black hole entropy in string theory

Black Hole Entropy in String Theory A window into the structure of quantum gravity

Journées de la FRIF February 2008

Dourdan

Atish Dabholkar

Atish Dabholkar Quantum Black Holes 2

Collaborators

Frederik Denef, Davide Gaiotto, Joao Gomes,

Renata Kallosh, Nori Iizuka,

Ashik Iqubal, Alex Maloney,

Greg Moore, Suresh Nampuri, K Narayan,

Sameer Murthy, Ari Pakman,

Boris Pioline, Sandip Trivedi,

Ashoke Sen, Masaki Shigemori


Quantum Gravity

• Physical regime where all fundamental constants of nature are simultaneously

important ~ , c, G

• Quantum field theory of gravity is however non-renormalizable. Uncontrolled divergences because G is dimensionful unlike the fine structure constant in QED.


Superstrings

• Superstring theory includes Einstein gravity in the low energy limit.

• Perturbation theory is finite. No short distance divergences (1980s).

• How about nonperturbative effects?

• Black Holes provide an invaluable laboratory and Duality the tools for testing the theory in the regime of strong gravity.


Plan• Black Hole Entropy and Microstates

• Precision Counting of Black Holes

• Quantum cloak for a classical singularity

• Small Black Holes and Rings

• AdS/CFT Holography and its tests

• Fundamental String Hologram


Quantum Gravity

General Relativity Quantum Mechanics

Macroscopic Microscopic

Thermodynamics Statistical Mechanics

Black Hole Quantum Soliton

Geometry Counting

AdS CFT


Consistency Tests of String Theory

Black holes provide a very useful thermodynamic context where quantum gravitational effects are calculable and highly precise tests are possible. They attest in a nontrivial way to the nonperturbative consistency of string theory as a quantum theory of gravity.

• Microscopic explanation of entropy• Dynamical tests of Holography


Black Hole

• Spacetime around a gravitationally collapsed star. Solution of Einstein’s eqn.

• The force of gravity is so strong that even light cannot escape its pull.

• Gravity around a black hole is completely specified by its Mass M, Charge Q, Spin J

No Hair Theorem• There is a singularity inside clothed by an

Event Horizon.


Event Horizon

• One way surface from behind which you cannot come out but will surely meet a singularity where tidal forces are infinite.


Geometric Parameters

• Radius of the horizon R

• Area of the horizon A = 4π R2

• Surface gravity κ = Acceleration at the horizon (e. g. )

For example, for a black hole of mass M

R = 2 GM A = 4π R2 κ= 1/4GM


• Second law of Thermodynamics

• Boltzmann: Entropy S is the logarithm of the total number Ω of microstates of the system at volume V and energy E

Entropy


• The significance of entropy stems from the fact that one can draw important conclusions about the microstructure without using a microscope, from gross thermodynamic, macroscopic properties,.

• Very much as one would do for a piece of metal in condensed matter physics, we would like to use black hole entropy to learn about the microstructure.

Entropy as a window into the microstructure.


Thinking about entropy and the second law in the context of black holes has yielded surprising new insights—

1) Black Hole Entropy 2) Holography—the number of

fundamental degrees of freedom of quantum gravity is vastly smaller and scales with area of space instead of volume. AdS/CFT

Follows from robust considerations of thermodynamics, quantum mechanics, and general relativity.


Entropy of Oxygen Gas

Ω is the number of ways to distribute N molecules of de Broglie wavelengh λ in volume V for dilute gas. Correct Entropy!


Quantum Structure of Matter

• Some aspects of quantum microstructure of matter could be deduced from thermodynamics.

• Gibbs deduced the N! for identical particles without knowing about quantum mechanics or spin statistics theorem. No `microscope’

• Maxwell and Jeans deduced quantum freezing of degrees of freedom and failure of classical equipartition of energy from specific heat of polyatomic gases as early as 1860.


Quantum Structure of Spacetime

• What happens if you throw a bucket of hot water into a black hole? Bekenstein

• The entropy of world outside the black hole would decrease violating second law of thermodynamics. Unless the black hole also has entropy.

• He noted that the area of the black hole behaves like entropy in many ways and always increases.


• If a black hole has energy and entropy it must also have temperature, dE = T dS.

• Because of pair creation near the horizon, a black hole has temperature with a black body spectrum. The Hawking Temperature of a black hole is given by


Black Hole entropy

• A remarkably general formula that involves three fundamental constants of nature. The length is the Planck length 10-33 cm.

• An important clue about the microscopic structure of quantum theory of gravity. There are calculable quantum corrections..


Is S = k log (Ω) ?

• What are the microstates of the black hole that can account for this thermodynamic entropy?

• Note that the entropy of a solar mass black hole is enormously bigger compared to the entropy of the sun itself.

• This was an open problem for over two decades after Hawking’s discovery.


String Theory

• For some supersymmetric black holes with large classical area, one can explain the thermodynamic entropy in terms of microscopic counting. For example, for a black hole with three charges Q1, Q2, Q3

Macroscopic Microscopic Bekenstein-Hawking Strominger-Vafa


Can we compute Corrections?

• Macroscopic (from thermodynamics)

• Microscopic (from counting states)


Solution and Entropy

• Bekenstein-Hawking entropy formula and the laws of black hole thermodynamics follow from Einstein equations.

• These equations follow from an action involving only two derivatives.

• To compute corrections, we need to take into account higher derivative terms and compute their effect both on the solution and the entropy formula.


Einstein-Hilbert Action

• Classical dynamics of a gravitational field is also governed by a two derivative action.

• Here gab, , the spacetime metric, is a dynamical variable like the coordinate y. Riemann tensor Rabcd involves two derivatives of the metric.


Higher Derivative Actions

• Point particle

• General Relativity


Tall Order….

• Action: Find the relevant higher derivative corrections to the action by computing the coefficients C1, C2 as functions of scalars.

• Solution: Solve the resulting nonlinear, higher derivative, partial differential equations to find the solution.

• Wald Entropy: Given the action, find corrections to the Bekenstein-Hawking formula and then evaluate for the solution.


Counting

• Count the states in the quantum Hilbert space of string theory with the same charges and mass as the black hole.

• In general, a rather complicated problem of determining the number of bound states of a collection of solitons.

• In a number of cases this problem can be solved exactly!


Electric States

• Electric charge vector Q

• Degeneracy given by Fourier coefficients of modular forms. For example,

Z(q) =1

´24(q)=

1

qQ(1¡ qn)24 =

Xc(N)qN

(Q) = c(Q2=2)


Dyonic States

• Electric charge Q, magnetic charge P

• Degeneracy given in terms of Fourier coefficients of Siegel Modular forms.

Z(q; p; y) =1

©k(q; p; y)=X

c(N;M;L)qN pM yL

(Q;P ) = c(Q2=2; P2=2; Q ¢ P )


Entropy(Wald)

Action(Topological

String)

Solution(Attractor)

States (Counting)

Black HoleEntropy

Micro

Macro


Dyonic Black Holes

• Compute the entropy from the effective action including the subleading corrections.

• Compute the degeneracy from the asymptotic expansion of the Fourier coefficients for large charges.

• Compare these two completely different computations.


Both obtained by the minimum value of the

same function F of two variables a and

F = ¼2[ a

2+¾2

¾P 2+ 1

¾Q2¡ 2 a

¾Q ¢ P + 128¼Á(a; ¾) + : : :]

Á(a; ¾) = ¡ 164¼2f(n+ 2) log¾ + log jf (n)(a+ i¾)j2g

¾


Small Black Holes

• Electric black holes have `small’ area. In fact classical spacetime is singular. Classical area Ac and hence the Bekenstein-Hawking entropy S is zero. But the degeneracy states is nonzero. How can S equal ??

• Quantum corrections to geometry are essential. Quantum corrected spacetime has a horizon and nonzero area Aq.

k log


Penrose diagram

Classical Geometry Quantum Geometry


Quantum Clothing

• Thus, the spacetime geometry is singular

if we keep only Einstein-Hilbert action.

• Inclusion of higher derivative corrections

cover the nakedness of the singularity and

clothe it with an event horizon.

• Bekenstein-Hawking-Wald entropy can

then be computed for this event horizon.


• We find that microscopic counting gives

• This matches perfectly with macroscopic entropy S (with an assumption about the ensemble). Quantum area

log(Q2=2) » 4¼pQ2=2¡ 27

2logpQ2=2

logp2¡ 675

32¼pQ2=2¡ 675£ 9

2048¼2Q2¡ : : :

Aq = 8¼pQ2=2


Holography

• If black hole has entropy that scales as area, then the number of degrees of freedom of quantum gravity in a volume V must scale with the area A and not the volume. Otherwise, in a gravitational collapse to form a black hole, the entropy would decrease spontaneously.


AdS/CFT

• This heuristic reasoning has found its precise realization in holographic duality.

• A string theory in (d+1) dimensional AdS spacetime is quantum equivalent to a conformal field theory in d-dimensional spacetime. Anti deSitter spacetime is a symmetric space with constant negative curvature. Geometry near the horizon.


Dynamical Tests of Holography

• String Theory on 3d AdS space

is holographically dual to a 2d conformal field theory with target space

AdS3 £ S3 £ T4

(T4)N =SN


Interactions in the CFT

• Set of chiral operators in the boundary theory of weight = charge

• Three point correlators

fOhg

hO(0;0)yh3

O(0;0)h2

O(0;0)h1

i =µ1

N

¶1=2 · (2h3¡ 1)3(2h1¡ 1)(2h2¡ 1)

¸1=2


Interactions in AdS space

• Construct the chiral operators from operators in WZW models based on SL(2, R) and SU(2) at level k (related to N) by imposing BRST invariance etc..

• Compute three point correlators in each of these WZW models

• Put it all together taking care of superconformal ghosts, picture changing...


Three point functions

• For example, for the SL(2, R) factor

• But combining everything dramatically simplifies the answer. Agreement!!

C(a1;a2;a3) =

p¡(b2)b

12 ¡ b

2

¨(b)¨(a+ 2b)

3Y

i=1

¨(a¡ 2ai + b)[¨(2ai + b)¨(2ai + 2b)]1=2

log¨(x) =

Z 1

0

dt

t

"µQ

2¡ x

¶2

e¡ t ¡ sinh2((Q2 ¡ x) t2 )

sinh bt2 sinh

t2b

#


Small Black holes and Higher Derivatives

• The analysis described earlier was using

only four derivative terms in the effective

action. This is enough for computing the

entropy because of an anomaly argument.

• But the geometry is highly curved on string

scale. Six or eight derivative terms as

important for the geometry.


Fundamental String As a Hologram

• We have proposed an exact conformal field theory description that sums the derivative expansion.

• The conformal field theory has the right symmetries, correct entropy, as expected for the small black hole.

• AdS/CFT holography for small black holes. Further checks are needed.


Conclusions

• String theory gives a remarkably consistent framework for quantum gravity where black holes can be regarded simply as thermodynamic ensembles of quantum states like in ordinary statistical physics.

• Entropy matches with great precision even to subleading order in several cases.

• Nontrivial dynamical tests of holography.

black hole entropy in string theory

Documents