Strings and Black Holes

Erik VerlindeInstitute for Theoretical Physics

University of Amsterdam

General Relativity

12 8R Rg GTµν µν µνπ− =

Gravity = geometry

Einstein: geometry => physics

Strings: physics => geometry

• 1916 Schwarzschild solution.

Einstein: ``I had not expected that the exact solution to the problem could be formulated. Your analytical treatment to the problem appears to me splendid.''

• 1939 Gravitational collapse: “frozen” or “black” stars.

• 1959~ Global structure, exact solutions.

• 1967 “Black hole” singularity theorems.

• 1975 Black hole thermodynamics.

• 1994 Holographic principle.

. 1996 Microscopic origin of entropy.

A Brief History of Black Holes

Oppenheimer-Volkov

Penrose-Hawking

Bekenstein-Hawking

‘t Hooft- Susskind

Strominger-Vafa

Kruskal, Kerr, Newman

• 1968 Veneziano amplitude, dual models

• 1971 Bosonic and fermionic string theory

• 1976 Superstrings as theory of quantum gravity

• 1984 Anomaly cancellation, heterotic strings

• 1995 String dualities, D-branes

• 1996 Microscopic origin of black hole entropy

• 1997 AdS-CFT correspondence and holography.

A Brief History of String TheoryNambu-Goto

Neveu-Schwarz-Ramond

Witten, Polchinski

Strominger-Vafa

Green-Schwarz

Scherk-Schwarz

Maldacena

Black Holes

• Hawking Radiation • Information Paradox • The Holographic Principle

Black Holes in String Theory

• Strings and D-branes• Microscopic Origin of Black Hole Entropy• Holography in String Theory

Outline

Hawking Radiation

3

8cT

kGMπ=

h

Black Holes emit thermal radiationwith temperature

Note: depends on 4 fundamental constants of Nature!!

2

GMaR

=2exp exp c

kT aω πω⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠h

Boltzman factor determined by acceleration at horizon

Bekenstein-Hawking Entropy

2, 44cAS A R

Gπ= =

h

What is the origin of this entropy?

Hawking 1975

Black Hole ThermodynamicsCharged Black Holes:

2

2

2( ) 1 M QH rr r

= − +

( )2

2 2 2 2 2 2( ) sin( )

drds H r dt r d dH r

θ θ φ= − + + +

Reisner-Nordstrom

22 2 2 2( )

( )drds H r dt r d

H r= − + + Ω

Inner and outer horizon

2 2r M M Q± = ± −

Electric Potential, Entropy and Temperature

( )2

14

T r rrπ + −+

= −2S rπ +=2

2

Qr+

Φ =

obey the first law of thermodynamics

TdS dM dQ= + Φ dJ+Ω

2

1 Qr

⎛ ⎞→ −⎜ ⎟⎝ ⎠

Extremal Black Holes

M Q=

0T →

Einstein equation as equation of state

Raychaudhuri equation (no shear or vorticity)

212

vv

d R k kd

µµ

θ θλ= − − ;k µ

µθ =

can be integrated using the Einstein equation

21 ( )8

vvT k k O

Gµ

µθ λ λπ

= +

0θ =

horizon

0k kµ µ =

Ted Jacobson

= local version of 1st law of thermodynamics

Tds ρ=

The Information Paradox (early ’90’s)

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The Holographic Principle (1994)

Planck Area

‘t HooftSusskind

Strings, D-branes and GravityOpen strings: gauge interactions

Closed strings: gravity

, IA Xµ

, ,g Bµν µν Φ

D-branes: gauge theory on worldvolume.source for gravity:

gravity induced byopen string loops

D-branes as Black Holes

D-branes wrap certain “cycles” insidecompactification manifold. They becomecharged massive objects: (extremal) black holes.

Their world-volume theory gives a microcopicdescription of the states associatedwith the black holes.

( )2 2 2 2 21 ( )( )

ds dt H r dr r dH r

= − + + Ω

2

( ) 1 QH rr

⎛ ⎞= +⎜ ⎟⎝ ⎠

iiQ Qφ=

Microscopic Origin of Entropy

2 2sM N −= l

Excited string states have high degeneracy

#states exp cNπ≈

2 2exp exps pM Mπ πl l

But not enough to explain black hole entropy

2#states exp Qπ≈

For extremal black holes

D-brane described microscopically by “gas of strings” with

2c Q= 2N Q=

Exact counting

2 2( , )4AS Q P P Qπ= = 2 2( , ) log ( , )S Q P D Q P=

Extremal “dyonic” black holes:

Generating function:

( ) ( )

, ,

( , ) 1c nmtN sM nt ms

N M n m

D N M e e−− − − −= −∑ ∏

41( )1

nN

nn n

qc N qq

⎛ ⎞+= ⎜ ⎟−⎝ ⎠

∑ ∏

with:

AdS/CFT CorrespondenceAnti-de Sitter- Conformal FieldTheory

Near horizon geometry of a D3-brane

22 2 2 2 2

2(1 )1drds r dt r d

r= − + + + Ω

+

55AdS S×

AdS metric

Strings on

= dual to a CFT: N=4 SuperYang-Mills

( , ) ( , )CFT stringZ g Z gφ φ=

22 2 2 2( )

( )drds H r d t r d

H r= − + + Ω

AdS black hole = thermal CFT

22( ) 1 MH r r

r= − +

quarks

Boundary: 4D Minkowski space

Hawking radiationWilson loops

Bulk: 5D anti-de Sitter space

Black holes

strings

4D Attractor Black Holes and Entropy

( ) 2, 2Im ( )iF Q F QΦ = + Φ( ) ( ) ( ), , ,FS Q P F Q QΛ

Λ

∂= Φ − Φ Φ

∂Φ

( ),FP QΛ Λ

∂= Φ∂Φ

Ooguri, Strominger, VafaEntropy as Legendre transform

( )( ),

Re

Re

X Q

F P

Λ Λ

Λ Λ

=

=( ) , ,

,,

Q PS Q P X F X F

ΛΛΛ Λ

⎡ ⎤= −⎢ ⎥⎣ ⎦

Semiclassical entropy

( ) ( )expX iF XΨ =

Mixed partition function factorizes as

( ) ( )2

, P

pQ P e Q i− ΦΩ = Ψ + Φ∑

( , ) log ( , )S Q P D Q P=

, ,( , ) ( ) ( )P Q P QP Q dΩ = ΦΨ Φ Ψ Φ∫, ( ) ( )i P

P Q e QΦΨ Φ = Ψ +Φ

• Flux vacua as wave functions on moduli space

• Relative probability determined by entropy

Flux Wave Functions

. ..

.. .

...

.. .

,P QX

( ),P Q XΨ

• Moduli fixed by fluxes : discrete points.

The Entropic Principle

Flux Vacua

, , exp ( , )P Q P Q S P QΨ Ψ ≈

Entropic Principle

• Nature is (most likely) described by state of maximal entropy

• Constructive way to select vacua (in contrast with “Anthropic Principle”)