The Black Hole Information Paradox Mathur

7/30/2019 The Black Hole Information Paradox Mathur

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The black hole information paradox

Samir D. Mathur


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Gravity

Thermodynamics Quantum Theory

BlackHoles


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Black hole entropy

Gas with

entropy S

Suppose we throw a box of gas into a black holeThe gas disappears, and we seem to have reduced the

entropy of the Universe

Have we violated the second law of thermodynamics ?


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dSmatter

dt+

dShole

dt 0

So we save second law of thermodynamics ...

Suppose we assume that the hole

has an entropy

Then

S=c3

A

4G A

4G(c = = 1)

(Bekenstein 72,

Hawking 74)


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The entropy puzzle

The black hole has entropy

Statistical Mechanics says that

S = ln(# states)

Thus the black hole should have stateseS

Solar mass : states101077

Where are these states ?

S=

c3

A

4G


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Fix M, look for small distortions of the hole ...

Find NO allowed distortions:

Black hole geometry completely fixed byits conserved quantum numbers:

mass, charge, angular momentum

This would mean S = ln 1 = 0

Where are the states of the black hole ?

Black holes have no hair (John Wheeler)


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Fix M, look for small distortions of the hole ...

Find NO allowed distortions:

Black hole geometry completely fixed byits conserved quantum numbers:

mass, charge, angular momentum

This would mean S = ln 1 = 0

Where are the states of the black hole ?

Black holes have no hair (John Wheeler)


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A more serious (but related) problem: The information paradox


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+_

e+e-

Schwinger pair production process(interesting, but not a problem !)

State of created quanta is entangled

Sent = ln 2

Entanglement entropy

After steps

Sent = N ln 2

N


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The information problem

|02|02 + |12|12

|0n|0n + |1n|1n

. . .

|01|01 + |11|11

M

Schwinger processin the gravitational

field


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Possibilities

Sent = N ln 2

To have this entanglement, the remnant

should have at least internal states2N

But how can we have an unbounded

degeneracy for objects with a given

mass ?

Planck massremnant

Sent = N ln 2

Complete

evaporation

The radiated quanta are in an entangled

state, but there is nothing that they are

entangled with !

They cannot be described by any

wavefunction, but only by a density matrix

failure of quantum mechanics


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The two problems are related ...

A normal body emits radiationfrom its surface, so that theradiation depends on themicrostate. In the black hole theradiation is pulled from the

vacuum

A normal body has visiblemicrostates, so the entropycan be found by countingthem


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What do string theorists say ?


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The entropy problem :

Does represent a count of states ?Sbek =A

4G

(A) There are 10 - 4 = 6 compact directions in string theory

(B) The coupling constant is a free modulus that we can varyg

(C)If we look at states with mass = charge, then their numberdoes not change with (supersymmetric protection of BPS

states)

g


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g 0 g large

All charges are obtained by

wrapping strings, branes etc

along compact directions

Smicro = 2n1n2n3

Entropy is given through number

of intersection points

Expect that gravitational field will

extend around branes to make ablack hole

Find

Sbek A

4G= 2n1n2n3

(Susskind 93, Sen 95,Strominger-Vafa 96 ...)

Thus is a count of statesSbek


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The information problem : What happens to the entanglement ?

|02|02 + |12|12

|0n|0n + |1n|1n

. . .

|01|01

+ |11|11

M

String theorists: String theory is unitary, so there

should be no problem, really

GR folks: But what is the resolution to the problem?Hawking has an explicit computation, and you have notshown us what is wrong with it !!


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|02|02 + |12|12

|0n|0n + |1n|1n

|01|01 + |11|11

M

+ 1

+ 2

+ n

There will of course be small corrections tothe leading order Hawking computation

Small correction to a large number of pairswill (hopefully) disentangle the inner and outer quanta

Stringtheorists


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You must think we are such fools.

We know there will always be small corrections from

quantum gravity.

If that could remove the entanglement, why would

we be worrying about the information paradox for

35 years ?

|02|02 + |12|12

|0n|0n + |1n|1n

. . .

|01|01 + |11|11

M

+ 1

+2

+ n

GR folks


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KipThorne

JohnPreskill

StephenHawking

In 2005, Stephen Hawking surrendered his bet to John Preskill,based on such an argument of small corrections ...

(Subleading saddle points in a Euclidean path integral giveexponentially small corrections to the leading order evaporationprocess)

But Kip Thorne did not agree to

surrender the bet ...


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So who is right ?

+_

e+e-

entangled

pairs

12() 1

2() . . .

+ corrections

Sent N ln 2

Schwinger process


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For the surface is spacelike

ds

2

= (1

2M

r )dt2

+

dr2

(1 2Mr

) + r2

(d2

+ sin2

d2

)

The black hole is described by the Schwarzschild

metric

The black hole

For the surface is spaceliker = constant

Crucial point about the black hole:

t = constant

r < 2M

r > 2M


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r=0 horizon

t=constant

r=constant

We have to draw spacelike slices to foliate spacetime

(no time-independent

slicing possible)


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Entangled pairs

The Hawking process

Stretching of spacetimecauses field modes toget excited


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r=0 horizon

correlated pairs

Older quanta move apart

initial matter

Hawking state

|1 = 12

|0|0+ |1|1

10 light years77

(We will use a discretized

picture for simplicity; for

full state see e.g. Giddings-

Nelson)


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Hawkings argument

SN+1 = SN + ln 2

SN: Entanglement entropy

after N pairs have

been created

The radiation state (green quanta) are highly entangled with the infallingmembers of the Hawking pairs (red quanta)

S= Ntotal ln 2

|1 = 12

|0|0+ |1|1

|1 = 1

2

|0|0+ |1|1

|1 = 12

|0|0+ |1|1

E l d


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Entangled state

If the black holeevaporates away,

we are left in aconfiguration whichcannot be described

by a pure state

(Radiation quanta areentangled, but there isnothing that they areentangled with)

We can get a remnant

with which the radiationis highly entangled

S h ll d ?


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So what can small corrections do ?

Rules of the game:

(a) The region where pairs are being produced has normal evolution;

i.e. vacuum modes evolve as expected upto corrections of order

(b) The stuff inside the hole can be reshuffled in any way we want, but

the quanta that have left cannot be altered

1

Making rigorous the

statement that

Nothing happens

at the horizon

12

|01|01 + |11|11

+ 12

|01|01 |11|11


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Theorem: Small corrections to Hawkings leading ordercomputation do NOT remove the entanglement

(SDM 09)Sent

Sent< 2

Bound does not depend on the number of pairs N

A

B C

D

S(A + B) + S(B + C) S(A) + S(C)

S(A) = Tr[A ln A] etc.

Basic tool : Strong Subadditivity(Lieb + Ruskai 73)

C l i


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In Schwinger process

or in black hole:Entanglement rises with

each emission

Cannot resolve theproblem as long as

corrections to lowenergy dynamics aresmall at the horizon

Conclusion:


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An order unity correction at the horizon means that we need hair ..

Thus we have a conflict between two theorems:

(a) The Hawking argument, supplements by the inequality thatshows its robustness to small corrections

(b) The no hair theorem, which encodes our failure to findany alternative to the black hole geometry


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So, what is the resolution?

(Avery, Balasubramanian, Bena, Chowdhury, de Boer,Gimon, Giusto, Keski-Vakkuri, Levi, Maldacena, Maoz,

Park, Peet, Potvin, Ross, Ruef, Saxena, Skenderis,Srivastava, Taylor, Turton, Warner ...)


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The traditional expectation ...

lp

N lp

weak

coupling

strong

coupling

But it seems in string theory the opposite happens ...

lp

weak

couplingstrong

coupling

Size of bound states grows with coupling, number of branes ...

(SDM 97)

Recall that weak coupling states are described by intersecting branes


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Start with the simplest microstates:

weakcoupling

strong

coupling

Recall that weak coupling states are described by intersecting braneswrapped on compact directions

No horizon, no singularity ... so we do not get the traditional hole ...

Recall that weak coupling states are described by intersecting branes


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Start with the simplest microstates:

weakcoupling

strong

coupling

Recall that weak coupling states are described by intersecting braneswrapped on compact directions

No horizon, no singularity ... so we do not get the traditional hole ...


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Many such constructions have been done ...

General lesson: Eigenstates in string theory do not have a

traditional horizon

That is, there is no smooth region straddling the horizon wherelow energy pair modes evolve as expected on gently curvedspacetime

Geometry can be different from the traditional hole, or moregenerally, there is no geometry, just a quantum fuzz

fuzzballs


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Thus we have finally found the hair for black holes ...

Nature of the hair:

Compact directions makelocally nontrivial fibrations overthe noncompact directions

small compactdirection circle

Angular sphere of

noncompact directions

eSbek states upon quantization

Thus the hair are fundamentally a nonperturbative construct

involving the extra dimensions ...


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|ktotal = (J,total

(2k2))n1n5(J,total

(2k4))n1n5 . . . (J,total

2 )n1n5 |1total

AdS3 S3 T4

Geometry for simple

state (winding =1)

Generic D1D5P CFT stateSimple states: all components the same,

excitations fermionic, spin aligned


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ds2 = 1h

(dt2 dy2) + Qphf

(dt dy)2 + hf

dr2Nr2N + a

2+ d2

+ h

r2N na2 +

(2n + 1)a2Q1Q5 cos2

h2f2

cos2 d2

+ h

r2N + (n + 1)a

2 (2n + 1)a2Q1Q5 sin

2

h2f2

sin2 d2

+a22Qp

hf

cos2 d + sin2 d

2

+

2aQ1Q5

hfn cos

2

d (n + 1) sin2

d

(dt dy)

2aQ1Q5

hf

cos2 d + sin2 d

dy +

H1

H5

4i=1

dz2i

f = r2N a2 n sin2 + a2 (n + 1) cos2 h =

H1H5, H1 = 1 +

Q1

f, H5 = 1 +

Q5

f

Q1Q5Q1Q5 +Q1Qp +Q5Qp

(Giusto SDM Saxena 04)


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How does Hawking radiation arise ?


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The Black Hole


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Spinning star

causes frame-dragging

Far away light conesare close to normal

Close by, light cones tilt so much thatevery object MUST rotate

ERGOREGION

A curiousity:Ergoregions

Hawking radiation


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Hawking radiation

Traditional picture

Ergoregion

Actually radiation comes outjust like from any other

object, not from the vacuum

Hawking radiation


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Hawking radiation

Traditional picture

Ergoregion



Hawking radiation


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Hawking radiation

Traditional picture

Ergoregion



Hawking radiation


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Hawking radiation

Traditional picture

Ergoregion



Hawking radiation


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Hawking radiation

Traditional picture

Ergoregion



Hawking radiation


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Hawking radiation

Traditional picture

Ergoregion




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Why does semiclassical intuition fail ?

Shell collapses to make a black hole


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??

Shell collapses to make a black hole ...


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Consider the amplitude for the shell to tunnel to a fuzzball state

Amplitude to tunnel is very small

But the number of states that one can tunnel

to is very large !

Toy model: Small amplitude to tunnel to a neighboring well but there


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Toy model: Small amplitude to tunnel to a neighboring well, but there

are a correspondingly large number of adjacent wells

In a time of order unity, the wavefunction in the central well becomes a

linear combination of states in all wells (SDM 07)

1 [ ]Path integral


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Z=

D[g]e

1

S[g]

Measure hasdegeneracy of states Action determinesclassical trajectory

For traditional macroscopic objects the measure is order while theaction is order unity

Path integral

But for black holes theentropy is so large that the

two are comparable ...

We have a failure of thesemiclassical approximation ...

1 [ ]Path integral


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Z=

D[g]e

1

S[g]

Measure hasdegeneracy of states Action determinesclassical trajectory

For traditional macroscopic objects the measure is order while theaction is order unity

Path integral

But for black holes theentropy is so large that the

two are comparable ...

We have a failure of thesemiclassical approximation ...

Wh t h if f ll i t bl k h l ?


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What happens if someone falls into a black hole ?

(SDM+Pumberg 2011)

(a) Low energy dynamics (E kT)

No horizon, radiation from ergoregions, so radiation

like that from any warm body

no information loss since radiation depends on choiceof microstate

(b) Correlators in high energy infalling frame (E kT)

k

k

Collective oscillations of fuzzball gives Greens functions that

equal Greens functions in empty space ....


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Summary

( ) f


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(A) The black hole information paradox is a serious problem: It does not

allow GR to be consistent with quantum unitarity

|02|02 + |12|12

|0n|0n + |1n|1n

|01|01 + |11|11

M

Inequality shows that there is no way around this problem unless we

have an order unity change to low energy evolution at the horizon

(B) Th bl th i th h i th hi h t d f


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(B) The problem then is the no hair theorem which prevented us from

finding any significant change to the evolution at the horizon

(C) In String theory we can now make explicit constructions of the

microstates of the black hole. It turns out that they do have hair;

in fact a horizon never forms

N lp

lp

weak

couplingstrong

coupling

(D) O h k h i l i l i i i f il d


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(D) One can then ask how semiclassical intuition failed.

The reason can be traced to the large entropy of gravitational

states of the black hole, which made the measure in the path integral

compete with the classical action

Z=

D[g]e1

S[g]

This should be a basic lesson for the behavior of quantum gravity in general,

in all situations where we have a sufficiently large density of quanta ...

(D) O h k h i l i l i i i f il d


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(D) One can then ask how semiclassical intuition failed.

The reason can be traced to the large entropy of gravitational

states of the black hole, which made the measure in the path integral

compete with the classical action

Z=

D[g]e1

S[g]

This should be a basic lesson for the behavior of quantum gravity in general,

in all situations where we have a sufficiently large density of quanta ...

(E) Cosmology


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S E3

4

S E

S

E2

S E9

2

?

?

?

Radiation

String gas/brane gas

Fractional brane gas:

The stuff insideblack holes

Matter is also

crushed to highdensities in theearly Universe ...

So these lessons onphase space mayradically change ourpicture of thedynamics there ...

(E) Cosmology


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The infall problem


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p

What happens to an object (E >> kT) that falls into the black hole?

What does an infalling observer feel ?

??

Low energy radiation

modes are corrected by

order unity, no information

loss in process of creation

Is it possible that the dynamics of high

energy infalling objects can approximated

by the traditional black hole geometry in

some way?

Now we know that black hole microstates are fuzzballs let us see if we can do


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Now we know that black hole microstates are fuzzballs. let us see if we can do

any better ...

Central part of eternal black hole diagram looks like a piece of

Minkowski spacetime, Horizons look like Rindler horizons

So complementarity looks as strange as asking that we get destroyed at a

Rindler horizon, and in a dual description we continue past the horizon

Rindler space: Accelerated observers see a thermal bath


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p

t = R sinh

x = R cosh

Minkowski spacetime

Right RindlerWedge

Left RindlerWedge

R = R0

= 0

t

x

R = R0An observer moving along

sees a temperature

T =

1

2R0

The Minkowski vacuum can be

written an an entangled sum ofRindler states

0M =

k

e

Ek

2 |EkL REk|

An observation


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If there is a scalar field ,

then the Rindler states will

have a bath of scalar quanta

If has a interaction,then this bath of scalar quanta

will be interacting

3

The graviton is a field that is

always present, so we will have

a bath of (interacting) gravitons

Thus expect fully nonlinear

quantum gravity near Rindler

horizon


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Rindler coordinates end at

the boundary of the wedge

Thus it is logical to expect that the gravity

solution for Rindler states should also end

But this is exactly what fuzzball microstates do !

0

M =

k

e

Ek

2

|Ek

L REk

|

=

Thus we expect :

Black Holes : Israel (1976): The two sides of the eternal black hole


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Black Holes : Israel (1976): The two sides of the eternal black holeare the two entangled copies of a thermal system inthermo-field-dynamics

Re[t]Im[t]

Maldacena (2001): This implies that the dual to theeternal black hole is two entangled copies of a CFT

Van Raamsdonk (2009): CFT states are dual togravity solutions ... so we should be able to writean entangled sum of CFT states as an entangledsum of gravity states ...

Thus we can expect that summing over fuzzball microstates will generate the


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=

p g g

eternal black hole spacetime

(SDM + Plumberg2011)

Is it reasonable to expect that sums over (disconnected) gravitationalsolutions can be a different (connected) gravitational solution ?

Something like this happens in 2-d Euclidean CFT ...

The fuzzball microstates do not have horizons, but the eternal black

hole spacetime does ...

Sewing process in CFT


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k k

k

ehkhk =

(a) Low energy dynamics (E kT)


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No horizon, radiation from ergoregions, so radiation

like that from any warm body

no information loss since radiation depends on choice

of microstate

(b) Correlators in high energy infalling frame (E kT

)

k

k

for generic

statesk

m

m m

k|O1O2|k

m

eEmm|O1O2|m


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The Black Hole Information Paradox Mathur