[PPT] - Resolving the Structure of Black Holes IHES September 19, 2013 PowerPoint Presentation

SLIDE 1

Resolving the Structure of Black Holes

IHES September 19, 2013

SLIDE 2

Recent work with:

Iosif Bena, Gary Gibbons

N. Bobev, G. Dall’Agata, J. de Boer, S. Giusto, Ben Niehoff,
M. Shigemori, A. Puhm, C. Ruef, O. Vasilakis, C.-W. Wang

Outline

Motivation: Solitons and Microstate Geometries
Smarr Formula: “No Solitons without Horizons”
Topological stabilization: “No Solitons without Topology”
Conclusions

Based on Collaborations with:

New scales in microstate geometries/black holes
Microstates and fluctuations of microstate geometries

SLIDE 3

Solitons versus Particles

Electromagnetism:

Self-consistency/Completeness: Motion of particles should follow

from action of electromagnetism ...

Divergent self-energy of point particles ...

★ Replace point sources by smooth “lumps” of classical fields

Mie, Born-Infeld: Non-linear electrodynamics

⇒

General Relativity Non-linearities ⇒ new classes of solitons? Four dimensional GR, electromagnetism + asymptotically flat: “No Solitons without horizons” Yang-Mills

Non-abelian monopoles and Instantons

Nearest thing: Extreme, supersymmetric multi-black-hole solutions

SLIDE 4

Hawking Radiation versus Unitary Evolution of Black Holes

Black hole uniqueness ⇒ Universality of Hawking Radiation Independent of details and states of matter that made the black hole Entangled State of Hawking Radiation

0 0 H 1 1 H

0 0 1 1

+

√2 1 (

)

Evaporation of the black hole: Sum over internal states ⇒ Pure state → Density matrix Complete evaporation of the black hole ⇒

Loss of information about the states of matter that made the black hole

Entanglement of N Hawking quanta with internal black hole state = N ln 2 Complete evaporation + Entanglement ⇒

Hawking radiation cannot be described by a simple wave function

Tension of Black hole uniqueness and Unitarity of Quantum Mechanics

SLIDE 5

Mathur (2009): Corrections cannot be small for information recovery ⇒ There must be O(1) to the Hawking states at the horizon.

Fix with small corrections to GR?

0 1 1 0

+ ε2 + (ε1

)

+ ε (

)

0 0 1 1

0 0

1 1

+

√2 1 (

)

Restore the pure state over vast time period for evaporation? Entangled State of Hawking Radiation

Is there a way to avoid black holes and horizons in the low energy

(massless) limit of string theory = supergravity?

Can it be done in a manner that looks like a black hole on large

scales in four dimensions? Are there horizonless solitons? New physics at the horizon scale?

SLIDE 6

Microstate Geometries:

Definition

Smooth, horizonless solutions with the same asymptotic structure as a

given black hole or black ring

Solution to the bosonic sector of supergravity as a low energy limit
f string theory

Simplifying assumption:

Singularity resolved; Horizon removed

Time independent metric (stationary) and time independent matter

Smooth, stable, end-states of stars in massless bosonic sector of string theory? This is supposed to be impossible because of many no-go theorems: Intuition: Massless fields travel at the speed of light ... only a black hole can hold such things into a star. “No Solitons without horizons”

SLIDE 7

The Komar Mass Formula

M = − 1 16πGD (D − 2) (D − 3) Z

SD−2 ∗dK

In a D-dimensional space-time with a Killing vector, K, that is time-like at infinity

ne has

where SD-2 is (topologically) a sphere near spatial infinity in some hypersurface, Σ.

SD-2

∗dK ≈ − (∂ρg00) ∗ (dt ∧ dρ) K = ∂ ∂t

g00 = − 1 + 16πGD (D − 2) AD−2 M ρD−3 + . . .

d ∗ dK = − 2 ∗ (KµRµνdxν) More significantly

Rµν = 8πGD ⇣ T µν − 1 (D − 2) T gµν ⌘

≈ Z

Σ

T 00 dΣ0

Σ

If Σ is smooth with no interior boundaries:

linearized

M = 1 8πGD (D − 2) (D − 3) Z

Σ

KµRµν dΣν

SLIDE 8

Smarr Formula I SD-2

Σ

H1 H2

More generally, Σ will have interior boundaries that can be located at horizons, HJ.

⇠ = K + ~ ΩH · ~ LH

8πGD (D − 3) (D − 2) M = Z

e Σ

RµνKµ dΣν +

1 2

X

HJ

Z

HJ

∗dK Excise horizon interiors:

Σ → e Σ

Null generators of Kerr-like horizons:

ξara ξb = κ ξb

Surface gravity

f horizon, κ

⇒

1 2

Z

H

∗dξ = κ A

1 2

X

HJ

Z

HJ

∗dK = X

HI

h HI AHI + 8⇡GD ~ ΩHI · ~ JHI i

⇒

Vacuum outside horizons:

8⇡GD (D − 3) (D − 2) M = X

HI

h HI AHI + 8⇡GD ~ ΩHI · ~ JHI i

SLIDE 9

Smarr Formula II: No Solitons Without Horizons

Goal: Show that

Z

Σ

RµνKµ dΣν

= Boundary term (with no contribution at infinity) If Σ is a smooth space-like hypersurface populated only by smooth solitons (no horizons) the one must have:

⇒ M ≡ 0

Space-time can only be globally flat, R4,1

⇒ “No Solitons Without Horizons” ....

Positive mass theorems with asymptotically flatness: ✦ Not true for massive fields ... but (almost) true for massless fields

SD-2

Σ

SD-2

Σ

8πGD (D − 3) (D − 2) M = Z

Σ

RµνKµ dΣν

If Σ is smooth with no interior boundaries:

SLIDE 10

It all comes down to: ∗(KµRµνdxν) = d(γD−2) M = 1 8πGD (D − 2) (D − 3) Z

Σ

∗D (KµRµνdxν) and “No solitons without horizons” requires showing that for some global (D-2)-form, γ.

SLIDE 11

Bosonic sector of a generic massless supergravity

Graviton, gμν
Tensor gauge fields, F(p)K
Scalars, ΦA

Scalar matrices in kinetic terms: QJK(Φ), MAB(Φ) Equations of motion: d❋(QJK(Φ) F(p)K) = 0 Bianchi: d(F(p)K ) = 0 Einstein equations:

Rµν = QIJ h F I

µρ1...ρp−1 F J ν ρ1...ρp−1 − c gµν F I ρ1...ρpF J ρ1...ρpi

+ M AB h ∂µΦA ∂νΦBi

Define: GJ,(D-p) ≡ ❋ (QJK(Φ) F(p)K) and QJK by QIK QKJ = δIJ then: d(F(p)K) = 0 and d(GJ,(D-p)) = 0 = a QIJ F I

µρ1...ρp−1 F J ν ρ1...ρp−1

+ b QIJ GI µρ1...ρD−p−1 GJ ν

ρ1...ρD−p−1

for some constants a,b and c

+ M AB h ∂µΦA ∂νΦBi

SLIDE 12

Time Independent Solutions

Killing vector, K, is time-like at infinity Assume time-independent matter: LKΦA = 0

LKF I = 0 ,

⇔

RµνKµ

Scalars drop out of ⇒

Kµ∂µΦA = 0

LKΦA = 0

LKGI = 0

⇒

+ M AB h Kµ ∂µΦA ∂νΦBi + b QIJ Kµ GI µρ1...ρD−p−1 GJ ν

ρ1...ρD−p−1

a QIJ Kµ F I

µρ1...ρp−1 F J ν ρ1...ρp−1

KµRµν =

Cartan formula for forms:

LKω = d(iK(ω)) + iK(dω)

d(F(p)I) = 0, d(GJ,(D-p)) = 0 ⇒ d(iK(F(p)I)) = 0, d(iK(GJ,(D-p))) = 0

Ignore topology:

iK(F(p)I) = dα(p-2)I , iK(GJ,(D-p)) = dβJ,(D-p-2)

Define (D-2)-form, γD-2 = a α(p-2)J ∧ GJ,(D-p) + b βJ,(D-p-2) ∧ F(p)J

∗(KµRµνdxν) = d(γD−2)

Then:

⇒ M = 0 ⇒ “No Solitons Without Horizons” ....

SLIDE 13

Omissions:

Topology
Chern-Simons terms

d❋(QJK(Φ) F(p)K) = Chern-Simons terms

Equations of motion in generic massless supergravity: ⇒ d(GJ,(D-p)) = Chern-Simons terms

∗(KµRµνdxν) = d(γD−2)

⇒ + Chern-Simons terms

⇒ M ~ Topological contributions + Chern-Simons terms

SLIDE 14

Five Dimensional Supergravity

N=2 Supergravity coupled to two vector multiplets

S = Z √−g d5x ⇣ R − 1

2QIJF I µνF Jµν − QIJ@µXI@µXJ − 1 24CIJKF I µνF J ρσAK λ ¯

✏µνρσλ⌘

QIJ = 1 2 diag

(X1)−2, (X2)−2, (X3)−2

Three Maxwell Fields, F I, two scalars, X I, X 1X 2X 3 = 1

Rµν = QIJ h F I

µρ F J ν ρ − 1 6 gµν F I ρσF J ρσ + ∂µXI ∂νXJi

Einstein Equations:

ds2

5 =

− Z−2 (dt + k)2 + Z ds2

4

Generic stationary metric: Four-dimensional spatial base slices, Σ:

Assume simply connected
Topology of interest: H2(Σ,Z) ≠ 0

SLIDE 15

d(iK(F I)) = 0

iK(F I) = dλI

Kµ QIJ F I

µρ F J ν ρ

= rρ

QIJ I F J ρν

+

1 16 CIJK ✏ναβγδ I F J αβ F K γδ

= Boundary term + Chern-Simons contribution for some functions, λI Simple connectivity

⇒

Dual 3-forms: GI = *5 QIJ FJ

d(GI) = d ∗ (QIJ F J) ∼ CIJK F J ∧ F K

Use iKF J = dλJ ⇒

iK

CILM F L ∧ F M

∼ CILM d

λLF M

Therefore where βI are global one-forms and HI are closed but not exact two forms ... ⇒

d ⇣ iK(GI) +

1 2 CIJK λJ F K⌘

= 0

iK(GI) = dβI −

1 2 CIJK λJF K + HI

Kµ QIJ GI µρσ GJ

νρσ

= 2 rρ

QIJ I σ GJ

ρνσ

1

4 CIJK ✏ναβγδ I F J αβ F K γδ

+ QIJ Hρσ

I GJ ρσν

≠ 0

d(iK(GI)) = − iK(d(GI))

Cartan:

SLIDE 16

Kµ∂µXI = 0

Kµ QIJ F I

µρ F J ν ρ

= rρ

QIJ I F J ρν

+

1 16 CIJK ✏ναβγδ I F J αβ F K γδ

+ QIJ Hρσ

I GJ ρσν

⇒

boundary terms cohomology

Generalized Smarr Formula

M = 3 16πG5 Z

Σ

KµRµν dΣν

= 1 16πG5 Z

Σ

HJ ∧ F J

⇒ Chern-Simons contributions cancel!

Rµν = QIJ h

2 3 F I µρ F J ν ρ + ∂µXI ∂νXJi

+

1 6 QIJ GI µρσ GJ ν ρσ

Kµ QIJ GI µρσ GJ

νρσ

= 2 rρ

QIJ I σ GJ

ρνσ

1

4 CIJK ✏ναβγδ I F J αβ F K γδ

KµRµν = 1

3 rµ ⇥

2 QIJ λI F J

µν + QIJ βI σ GJ µνσ

⇤ +

1 6 QIJ Hρσ I GJ ρσν

iK(GI) = dβI −

1 2 CIJK λJF K + HI

SLIDE 17

No Solitons without Topology

M = 3 16πG5 Z

Σ

KµRµν dΣν

= 1 16πG5 Z

Σ

HJ ∧ F J

If Σ is a smooth hypersurface with no interior boundaries The mass can topologically supported by the cohomology H2(Σ,R)

SD-2

Σ

Stationary end-state of star held up by topological flux ...

Black-Hole Microstate?
A new object: A Topological Star

Only assumed time independence: Not simply for BPS objects

SLIDE 18

A Class of BPS Examples

Large families of BPS solutions where the four-dimensional spatial base, Σ, is a circle fibration over flat R3:

R3 S1 y(i) y(j) y(k) R3 S1 y(i) y(j) y(k)

Δij Δjk

R3 S1

Non-trivial 2-cycles, Δij: S1 fiber along any curve from y(i) to y(j)
Fiber pinches off at special points, y = y(i)
Intersection matrix computed from orientations at intersection points y(i)

σAK ≡ σijK = Flux of FK through Ath cycle in H2(Σ,R)

y(i) y(j)

Δij

σij

K

σij

K ≡

Z

∆ij

F K

Fluxes:

SLIDE 19

Charge and Mass

Chern-Simons Interaction:

rρ

QIJF J ρ

µ

=

1 16 CIJK ✏µαβγδ F J αβ F K γδ

Electric Charge, QI ~ Intersection of Magnetic fluxes FJ ⋀ FK

IAB ≡ Inverse of the Intersection Form

QI = − CIJK IAB σJ

A σK B

M = − 1 32πG5 CIJK αI Z

Σ

F J ∧ F K

M = 1 16πG5 Z

Σ

HJ ∧ F J

Mass: BPS where

αI ≡ Z (XI)−1

∞

= normalization of U(1) couplings

M = αI QI

⇒ BPS condition Mass formula is more general for non-BPS examples

SLIDE 20

Resolving Black Holes by Geometric Transitions

Chern-Simons Interaction is also the “dynamical key” to the geometric transition that resolves the singularity and removes the horizon of a black hole

E ~ Q

E ~ (σ)2 σ Electric Charge, QI

Singular charge source Smooth cohomological fluxes blowing up 2-cycles phase transition

~ Magnetic fluxes σJ ⋀ σK

rρ

QIJF J ρ

µ

=

1 16 CIJK ✏µαβγδ F J αβ F K γδ

Standard description of black holes in string theory: Singular brane sources Geometric transitions: Singular brane sources Smooth fluxes New phase of black-hole physics?

SLIDE 21

Spin Systems and Bubble Equations

Each 2-cycle, or bubble, has an intrinsic angular momentum:

Γij ≡ ± 1

6 CIJK σI ij σJ ij σK ij

∼

QK σK

ij

~ Jij = 8 Γij ~ yi − ~ yj |~ yi − ~ yj|

y(i) y(j)

σij

K

Jij

The Bubble Equations No closed time-like curves near special points, y = y(i)

X

j6=i

Γij |~ yi − ~ yj| = 1

i

(excluding center of mass). Fixed fluxes + N points: (N-1) constraints on 3(N-1) variables, y(i) 2(N-1) dimensional moduli space ⇒

SLIDE 22

y(i) y(j)

Δij

Rough picture of the classical moduli space

Bubbles + Flux ⇒ Expansion force ⇒ Equilibrium BPS Configuration Gravity tends to try to collapse the 2-cycles ... Size of bubble = Separation of points y(i) when attraction balances fluxes expansion

y(1) y(2) y(3) y(4) y(5) y(6)

L

R3 ϑ,φ

Fluxes fix N-1 lengths, “L”
2(N-1) moduli: θ, ϕ ...

σij

K

X

j6=i

Γij |~ yi − ~ yj| = 1

i

SLIDE 23

Four Dimensions: Multi-Black Hole Solutions

If the S1 fiber scale remains finite at infinity then Σ ~ R3 × S1 and one can compactify to an effective four-dimensional description The fixed points, y(i), of the S1 action are singular from a four-dimensional perspective. ⇒ Multi-Black-Hole Solutions

y(i) y(j) y(k) R3

Denef: Quiver Quantum Mechanics

X

j6=i

Γij |~ yi − ~ yj| = 1

i

“Integrability conditions”

Classes of marginally bound configurations
Walls of marginal stability where some black hole centers fly off to infinity
Wall crossing formulae for degeneracies of states in dual quiver
Classes of bound solutions with no walls of marginal stability

Rich and complex moduli space: Higher-dimensional geometric significance was not appreciated/visible ...

SLIDE 24

Scaling Solutions

Very important class of solutions to bubble equations where one can have:

X

j6=i

Γij |~ yi − ~ yj| = 1

i

|~ yi − ~ yj| → 0

Simplest example: Three points where the Γij satisfy the triangle inequalities:

Γ13
≤
Γ12
+
Γ23
+ permutations

More generally, clusters can scale to zero size in R3: Apparently very singular ....

|~ yi − ~ yj| ≈

Γij
,

→ 0

y(1) y(3) y(2) λ"13 λ"23 λ"12

Homology cycles appear to be collapsing ... Singular corners of moduli space? Not from the five dimensional perspective ... Signs of Γij cause λ-1 terms to cancel in bubble equations

SLIDE 25

Solutions to bubble equations with clusters

f points converging:

r23 r34 r42 r14 r12 r13

y1 y2 y3 y4 r23 r34 r42 r14 r12 r13

y1 y2 y3 y4

r23 r34 r42 r14 r12 r13

y1 y2 y3 y4

rij ≡ |~ yi − ~ yj| → 0

These solutions look exactly like an extremal black hole with a long AdS throat capped off by “bubbled geometry.”

ds2

5 =

− Z−2 (dt + k)2 + Z ds2

4

Five Dimensional Geometry of Scaling Solutions

Five-dimensional metric has warp factors: In scaling limits, Z diverges in precisely the correct manner to open up an AdS2 × S3 (or AdS2 × S3) throat Closer scaling ⇔ Deeper throat Homology cycles limit to a fixed scale determined by horizon area

SLIDE 26

★ Bound states, no walls of marginal

stability:(?) Configurations are trapped in long AdS throat ...

★ Macroscopic solutions whose bubbles are much larger than

String/Planck scale ⇒ Supergravity approximation is valid

★ They look like classical black holes until

ne is very close to the horizon

★ The “foam” starts at/extends to the

scale of the classical black hole horizon

★ Failure of Black-Hole Uniqueness ★ Apply AdS/CFT in throat: These geometries describe black-hole microstates.

Comments

★ Far more general classes with bubbles

whose shapes fluctuate as functions of the extra dimensions ⇒ “Microstate Geometries” ⇒ Capture more black-hole microstate structure

★ Cycles/bubbles limit to finite size

SLIDE 27

New Parameters for Black Hole Physics?

Two new scales: Classically free parameters Geometry/Holographic Field Theory:

★ The Transition scale, λT = Scale of a typical 2-cycle

~ Flux quanta on typical 2-cycle × lp

★ The Depth of the Throat ~ Maximum Red/Blue shift, zmax

λgap = redshifted wavelength, at infinity of lowest mode of bubbles at the bottom of the throat.

★ The Depth of the Throat: Determines Energy gap in dual Field theory

Geometric transition represents a transition to a new infra-red phase e.g Holographic duals of N=1 gauge theories Transitioned geometry ➞ Confining phase; fluxes = gaugino condensate

★ The Transition Scale:

Fluxes =

VEV of Order parameter of new phase

Scale of Bubbles = New Dynamically Generated scale of field theory

Egap ~ (λgap)-1

SLIDE 28

Quantization of Geometries: The Energy Gap

The depth of the AdS throat is a very sensitive function of the orientations of these angular momenta and quantization can make vast, macroscopic changes in geometry Semi-classical quantization of moduli space: ⇔ quantizing these angular momenta

⇒

y(i) y(j)

σij

K

Jij

Each bubble has an intrinsic angular momentum

Limits throat depth: Fixes Maximum Red/Blue shift, zmax, and sets

Egap in the dual field theory

Cuts off or “compactifies” phase-space volume of long throats.
Can wipe out vast regions of smooth geometry in which curvature is

small and supergravity is a good approximation Semi-classical quantization

Bena, Wang and Warner, arXiv:0706.3786 de Boer, El-Showk, Messamah, Van den Bleeken, arXiv:0807.4556 arXiv:0906.0011

The y(i) cannot be precisely localized

⇒

SLIDE 29

Non-BPS Microstate Geometries

Many examples of extremal, non-BPS microstate geometries A handful of non-extremal microstate geometries ...

Jejjala, Madden, Ross and Titchener, hep-th/0504181

Non-extremal microstate geometries: A completely open problem ... Many five-dimensional axi-symmetric BPS examples: U(1)2 × R symmetry Effectively a two-dimensional problem: Can be reduced to a scalar coset

Apply inverse scattering methods?
Care with topology + Chern-Simons terms

Five-dimensional axi-symmetric, non-extremal solutions with U(1)2 × R symmetry? Smarr formula in five dimensions:

Q = |σ+|2 − |σ−|2

M = |σ+|2 + |σ−|2

Self-dual fluxes, σ+, anti-self-dual fluxes, σ- BPS ⇔ purely self-dual or purely anti-self-dual cohomology ..

SLIDE 30

BPS Fluctuating Bubbled Geometries

The geometric transition stabilizes a fuzzball against gravity and makes microstate geometries possible ... this happens at scales ~ λT Bubbled geometries can have BPS shape fluctuations that depend upon “transverse/internal dimensions.” These shape fluctuations can go down to

Egap and/or the Planck scale, lp.

Extensive work in five-dimensions: BPS shape fluctuations on 2-cycles depend upon functions of one variable: Huge amount of entropy lies in the shape fluctuations... Is it enough to give a semi-classical picture of the black-hole entropy? Expect entropy like that of a supertube

S ∼ p Q1Q2 ∼ Q

shape mode

S ~ Q3/2 BPS black holes in five-dimesions: Such fluctuating geometries as functions one variable cannot capture the sufficient of dynamics underpinning the black hole entropy ...

SLIDE 31

BPS Microstate Geometries in Six Dimensions

The superstratum: Conjectured object

Bena, de Boer, Shigemori and Warner, 1107.2650

Extra circle is now fibered over every five-dimensional 2-cycle ⇒ 3-cycle. Make the fluctuating cycles in five-dimensions also depend upon new U(1) fiber ... and still be a BPS state? Completely new class of BPS soliton is six dimensions

Completely smooth (microstate geometry)
Defined by a topological 3-cycle

fluctuates as functions two variables S ~ Q3/2 ???

New class of solitonic bound state in string theory

Construction of examples?

SLIDE 32

Final Comments

Microstate Geometry program: Classify and study smooth, horizonless

solutions to supergravity. A much richer subject than previously expected

Emerge from geometric transitions:

Singular brane sources → Smooth cohomological fluxes New phase of black hole ... bubbles start before horizon forms

Fluctuations of transitioned geometries: Scale Egap. Capture the entropy?

Miraculous existence through spatial topology and Chern-Simons terms

Mechanism for supporting matter before a horizon forms
Transition scale, λT = Scale of individual bubbles:

Not fixed classically, large values entropically favored? λT >> lp ?

Generalized “no go” theorem for semi-classical solitons in string theory:

If the space-time is even remotely classical, then only topological fuzz at the horizon scale can support a soliton: No Solitons without Topology

Multiple scales: The Horizon scale, M; The Transition scale, λT;