Resolving the Structure of Black Holes

IHES September 19, 2013

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

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

Hawking Radiation versus Unitary Evolution of Black Holes

Black hole uniqueness ⇒ Universality of Hawking RadiationIndependent of details and states of matter that made the black hole

Entangled State of Hawking Radiation0 0H

1 1H

0 0 1 1+√21 ( )

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

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+√21 ( )

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?

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 of string theory

Simplifying assumption:

Singularity resolved; Horizon removed

‣ Time independent metric (stationary) and time independent matterSmooth, 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”

The Komar Mass Formula

M = � 116⇡GD

(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 one has

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

⇤dK ⇡ � (@⇢g00) ⇤ (dt ^ d⇢)

K =@

@t

g00 = � 1 +16⇡GD

(D � 2)AD�2M

⇢D�3+ . . .

d ⇤ dK = � 2 ⇤ (KµRµ⌫dx⌫)

More significantly

Rµ⌫ = 8⇡GD⇣Tµ⌫ �

1

(D � 2) T gµ⌫⌘

⇡Z

⌃T 00 d⌃

0

Σ

If Σ is smooth with no interior boundaries:

linearizedM =

1

8⇡GD

(D � 2)(D � 3)

Z

⌃KµRµ⌫ d⌃

⌫

Smarr Formula I

SD-2Σ

H1 H2More generally, Σ will have interior boundaries that can be located at horizons, HJ.

⇠ = K + ~⌦H · ~LH

8⇡GD (D � 3)(D � 2) M =

Z

e⌃Rµ⌫K

µ d⌃⌫ + 12X

HJ

Z

HJ

⇤dK

Excise horizon interiors: ⌃ ! e⌃

Null generators of Kerr-like horizons:

⇠ara ⇠b = ⇠bSurface gravity of horizon, κ ⇒12

Z

H⇤d⇠ = A

12

X

HJ

Z

HJ

⇤dK =X

HI

hHI AHI + 8⇡GD ~⌦HI · ~JHI

i⇒

Vacuum outside horizons:

8⇡GD (D � 3)(D � 2) M =

X

HI

hHI AHI + 8⇡GD ~⌦HI · ~JHI

i

0

Smarr Formula II: No Solitons Without Horizons

Goal: Show that

Z

⌃Rµ⌫K

µ d⌃⌫ = 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 ΣSD-2 Σ

8⇡GD (D � 3)(D � 2) M =

Z

⌃Rµ⌫K

µ d⌃⌫If Σ is smooth with no interior boundaries:

It all comes down to:

⇤(KµRµ⌫dx⌫) = d(�D�2)

M =1

8⇡GD

(D � 2)(D � 3)

Z

⌃⇤D (KµRµ⌫dx⌫)

and “No solitons without horizons” requires showing that

for some global (D-2)-form, γ.

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) = 0Bianchi: d(F(p)K ) = 0

Einstein equations:

Rµ⌫ = QIJhF Iµ⇢1...⇢p�1 F

J⌫⇢1...⇢p�1 � c gµ⌫ F I⇢1...⇢pF

J ⇢1...⇢pi

+ MABh@µ�

A @⌫�Bi

Define: GJ,(D-p) ≡ ❋ (QJK(Φ) F(p)K) and QJK by QIK QKJ = δIJthen: d(F(p)K) = 0 and d(GJ,(D-p)) = 0

= aQIJ FIµ⇢1...⇢p�1 F

J⌫⇢1...⇢p�1

+ bQIJ GI µ⇢1...⇢D�p�1 GJ⌫⇢1...⇢D�p�1

for some constants a,b and c

+ MABh@µ�

A @⌫�Bi

Time Independent SolutionsKilling vector, K, is time-like at infinity

Assume time-independent matter: LK�A = 0LKF I = 0 ,

⇔ Rµ⌫KµScalars drop out of ⇒Kµ@µ�A = 0LK�A = 0

LKGI = 0⇒

+ MABhKµ @µ�

A @⌫�Bi

+ bQIJ Kµ GI µ⇢1...⇢D�p�1 GJ⌫⇢1...⇢D�p�1

aQIJ Kµ F Iµ⇢1...⇢p�1 F

J⌫⇢1...⇢p�1KµRµ⌫ =

•

0

• Cartan formula for forms: LK! = d(iK(!)) + iK(d!)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

⇤(KµRµ⌫dx⌫) = d(�D�2)Then:

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

0 0

Omissions:

• Topology

• Chern-Simons terms

d❋(QJK(Φ) F(p)K) = Chern-Simons termsEquations of motion in generic massless supergravity:

⇒ d(GJ,(D-p)) = Chern-Simons terms

⇤(KµRµ⌫dx⌫) = d(�D�2)⇒ + Chern-Simons terms

⇒ M ~ Topological contributions + Chern-Simons terms

Five Dimensional Supergravity

N=2 Supergravity coupled to two vector multiplets

S =

Z p�g d5x

⇣R� 12QIJF

Iµ⌫F

Jµ⌫ �QIJ@µXI@µXJ � 124CIJKFIµ⌫F

J⇢�A

K� ✏̄

µ⌫⇢��⌘

QIJ =1

2diag

�(X1)�2, (X2)�2, (X3)�2

�

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

Rµ⌫ = QIJhF Iµ⇢ F

J⌫⇢ � 16 gµ⌫ F

I⇢�F

J ⇢� + @µXI @⌫X

JiEinstein Equations:

ds25 = � Z�2 (dt+ k)2 + Z ds24Generic stationary metric:

Four-dimensional spatial base slices, Σ:

• Assume simply connected

• Topology of interest: H2(Σ,Z) ≠ 0

d(iK(FI)) = 0 iK(F

I) = d�I

Kµ�QIJ F

Iµ⇢ F

J⌫⇢�

= �r⇢�QIJ �

I F J⇢⌫�

+ 116 CIJK ✏⌫↵��� �I F J↵� F

K��

= Boundary term + Chern-Simons contribution

for some functions, λISimple connectivity

⇒

Dual 3-forms: GI = *5 QIJ FJ

d(GI) = d ⇤ (QIJ F J) ⇠ CIJK F J ^ FK

Use iKF J = d�J ⇒ iK�CILM F

L ^ FM�

⇠ CILM d��LFM

�

Therefore

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

⇒

d⇣iK(GI) +

12 CIJK �

J FK⌘

= 0

iK(GI) = d�I � 12 CIJK �JFK + H

I

Kµ�QIJ GI µ⇢� GJ

⌫⇢��

= � 2r⇢�QIJ �I � GJ

⇢⌫��

� 14 CIJK ✏⌫↵��� �I F J↵� F

K��

+ QIJ H⇢�I GJ⇢�⌫

≠ 0

d(iK(GI)) = � iK(d(GI))Cartan:

Kµ@µXI = 0

Kµ�QIJ F

Iµ⇢ F

J⌫⇢�

= �r⇢�QIJ �

I F J⇢⌫�

+ 116 CIJK ✏⌫↵��� �I F J↵� F

K��

+ QIJ H⇢�I GJ⇢�⌫

⇒boundary terms cohomology

Generalized Smarr Formula

M =3

16⇡G5

Z

⌃KµRµ⌫ d⌃

⌫ =1

16⇡G5

Z

⌃HJ ^ F J⇒

Chern-Simons contributions cancel!

Rµ⌫ = QIJh23 F

Iµ⇢ F

J⌫⇢+ @µX

I @⌫XJi

+ 16 QIJ GI µ⇢� GJ ⌫

⇢�

Kµ�QIJ GI µ⇢� GJ

⌫⇢��

= � 2r⇢�QIJ �I � GJ

⇢⌫��

� 14 CIJK ✏⌫↵��� �I F J↵� F

K��

KµRµ⌫ = � 13 rµ⇥2QIJ �

I F Jµ⌫ + QIJ �I

� GJ µ⌫�⇤+ 16 Q

IJ H⇢�I GJ⇢�⌫

iK(GI) = d�I � 12 CIJK �JFK + H

I

No Solitons without Topology

M =3

16⇡G5

Z

⌃KµRµ⌫ d⌃

⌫ =1

16⇡G5

Z

⌃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

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 ΔjkR3

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

�ijK ⌘

Z

�ij

FK

Fluxes:

Charge and MassChern-Simons Interaction:

r⇢�QIJF

J⇢µ

�= 116 CIJK ✏µ↵��� F

J ↵� FK ��

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

IAB ≡ Inverse of the Intersection Form

QI = � CIJK IAB �JA �KB

M = � 132⇡G5

CIJK ↵I

Z

⌃F J ^ FKM =

1

16⇡G5

Z

⌃HJ ^ F J

Mass:

BPS

where ↵I ⌘ Z (XI)�1��1 = normalization of U(1) couplings

M = αI QI⇒ BPS condition Mass formula is more general for non-BPS examples

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 fluxesblowing up 2-cycles

phase transition

~ Magnetic fluxes σJ ⋀ σK r⇢

�QIJF

J⇢µ

�= 116 CIJK ✏µ↵��� F

J ↵� FK ��

Standard description of black holes in string theory: Singular brane sources

Geometric transitions: Singular brane sources Smooth fluxes

New phase of black-hole physics?

Spin Systems and Bubble Equations

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

�ij ⌘ ± 16 CIJK �Iij �

Jij �

Kij

⇠�QK �

K�ij

~J ij = 8�ij~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

j 6=i

�ij|~yi � ~yj |

= �1i

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

2(N-1) dimensional moduli space⇒

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

j 6=i

�ij|~yi � ~yj |

= �1i

Four Dimensions: Multi-Black Hole SolutionsIf 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

j 6=i

�ij|~yi � ~yj |

= �1i “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 ...

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

X

j 6=i

�ij|~yi � ~yj |

= �1i

|~yi � ~yj | ! 0

Simplest example: Three points where the Γij satisfy the triangle inequalities:���13

�� ���12

�� +���23

�� + permutations

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

|~yi � ~yj | ⇡ ����ij

�� , � ! 0

y(1)y(3)

y(2)

λ"13λ"23λ"12

y(1)y(3)

y(2)

λ"13λ"23λ"12

y(1)y(3)

y(2)

λ"13λ"23λ"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

Solutions to bubble equations with clusters of points converging:

r23

r34r42

r14

r12

r13y1

y2

y3

y4

r23

r34r42

r14

r12

r13y1

y2

y3

y4

r23

r34r42

r14

r12

r13y1

y2

y3

y4rij ⌘ |~yi � ~yj | ! 0

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

ds25 = � Z�2 (dt+ k)2 + Z ds24

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

★ 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 one 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

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

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

JijEach 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⇒

Non-BPS Microstate GeometriesMany 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 = |�+|2 � |��|2M = |�+|2 + |��|2Self-dual fluxes, σ+, anti-self-dual fluxes, σ-

BPS ⇔ purely self-dual or purely anti-self-dual cohomology ..

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

Q1 Q2 ⇠ 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 ...

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?

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; The Energy Gap, Egap = (λgap)-1; The Planck Length, lp.

Microstate Geometries give a beautiful geometric realization of these ideas