roberto emparan- blackfolds
TRANSCRIPT
Blackfolds
Roberto EmparanICREA & U. Barcelona
Based on RE, T. Harmark, V. Niarchos, N. Obers to appear
Taiwan String Theory Workshop 2009
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→∞
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"
• 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
• 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
• 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
• 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
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)
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)
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 ,
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)
• 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
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
• 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
– 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
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⌋
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
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
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
∆
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
• 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⊥
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
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
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)
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
]
x = const
y = const
"Ring coordinates" x, y
y = yH
ψ
φ
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
Other properties
J J
T Ωthin
thin
fat
fat
MP bh
MP bh
fat rings ~ ("drilled-through") MP black holesthin rings ~ (circular) black strings
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
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
Questions?
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
• 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
• 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
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
• 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
• 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)
• 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
iν
](t, zi) = σ
µ , σµ → aµνσν , aµν ∈ O(1, p)
• 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)
A Blackfold Bestiary
"Bestiary: a descriptive or anecdotal treatise on various real or mythicalkinds of animals, esp. a medieval work with a moralizing tone.''
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)
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
• 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)
• 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
• 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
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
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
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
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)
• 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
• 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
~
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)
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
• 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
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
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
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