 ∗ − 𝑠0)

𝑠

𝑠

∗

𝐸 → ∞

Massless scalar field

infty horizon 𝜕 >

𝐸 2𝑠0 : perfectly absorbed

𝜕 = 𝒫(𝐸0)/𝑠

0 : perfectly reflected

𝑠

𝑠

∗

𝐸 2𝑠

2

𝑊 𝑠

∗ → 𝐸2

4𝑠

∗ 2 Θ(𝑠 ∗ − 𝑠 0)

Schwarzschild bh grav perturbations

Gravitational scalar, vector, tensor modes

𝑇𝑃(𝐸 − 1) reps

Kodama+Ishibashi

𝑊(𝑠

∗)

𝑠

∗

𝐸 = 7 ℓ = 2

Schwarzschild bh grav perturbations

scalar vector tensor 𝐸 = 500 ℓ = 500 Potential seen by 𝜕𝑠0 = 𝒫(𝐸)

Schwarzschild bh grav perturbations

scalar vector tensor 𝐸 = 1000 ℓ = 2 Potential seen by 𝜕𝑠0 = 𝒫(1) ℓ = 𝒫 1

𝑠

∗

𝑊

horizon infty

Quasinormal modes

Free, damped

• scillations of

black hole

• utgoing

ingoing

𝑠

∗

𝑊 −𝑊

QNMs as bound states in inverted potential

horizon infty

Quasinormal modes

analytic continuation

𝜕𝑠

0 = 𝒫(𝐸) QNMs

𝑠

∗

𝑠

𝑊 𝑠

∗ → 𝐸2

4𝑠

∗ 2 Θ(𝑠 ∗ − 𝑠 0)

𝜕𝑠0 = 𝒫 𝐸 high-frequency (‘scaling’ modes)

𝑠

∗

𝑠

Holes in flat space

Universal structure ∀ static, AF bhs

𝑊 𝑠

∗ → 𝐸2

4𝑠

∗ 2 Θ(𝑠 ∗ − 𝑠 0)

𝜕𝑠

0 = 𝒫(𝐸) QNMs

𝑠

∗

𝑠

𝑊 −𝑊

Triangular well → Airy wavefns

𝜕𝑠

0 = 𝒫(𝐸) QNMs

𝑊

𝑠

∗

𝑠

−𝑊

Airy zeroes

⇒ 𝜕(ℓ,𝑙) 𝑠0 = 𝐸 2 + ℓ − 𝑓𝑗𝜌 2 𝐸 2 + ℓ

1 3

𝑏𝑙

𝑙 = 1 𝑙 = 2

𝜕𝑠

0 = 𝒫(𝐸) QNMs

Universal spectrum @ large D

𝜕(ℓ,𝑙)𝑠0 = 𝐸 2 + ℓ − 𝑓𝑗𝜌 2 𝐸 2 + ℓ

1 3

𝑏𝑙 Depends only on bh radius 𝒔𝟏 Same spectrum for:

• any charges, dilaton coupling etc
• scalar, vector, tensor perturbations

Universal spectrum @ large D

𝜕(ℓ,𝑙)𝑠

0 = 𝐸

2 + ℓ − 𝑓𝑗𝜌 2 𝐸 2 + ℓ

1 3

𝑏𝑙

spectrum of scalar

• scillations of a hole

in space

Im𝜕 Re𝜕 ∼ 𝐸−2 3 → 0:

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

𝝏𝒔𝟏 = 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

𝝏𝒔𝟏 = 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:

−Im 𝜕𝑠

ℓ

− 4D exact − Large D

‘algebraically special’ mode

Quantitative accuracy

𝝏𝒔𝟏 = 𝓟(𝑬) modes Re 𝜕𝑠0 = 𝐸

2 + ℓ : good at moderate 𝐸

Im 𝜕𝑠0 ∼ 𝐸1 3

: only good at very high 𝐸

𝐸 ≳ 300 (!)

Re 𝜕𝑠 𝐸 ℓ = 2

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

𝑏 >

3 2 𝑠+

• Non-axisymmetric instability for

𝑏 >

1 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: 𝜕𝑠0 = 𝒫(𝐸) : non-decoupled modes

– scalar field oscillations of a hole in space – universal normal modes

𝜕𝑠0 = 𝒫(𝐸0) : decoupled modes

– localized in near-horizon region