String theory and black holes

Jan de Boer, Amsterdam

Padova, September 8, 2011

XVII European XVII European Workshop on Workshop on String Theory String Theory 20112011

Outline:

1. Introduction

2. Entropy computation

3. Microscopic degrees of freedom

4. Non-locality

5. Quantization of gravity

6. Dynamics

7. Instabilities

8. Other stuffPre-emptive apology for all missing references

Introduction

String theory has many black holes solutions.

At low energies string theory reduces to general relativity, so the main interest in studying black holes in string theory is to do things beyond GR.

In particular, questions like:

-microscopic interpretation of the entropy

-information loss paradox

-is gravity emergent and how?

-how to resolve the singularity?

will require going beyond classical GR.

Entropy Computations

The entropy of a black hole is to leading order given by

S = A4G

There is a lot of beautiful work systematically correcting this with higher order gravity/stringy corrections. (Wald entropy, Sen’s entropy function formalism, etc).

This does not provide a microscopic description of the entropy though.

For two classes of black holes, one can hope to unravel the microscopic degrees of freedom:

1. Black holes that are bound states of branes (or U-dual to bound states of branes or perturbative string states).

2. Black holes that can be put in AdS.

Case 1 reduces to the problem of counting the number of bound states of branes. Often a difficult problem.

Asymptotic Minkowski Space-Time

AdSThroat

Near HorizonGeometry

Black Hole

Asymptotic Minkowski Space-Time

AdSThroat

Near HorizonGeometry

Black Hole

Asymptotic AdS Space-Time

Near HorizonGeometry

Black Hole

In case 2, we can count the number of states in the dual field theory if

(i) We know the precise dual description of the black hole

(ii)We can determine the number of states in the dual field theory at strong coupling.

Because of (ii), black holes in AdS3 are great, because the growth of the number of states in the dual field theory is universal and given by the Cardy formula.

Subleading corrections are not universal and require more detailed knowledge of the relevant CFT.

The first counting of black hole entropy in string theory (Strominger+Vafa) is of this type:

The first counting of black hole entropy in string theory (Strominger+Vafa) is of this type:

This is just the metric of BTZxS3. BTZ is the three-dimensional AdS3 black hole. Can now use 2d CFT to do the counting.

In view of this, restrict to AdS black holes for remainder of talk.

Supersymmetric black holes are much easier than non-supersymmetric black holes. To do the counting, can try to use an index similar to Witten index

I = TrH(¡1)F

The index does not depend on the coupling constant and can e.g. be computed in free field theory.

The lore is that “everything that can be lifted will be lifted”.

λ0 ∞

E

bosonfermion

However, for 1/16-BPS black holes in AdS5xS5 one finds that

< black hole entropy < Total number of 1/16-BPS states at zero coupling

Suggests an extra dynamical mechanism that keeps states massless???

Similar issues with several other black holes (even in AdS3…)

Another puzzle: only a four-parameter family of 1/16-BPS black holes is known whereas the BPS bound suggests there should be five parameters?

Index at zero coupling

Kinney, maldacena, minwalla, raju

kunduri, lucietti, reall

Bena, chowdhury, JdB, El-Showk, Shigemori

¢ = J1 + J2 + Q1 +Q2 +Q3

Microscopic degrees of freedomThe underlying quantum degrees of freedom of gravity in AdS are those of the dual field theory. To improve our understanding of quantum gravity it would be great to have direct access to the individual degrees of freedom.

One can canonically quantize small fluctuations around the vacuum AdS solution and one finds special degrees of freedom which one can identify in the field theory: usually, these correspond to states that preserve a certain amount of supersymmetry.

This is only true for vacuum AdS. Small fluctuations in a black hole background, for example, correspond to “quasi-particles/phonons”.

In some cases one can even do better: one can find families of exact solutions of (super)gravity that can be quantized and identify the corresponding states in the dual field theory.

Two examples:

1. LLM geometries in 10d supergravity = ½-BPS states in N=4 d=4 super Yang-Mills theory

2. LM geometries in 6d supergravity = ½-BPS states in some d=2 superconformal field theory (“fuzzball” geometries)

Within these examples, one can analyze the relation between microscopic degrees of freedom and geometries in considerable detail.

See e.g. the review arXiv:0811.0263 – Balasubramanian, JdB, El-Showk, Messamah

½-BPS states in AdS5 – geometries classified by Lin, Luninand Maldacena

ds2 = ¡h¡2(dt+ Vidxi)2 + h2(d´2 + dxidxi) + ´eGd­2

3 + ´e¡Gd~­23

h¡2 = 2´ coshG; ´@´Vi = ´ij@jV; ´(@iVj ¡ @jVi) = ²ij@´z

z = 12 tanhG; z(´; x1; x2) = ´2

¼

Rdy1dy2

12¡u(y1;y2)

[(x¡y)2+´2]

Smooth geometries: . Function u defines a droplet in the -plane.

u 2 f0; 1gx1; x2

x1

x2

½-BPS configurations for AdS3 x S3 x M4

½-BPS states: States in the Fock space of b1+b3 fermions and b0+b2+b4 bosons with L0=N. (bi=dim Hi(M4)).

½-BPS geometries (Lunin-Mathur):

ds2 = 1pf1f5

[¡(dt+A)2 + (dx1 +B)2] +pf1f5(dx

22 + : : :+ dx2

5) +q

f1f5ds2M

Ai = Q5

L

R L0

F i(s)ds

j~x¡~F (s)j2 ; dB = ¤4dA; Q1 = Q5

L

R L0j ~F (s)j2ds

f5 = 1 + Q5

L

R L0

ds

j~x¡~F (s)j2 ; f1 = 1 + Q5

L

R L0

dsj~F (s)j2

j~x¡~F (s)j2

~F (s) describes a curve in the spanned by x2; : : : ; x5R4

Some results:

§Most pure states do not have a well-defined classical geometrical description – only coherent states do and correspond to geometries without horizons.

§Most mixed states do not have well-defined classical geometrical description, but some do – thermal ensembles and generalizations thereof. Precise rules not clear.

§Generic pure states are virtually indistinguishable from thermal states with the same quantum numbers.

§Coherent states correspond to states with extended features, non-local if you wish.

§The standard local picture of gravity emerges only after averaging over the underlying non-local degrees of freedom. In this sense, gravity and spacetime are emergent phenomena.

Wave-function for the metric is narrowly peaked around its expectation value

One thus has a precise understanding of the microscopic degrees of freedom of quantum gravity in these toy examples. In particular, one can make sense of the notion of “adding” geometries, and show that

+ + +…

Unfortunately, this only works in situations with a lot of supersymmetry. No large macroscopic black holes which preserve that many supersymmetries exist. Of course, we would like to generalize this picture to large, macroscopic black holes.

Therefore, significant effort is directed towards finding examples with less or even without supersymmetry.

For generic states, one does not expect to be able to identify states simultaneously at weak coupling (in the field theory) and at strong coupling (in weakly coupled gravity).

Put these objects with charges at locations ¡i = (Pi; Qi) ~xi 2 R3

For large supersymmetric black holes in 4d N=2 supergravity one can construct large families of smooth solutions of the field equations of supergravity with the same quantum numbers as these black holes using distributions of electrically and magnetically charged objects in three dimensions: do these give rise to the microscopic degrees of freedom that make up the black hole?? (see later)

Progress in d=4:

hh;¡ii+P

j 6=ih¡j ;¡iij~xj¡~xij = 0

There are corresponding solutions of the field equations only if (necessary, not sufficient)

Here, is the electric-magnetic duality invariant pairing between charge vectors. The constant vector h determines the asymptotics of the solution.

h¡1;¡2i = P1 ¢Q2 ¡ P2 ¢Q1

~J= 14 Pi6=jh¡ i;¡j i~xi ¡~xjj~xi ¡~xjj

Progress in d=5: (lot of previous work of Bena, Warner and collaborators)

We proposed that one can construct “superstrata”, generalizations of supertubes. At each point P, locally one has a flat brane with some charges which is ½-BPS.

Susy of 1/8 BPS black hole = ∩P

Susy preserved at P

This analysis suggests there exist supersymmetricsuperstrata whose profile can depend on arbitrary functions of two variables. The supergravity description of these objects could possibly give rise to a large number of new microstates.

Bena, JdB, Warner, Shigemori

If true, will this provide a complete description of all microscopic degrees of freedom of a large supersymmetricblack hole?

Non-locality

Physics near black holes has to break down in a non-local way. In order to have no information loss, higher order corrections are not sufficient. A new type of breakdown of effective field theory is required, or one needs remnants.

This is nicely review in Mathur (arXiv:1012.2101).

Fuzzballs have the potential to do exactly this: each fuzzball geometry has large, extended, non-local features.

One should however not confuse black holes with fuzzballgeometries. Fuzzball or microstate geometries are highly atypical states. A typical state is a linear combination of a large number of fuzzballs which is well approximated by the black hole geometry.

We can see a more explicit example of breakdown of locality using the d=4 geometries described above.

In d=4 there are so-called scaling solutions: solutions where the constituents can approach each other arbitrarily closely.

In space-time, a very deep throat develops, which approximates the geometry outside a black hole ever more closely.

None of these geometries has large curvature: they should all be reliably described by general relativity.

However, this conclusion is incorrect!

The symplectic volume of this set of solutions is finite. Throats that are deeper than a certain critical depth are all part of the same ħ-size cell in phase space: wave-functions cannot be localized on such geometries.

Quantum effects become highly macroscopic and make the physics of very deep throats nonlocal.

This is a breakdown of effective field theory similar to what the information paradox requires.

Wave functions have support on all these geometries

The number of states that one gets by quantizing families of smooth supergravity solutions is so far not sufficient to account for the entropy of the large supersymmetric black hole (superstrata remain to be analyzed).

Try to find an upper bound: count the number of states in a gas of BPS supergravitons. Idea is that all smooth BPS solutions are obtained by taking a superposition of free BPS supergravitons and letting the system backreact. Because of the BPS bound, the energy of the system cannot become be lowered.

After all, classical solutions can be thought of as coherent superpositions of gravitons…

Quantization of gravity

We showed that one can do this computation and indeed find an upper bound on the number of smooth supergravitysolutions: the number is insufficient to account for the black hole entropy.

In particular, this suggests that all attempts to quantize gravity on its own are futile and will never lead to a consistent unitary theory with black holes.

This is of course perfectly fine: string theory was invented to yield a consistent quantum theory of gravity, so (as a string theorist) it would have been somewhat disappointing if we could get away with gravity alone.

Ssugra » Q13 ¿ SBH » Q

12

Van den Bleeken, JdB, El-Showk, Messamah

Recent work on 3d quantum gravity via Chern-Simons theory, and on higher spin theories a la Vasiliev seems to support this picture.

There are caveats: e.g. solutions with different topologies, one should include singular solutions, or perhaps it is completely the wrong thing to do if gravity is emergent?

We are also right now exploring the possibility of including so-called non-geometric solutions – generically these seem to be present.

In generality, one probably needs the full machinery of closed string field theory.

JdB, Shigemori

Gopakumar’s talk

Dynamics

Important dynamical questions are:

-Give a precise description of the process of black hole creation. In the field theory the corresponding process is thermalization. Describe how entropy is being produced.

-Give a precise description of what happens when you perturb a black hole, e.g. by throwing something in. Semiclassically, this is a dissipative process.

For black branes, this dissipative process can be described (for long wavelength perturbations) in terms of fluid-gravity duality. For Schwarzschild black holes, this does not work.

For both processes, there is the issue of the relevant timescales. It appears that the relevant time scales are very short (much shorter than in weakly coupled field theory).

For objects falling into a black hole, Susskind has proposed that black holes are the fastest “scramblers” of information that exist in nature. In other words, the timescale is the shortest one possible.

It is not clear how to prove something like this. Perhaps one can use ideas from information theory.

(perhaps related, η/s is also very small for black holes horizons, and there may be a fundamental lower bound for η/s as well)

To compute scrambling time, Susskind proposes to take a charged test particle, to drop it into the black hole and to measure how long it takes for the charge to spread out on the “stretched horizon”.

Susskind, Lindesay

Result: t ~ β log S

Related to classical/quantum chaos? (Barbon)

Field theory computation?

Short thermalization times can be experimentally measured.

What exactly do we mean by thermalization??

In the bulk this corresponds to black hole creation – butwhat is a good probe of black hole creation??

Therefore, entropy as a function of time should provide a good characterization of thermalization.

Do not know a proper bulk description of this.

One possibility: use apparent horizon as a probe. Works quite well in examples but is slicing dependent. (Heller, Janik, Witaszczyk)

Another possibility is to compute the entanglemententropy as a function of time.

An extremely simple case study: infalling shells in AdS.

Bhattacharyya, Minwalla

Lin, Shuryak

Abajo-Arrastia, Aparício, López

Albash, Johnson

Described by Vaidya metric:

ds2 = 1z2 [¡(1¡m(v)zd)dv2 ¡ 2dzdv + d~x2]

mass function

m(v) = M2

(1 + tanh vv0

)

Take translationally invariant shells.

Chesler, Yaffe

Vijay Balasubramanian, Alice Bernamonti, JdB, Neil Copland, Ben Craps, Ekso Keski-Vakkuri, Berndt Müller, Andreas Schäfer, Masaki Shigemori, Wieland Staessens

§Thermalization proceeds from UV to IR and propagates withthe speed of light (cf causality bound). Not with the speed of sound (as hydrodynamic modes). But what is propagating???

§Thermalization of a region of size 1/T proceeds at a ratecompatible with experiment.

§Thermalization time of the order of 1/T, the only scale in the problem for a strongly coupled CFT at finite temperature.

Features from entanglement entropy computations:

Instabilities I am probably out of time by now……

Black holes in AdS, when you make them smaller, undergo a phase transition to thermal AdS.

Rather than equilibriating with their Hawking radiation, they radiate away completely.

In the past few years, it has become clear that black holes have many other possible instabilities which are important for all kinds of applications of AdS/CFT.

Mechanisms for instabilities:

§Tachyon condensation leading to hairy black branes(Gubser) – holographic superconductors

§Superradiant instabilities: the presence of bosonicmodes with exponentially growing Boltzmann factors rather than decaying ones.

§Gregory-Laflamme type instabilities: black holes localizing in compact dimensions, even supersymmetrically

§Fermion pair production near the horizon plus backreaction.

Basu et al, Bhattarya et al

Van den Bleeken, JdB, et al; Bena, JdB et al

Piolone, Troost

§Instabilities that spontaneously break translation invariance: need a Chern-Simons like coupling (possibly something with a pseudo-scalar).

Domokos, Harvey;

Nakamura, Ooguri, Park

Dual field theories of all these phases very poorly understood:

Gregory-Laflamme instability in D1D5 system can be related to short string condensation.

QCD has known instabilities of this type, chiraldensity waves.

Bena, chowdhury, JdB, El-Showk, Shigemori

Deryagin, Grigoriev, Rubakov

Other stuff/things I did not do/conclusions

§Kerr/CFT and its cousins – duals of rotating black holes?

§Complementarity, infalling observers, probing behind the horizon – a better description of local observers.

§Study of cosmological singularities/horizons

§A better description of the breakdown of locality.

§……………………