Roberto Emparan- Blackfolds

58
Blackfolds Roberto Emparan ICREA & U. Barcelona Based on RE, T. Harmark, V. Niarchos, N. Obers to appear Taiwan String Theory Workshop 2009

Transcript of Roberto Emparan- Blackfolds

Page 1: Roberto Emparan- Blackfolds

Blackfolds

Roberto EmparanICREA & U. Barcelona

Based on RE, T. Harmark, V. Niarchos, N. Obers to appear

Taiwan String Theory Workshop 2009

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Why study D>4 gravity?• Main motivation (at least for me!):

– better understanding of gravity (i.e. of what spacetime can do)

• In General Relativity in vacuumRµν=0

∃ only one parameter for tuning: D

– BHs exhibit novel behavior for D >4– They may admit a simpler description as D→∞

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Why study D>4 gravity?• Also, for applications to:

– String/M-theory– AdS/CFT

(+its derivatives: AdS/QGP, AdS/cond-mat etc)

– Math– TeV Gravity (bh’s @ colliders…)– etc

• Many findings will be "answers waiting for a question"

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• After the advances of the 1960's-70's, classical black holes have been widely regarded as essentially understood...

• ...so the only interesting open problems should concern quantum properties of black holes

• But recently (~8 yrs ago) we’ve fully realized how little we know about black holes – even classical ones! – and their dynamics in D>4

E.g. we’ve learned that some properties of black holes are

• 'intrinsic': Laws of bh mechanics ...

• D-dependent: Uniqueness, Topology, Shape, Stability ...

• Black hole dynamics becomes richer in D=5, and much

richer in D r 6

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• Activity launched initially by two main results:– Gregory-Laflamme (GL)-instability, its

endpoint, and inhomogeneous phases– Black rings, non-uniqueness, non-spherical

topologies

• I’ll focus mostly on simplest set up: vacuum, Rµν = 0, asymptotically flat solutions

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• Why is D>4 richer?

– More degrees of freedom– Rotation:

• more rotation planes• gravitational attraction ⇔ centrifugal repulsion

– ∃∃∃∃ extended black objects: black p-branes

– Both aspects combine to give rise to horizons w/ more than one length scale:new dynamics appears when two scales differ

FAQ's

x2 x4 x6

x1 x3 x5…

O(D−1)V U(1)⌊(D−1)/2⌋

− GMrD−3

+J2

M2r2

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• Why is D>4 harder?

– More degrees of freedom– Axial symmetries:

• U(1)'s at asymptotic infinity appear only every 2 more dimensions (asymptoticaly flat – or AdS – cases)

• in D=4,5: stationarity+2 rotation symmetries is enough to reduce to integrable 2D σ-model (but only in vacuum, not in AdS)

• if D>5 not enough symmetry to reduce to 2D σ-model

– 4D approaches not tailored to deal with horizons with more than one length scale: need new tools

FAQ's

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What's the goal?

• In D=4:– 1960-70's: Golden Age of Black Holes (4D)

• remarkably simple objects

• amenable to detailed theoretical study

– Much progress prompted by Kerr's exact solution (1963)

– Complete classification: uniqueness theorems

• In D>4: initial aim: try to repeat 4D success– Construct exact solutions

– Classify all possible black holes (even if no uniqueness)

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What's the goal?

• In D=5: exact solns+ classification – quite successful in integrable sector w/ three commuting

isometries

• In Dr6: – Dynamics too rich to be captured by looking for exact solutions

– only simplest bhs

– Classification becomes increasingly hopeless – and less interesting

– Develop qualitatively new tools

– Focus on capturing new dynamics(analogy to DBI branes)

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4D black holes• Static: Schwarzschild

• Stationary: Kerr

Horizon: ∆(r hor)=0, r hor∈ R+ ⇒ ⇒ ⇒ ⇒ GM r a

⇒⇒⇒⇒ upper bound on J for given M: J b GM 2

a =J

M

µ = 2GM

ds2 = −(1− µ

r

)dt2 +

dr2

1− µ

r

+ r2(dθ2 + sin2 θdφ2

)

ds2 = −dt2 + µrΣ

(dt+ a sin2 θdφ

)2+Σ

∆dr2 + Σdθ2 + (r2 + a2) sin2 θdφ2

Σ = r2 + a2 cos2 θ , ∆ = r2 − µr + a2 ,

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Uniqueness theorem: End of the storyMulti-bhs not rigorously ruled out, but physically unlikely to be stationary

(eg multi-Kerr can't be balanced)

J

extremal Kerr J = GM 2

non-singular, finite area

AH Kerr black holes

GM 2

4D black holes

(fixed mass)

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• Kerr bound J b GM 2 ⇒⇒⇒⇒ 4D bh horizons

characterized by a single scale:

r0 ∼ GM ∼ a

• Horizons are always round ("plump"), i.e., w/out large distortions

4D black holes

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Black holes in D>4

• Schwarzschild is easy:

• Dynamics very similar to 4D:– unique among static AF bhs Gibbons+Ida+Shiromizu

– classically linearly stable Ishibashi+Kodama

Tangherlini 1963

ds2 = −(1− µ

rD−3

)dt2 +

dr2

1− µ/rD−3 + r2dΩ(D−2)

µ ∝M

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• Solves Rµν = 0

• Compactify: x i ∼ x i + Li

different length scales on horizon: r0, Li

Black strings & branes

black p-braneTake a d-diml black hole(Ricci-flat)

add p extra dimhorizon topologyR

px S d-2

• Not asymp flat bhs, but necessary to understand them

ds2d+p = ds2d(black hole) +

p∑

i=1

dxidxi

r0

L

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– Gregory-Laflamme instability

– Non-uniform (=inhomogeneous=lumpy) black strings

GL zero-frequency mode = static perturbation

new static inhomogeneous solutions

λGL ≃ r0GubserWiseman

L<r0

r0L GL instability

L>r0

stable unstable

Separate scales New phenomena

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Rotation in Dr5• Myers+Perry (1986): rotating black hole

solutions with angular momentum in an arbitrary number of planes

e.g. D=5,6: J1, J2

D=7,8: J1, J2, J3 etc

• They all have spherical topology

x2 x4 x6

SD−2

x1 x3 x5

J1 J2 J3 … J⌊(D−1)/2⌋

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Ultra-spinning black holes in Dr5(single spin case)

gravitationalcentrifugal

r2− 1 = − µ

rD−3+a2

r2

ds2 = −dt2 + µ

rD−5Σ

(dt+ a sin2 θ dφ

)2+Σ

∆dr2 +Σdθ2 + (r2 + a2) sin2 θ dφ2

+r2 cos2 θ dΩ2(D−4)

Σ = r2 + a2 cos2 θ , ∆ = r2 + a2 − µ

rD−5, µ ∝M

a ∝ J

M

Myers+Perry 1986

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Quadratic equation: fix µ, then acan't be too large for real root

4D: 2a b µ5D: a 2 b µ

⇒ upper bound on J for given M

Horizon: ∆=0 ∆ = r2 + a2 − µ

rD−5

D=4, 5:

r

a

r0

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r

∼ r 2

Dr6:

For fixed µ there is an outer event horizon for any value of a

⇒⇒⇒⇒No upper bound on J for given M

⇒⇒⇒⇒∃ ultra-spinning black holes

must have a zero ∀a

∼ −r −(D−5)

Horizon: ∆=0 ∆ = r2 + a2 − µ

rD−5

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JD = 5upper (extremal) limit

on J is singular

JD r 6 no upper limit on J

for fixed M

J

D = 5

extremal, singular:µ=a2 ⇒ r+=0

• Single spin MP black holes:

AH AH

J

D r 6

aH → 0 , j →∞

r2+ + a2 − µ

rD−5+

= 0

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• Consider a p r+ ∆(r =r+)=0

Ultra-spinning regimeds2 = −dt2 + µ

rD−5Σ

(dt+ a sin2 θ dφ

)2+Σ

∆dr2 +Σdθ2 + (r2 + a2) sin2 θ dφ2

+r2 cos2 θ dΩ2(D−4)

r+

a

A‖ =

∫dθdφ

√gθθgφφ|r+ = Ω2(r2+ + a2) ≃ Ω2a2 −→ ℓ‖ ∼ a

A⊥ =

∫dΩ(D−4)(r+ cos θ)

D−4 = ΩD−4(r+ cos θ)D−4 −→ ℓ⊥ ∼ r+

(a ∼ J/M ∼ j)

A7A⊥

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Ultra-spinning regime

T

a

T ∼ 1/r+

black membrane-like

Kerr-like

a ∼r+

• Black membrane limit: a →∞

finite (transverse) horizon: keep r+ finite

finite mass density: keep µ/a2 finiter+

a

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Multiple spins

• If all spins similar j1∼j2∼…∼j⌊(D-1)/2⌋⇒ similar to Kerr

• ∃ ultra-spinning regimes if one ji (even D) or two ji (odd D) are much smaller than the rest

• Qualitatively understood as in single-spin:p ultra-spins limit to black 2p-branes

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bend into a circlespin it up

black string:Schw4D x R

Horizon topology S 1x S 2

Exact solution available – and fairly simpleRE+Reall 2001

S2

The forging of the ring (in D=5)

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Metric ψ-rotation

S2G(ξ) = (1− ξ2)(1 + νξ)

S1 φ

x

ν ∼ R2/R1 → shape, λ/ν → velocity

equilibrium → λ(ν)

Parameters λ, ν, R

F (ξ) = 1 + λξ

ds2 = −F (y)F (x)

(dt−C(λ, ν)R 1 + y

F (y)dψ

)2

+R2

(x− y)2 F (x)[−G(y)F (y)

dψ2 − dy2

G(y)+dx2

G(x)+G(x)

F (x)dφ2

]

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x = const

y = const

"Ring coordinates" x, y

y = yH

ψ

φ

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5D: one-black hole phases

thin black ring

1

fat black ring

Myers-Perry black hole

(naked singularity)

(slowest ring)

3 different black holes with the same value of M,J

(fix scale M)

2732

AH

J 2

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Other properties

J J

T Ωthin

thin

fat

fat

MP bh

MP bh

fat rings ~ ("drilled-through") MP black holesthin rings ~ (circular) black strings

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Black ring to boosted black string

• Ultra-spinning limit:R→∞

finite thickness: νR ∼ r0 finite

finite velocity: λ/ν finite

⇒ boosted black string, with limiting boost velocity

v = tanhα =1√2

Rr0

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Hints of a pattern?• Ultra-spinning MP bhs and black rings:

– rotation unbounded: greatly distorts the horizon and introduces a second length scale J/M, much larger than the thickness r0

– limit to a (boosted) black string/brane

Ultra-spinning, black brane-like, two length scales

Moderately-spinning, Kerr-like, one length scale

D=5Ω

J

T

J

Dr6ΩT

AH

AH

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Questions?

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4D vs hi-D Black Holes: Size matters

• 4D BHs: not only unique, spherical, and stable, but also dynamics depends on a single scale: r0∼GM

– true even if rotating: Kerr bound J b GM 2

• Main novel feature of D>4 BHs: in some regimes they're characterized by two separate scales– No upper bound on J for given M in D > 4

Length scales J/M and (GM )1/(D−3) can differ arbitrarily

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• All this has led to realize that hi-D bhs have qualitatively new dynamics that was unsuspected from experience with 4D bhs

• 4D bhs only possess short-scale (∼r0) dynamics

• Hi-D bhs: need new tools to deal with long-distance (∼R pr0) dynamics

• Natural approach: integrate out short-distance physics, find long-distance effective theory

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• Two different (but essentially equivalent) techniques for this:– Matched Asymptotic Expansion (Mae)

• GR-minded approach, well-developed, produces explicit perturbative metric, but cumbersome

– Classical Effective Field Theory (Cleft)• QFT-inspired (Feynman diagrams), efficient for calculating

physical magnitudes, but not yet developed for extended objects

• To leading order, no difference

• Leading order = blackfold approach

Goldberger+RothsteinKol

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locally (r ` R) equivalent toLorentz-transformedblack p-brane

r0

Blackfolds: long-distance effective dynamics of hi-d black holes• Black p-branes w/ worldvolume = curved submanifold

of spacetime

Mpsn

Horizon: Mp x sn

R

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• Long-distance dynamics: find brane spacetime embedding x µ(σα)

• General Classical Brane Dynamics: Carter

Given a source of energy-momentum, in probe approx,

– Newton's force law: F=ma

– Nambu-Goto-Dirac eqns: Tµν = Tgµν Kρ=0: minimal surface

Long-distance theory

∇µTµν = 0 ⇒ T µνKµνρ = 0

extrinsic curvature

or, with external force: F ρ = T µνKµνρ

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• But, what is Tµν for a blackfold?

• Short-distance physics determines effective stress-energy tensor:

– blackfold locally Lorentz-equivalent to black p-brane of thickness (sn -size) r0

– In region r0 ` r field linearizes⇒ approximate brane by equivalent

distributional source Tµν(σα)

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• Black p-brane

Lorentz-transform:

(e.g., black string boosted black string)

ds2 = −(1− r

n0

rn

)dt2 +

p∑

i=1

dz2i +dr2

1− rn0

rn

+ r2dΩ2n+1

r0

ziTtt = rn0 (n+ 1)

Tii = −rn0

Tµν → Tµν = rn0

[(n+ 1)atµa

tν −

∑p

i=1 aiµa

](t, zi) = σ

µ , σµ → aµνσν , aµν ∈ O(1, p)

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• Position-dependent thickness r0(σα) and "boosts" aµ

ν(σα) ⇒ Tµν(σα)

• Impose global blackness condition:– Uniform surf gravity κ & angular velocities Ωi

(eliminate thickness and boost parameters)

• Solve Kµν ρ(σα)T µν (σα;κ, Ωi)=0

⇒ determine x µ(σα;κ, Ωi)

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A Blackfold Bestiary

"Bestiary: a descriptive or anecdotal treatise on various real or mythicalkinds of animals, esp. a medieval work with a moralizing tone.''

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Tune boost to equilibrium

Horizon S 1 x sD−3

"small" transverse sphere ∼ r0

• Simplest example: black rings in Dr5

KµνρTµν = 0

T11R

= 0

T11 = rD−30

[(D − 3) sinh2 σ − 1

]

⇒ sinh2 σ =1

D − 3

R

r0

(in D=5 reproduces value from exact soln)

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Tune boost to equilibrium

Horizon Tpx s n+1

"small" transverse sphere ∼ r0

• p-brane curved into Tp:

KµνρTµν = 0

T11R1

=T22R2

= · · · = 0

Tii = rn0 (n(a

ti)2 − 1)

⇒ ati = −1√n, att =

√n+ p

n

n=D−p−3

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• Axisymmetric blackfolds

• Simple analytic solutions:– even p : ultraspinning MP bh, with p/2 ultraspins

– odd p : round Sp, with all (p+1)/2 rotations equal

• I'll illustrate two simple cases of each

(possibly rotations along all axes)

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• Ultra-spinning 6D MP bh as blackfold

• Embed black two-brane in 3-space

ds2=dz2+dρ2+ρ2dφ 2, as z =const

to obtain planar blackfold P2 xs2

• Find size r0(ρ) of s 2 & boost α(ρ) of locally-equiv 2-brane:– Soln: α(ρ)→∞, r0(ρ)→ 0 at ρmax=1/Ω

• Disk D2 fibered by s 2 : topology S 4: like 6D MP bh!

• All physical magnitudes match those of the ultraspinning 6D MP bh

ρ

φ

locally equiv to boosted black 2-brane

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• S3 xsn+1 black hole as blackfold (n r 2)

• Embed three-brane in a space containing

ds2=dr2+r2dΩ2(3)

as r =R

• Solution exists if |Ω1|=|Ω2|=(3/(3+n))1/2R−1

size of sn r0= const

• If |Ω1|>|Ω2| then numerical solution for r =R(θ) : non-round S3

φ1

φ2S3

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In Progress:• Sphere products:

• Static minimal blackfolds:If no boost (no local Lorentz transf)Tij = −Pgij (spatial) ⇒ Kρ=0: minimal submanifold

D =∑

i pi + n+ 3

e.g.: hyperboloid

non-compact static blackfold

pi∈odd

Spi × sn+1

"small" transverse sphere ∼ r0

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Blackfold dynamics as 1st Law• For stationary blackfolds, compute M, Ji , AH , by

integrating stress-energy tensor Ttt , Tti , and horizon area element

• Consider M [x µ], Ji [x µ], AH [x µ] as functionals of embedding x µ(σα;κ, Ωi)

• Then eqs of motion Kµν ρT µν=0 are equivalent to

⇒ Stationary blackfold eqs = 1st Law

δM

δxµ− κ

8πG

δAHδxµ

−ΩiδJiδxµ

= 0

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AH

J

Blackfolds in Dr6 phase diagram(w/ one spin)

Black rings and Black Saturns as blackfolds

can be described as blackfold

(M = 1)

cannot be described as blackfold

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D=5

Blackfolds in AdS (w/ one spin)

AH

Jfixed M

J=MLAdS

Dr6

AH

J

"compressed" versions of Λ=0

J=MLAdS

BPS bound: MLAdS r Σi |Ji |

(could also include Black Saturns)

Thin rings: r0`min(R,LAdS)

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• Stability of blackfolds for long wavelength(λ p r0) perturbations can be analyzed within

blackfold approximation

• But black branes have short-wavelength G-L instabilities

• Expect blackfolds to be unstable – on quick time scales, Γ∼1/r0

Instabilities and non-uniform phases

λGL ≃ r0/.88r0

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• Non-uniform static black branes exist:• Expected to also be unstable below D* (∼13)

but stable above D*

• Use stable non-uniform branes as basis for blackfolds (~ wiggly cosmic strings)

• These would emit grav waves, but in a much longer time-scale than GL-instability⇒ long-lived wiggly blackfolds

~ effective Tµν

~

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Other blackfold applications• Curved strings & branes in gravitational potentials:

– black rings in AdS: ∃w/ arbitrarily large radius

– black rings in dS: ∃ static ones

– black rings in Taub-NUT (D0-D6 brane interactions)

– black saturns, charged and susy blackfolds

– Probe D-branes at finite temperature

• Dynamical evolution– Time-dependent evolution

– Coupling to gravitational radiation (quadrupole formula)

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Limitations• Stationarity not guaranteed at higher

orders (i.e. w/ backreaction)

• Misses 'threshold' regime of connections and mergers of phases, where scales meet r0 ∼R

resort to extrapolations, numerical analysis, and other information

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• Long-distance dynamics of hi-d black holes efficiently captured by blackfold approach (extends & includes DBI approach to D-branes)– black branes are elastic!

• Change focus:

– search for all Dr6 black hole solutions in closed analytic form is futile (some may still show up: p=D−4)

– classification becomes increasingly harder at higher D, but alsoless interesting

Investigate novel dynamical possibilities

Blackfolds as organizing framework for hi-d black holes

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A conjecture about uniqueness and stability of hi-d black holes• Non-uniqueness (=more than one solution for given asymptotic

conserved charges) and instabilities appear only as the two horizon scales begin to differ, ie when

Ji /M t (GM )1/(D−3)

(for at least one Ji )

• For smaller spins ∃ only MP bhs (as far as we know), with properties similar to Kerr, and there is no suggestion of their instability

• It really looks like it's the appearance of more than one scale that brings in all that differs from 4D

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A conjecture about uniqueness and stability of hi-d black holes• Conjecture:

MP black holes are unique and dynamically stable iff

∀i =1,…,⌊(D−1)/2⌋

|Ji | < αDM (GM )1/(D−3),

with αD=Ο(1) (to be determined, presumably numerically).

• Applies to stationary black hole solutions w/ connected non-degenerate horizons, and possibly to multi-black hole solutions in thermal equilibrium

• Presumably also extends to charged solutions

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Conclusions & Outlook• Hi-D black hole dynamics turns out to be surprisingly rich,

but we have made great progress in recent years

• Our understanding of vacuum 5D black holes is close to complete (main open problem: 1 or 2 rotation isometries?)

• In Dr6, blackfold approach captures much of the new physics. It remains to:– develop blackfold dynamics: we have the tools

– understand effects at threshold of effective description: connections between phases, GL-like instabilities and inhomogeneous phases. Need new tools, or numerical analysis

• In AdS, the cosmo length scale makes it difficult to analyze some regimes of interest – eg large black holes– again: new approaches, or numerics

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