Hawking’s puzzle: The Black Hole Information Paradox

Samir D. Mathur

The Ohio State University

NASA

General relativity

(Black holes)Quantum theory

Hawking’s work on black holes showed that General relativity and Quantum mechanics contradict each other

This puzzle is known as the Black Hole Information Paradox

(Einstein) (Heisenberg)

(Hawking)

General relativity

Galileo dropped two balls, one heavy and one light, from the leaning tower of Pisa

He wanted to know which would reach the ground first

He found that they reached the ground at the same time

y

t

Both balls have the same graph, so we say they follow the same trajectory on the y-t plane

y

This situation is very special, it holds only for gravity

If we have electrostatic forces, then different objects have different trajectories ...

+++++++++++++++++++++

+ -

y

t-

+

Since the graphs for gravity are special, we should be able to make a new kind of theory for gravity

Traditional approach: Particles have a straight line graph if there is no force

The graph is curved if there is a force

What we will do now is curve the graph paper instead ....

x

t

x

t

Flat graph paper: straight lines arethe usual graphs for ‘no force’

‘Straight line’ on curved surface is defined to be the shortest distance between two points

For example, the great circles on earth give the shortest distance between two points ...

Einstein’s idea: Suppose there is gravity in some region

We want to describe the graphs of particles moving in this region

Newton would have taken a flat graph paper, and drawn a curved trajectory on it

But we should instead take a curved graph paper, and draw a ‘straight’ line on it

x

t

x

t

x

t

No gravity: flat graph paper, path is straightline

Gravity as described by Newton: Graph paper flat, path is curved

Gravity as described by Einstein: Graph paper curved, path is ‘straight’

What is the advantage of doing it Einstein’s way?

Answer: It gives the actual paths correctly !!

Mass of sun

No masses around ... spacetime is flat ...particle moves in straight line

Sun curves the spacetime around it ...Particles (like earth) make orbits that are curves ...

In Newton’s theory, the orbit is a closed ellipse

In Einstein’s theory, it is not quite closed ....

Observations agree with Einstein’s theory !!43 arc seconds per century

Once we know that spacetime should be curved, we find many interesting effects:

(a) Black holes

Tear a hole is spacetime; what falls in cannot get out

(b) Wormholes

Fall in somewhere,come out somewhere else

(c) A part of spacetime pinches off into a new Universe ... can we ever go there ?

(d) Extra dimensions: We usually have the dimensions x, y, z, t

Can there be an extra direction w ?

If it is there, why don’t we see it ?

Maybe it is too small ....

Black hole atthe center ofthe Milky Way

Black hole radiation

An interesting effect (Schwinger effect)

+++

+ ----

Schwinger effect

Stronger attraction Weaker attraction

Black hole

Hawking radiation

The paradox

Hole disappears

Radiation left

A serious problem

No information about matter

Final state has no information about initial state

But in quantum mechanics the final state always has fullinformation about the initial state

λ ∼ R (15)

∆t ∼ R

c(16)

tevap ∼G2M3

�c2(17)

T =1

8πGM=

dE

dS(18)

tevap ∼ 1063 years (19)

eS (20)

S = ln(# states) (21)

101077(22)

S =c3

�A

4G→ A

4G(c = � = 1) (23)

S = ln 1 = 0 (24)

Ψ ρ (25)

A ∼ e−Sgrav , Sgrav ∼1

16πG

�Rd4x ∼ GM2 (26)

# states ∼ eS , S ∼ GM2 (27)

A = 0 Sbek =A

4G= 0 Smicro = ln(1) = 0 (28)

�F (y − ct) (29)

n1 np e4π√

n1np (30)

Smicro = 4π√n1np Sbek−wald = 4π√

n1np Smicro = Sbek (31)

Ψf = e−iHtΨi Ψi = eiHtΨf (32)

2

λ ∼ R (15)

∆t ∼ R

c(16)

tevap ∼G2M3

�c2(17)

T =1

8πGM=

dE

dS(18)

tevap ∼ 1063 years (19)

eS (20)

S = ln(# states) (21)

101077(22)

S =c3

�A

4G→ A

4G(c = � = 1) (23)

S = ln 1 = 0 (24)

Ψ ρ (25)

A ∼ e−Sgrav , Sgrav ∼1

16πG

�Rd4x ∼ GM2 (26)

# states ∼ eS , S ∼ GM2 (27)

A = 0 Sbek =A

4G= 0 Smicro = ln(1) = 0 (28)

�F (y − ct) (29)

n1 np e4π√

n1np (30)

Smicro = 4π√n1np Sbek−wald = 4π√

n1np Smicro = Sbek (31)

Ψf = e−iHtΨi Ψi = eiHtΨf (32)

2

Thus the process of formation and evaporation of a black hole cannot be described by any quantum Hamiltonian

This is known as the black hole information paradox

What did string theorists say ?

It tuned out that string theorists made a mistake (Maldacena 2001) ...

They argued that small quantum gravity effects would make small changes to the state of the emitted radiation ...

�

The correction is small for each emission, but the number of emitted particles is very large

Thus these small effects can encode the information of the matter that fell into the hole (apples vs oranges)

If this was true, then the Hawking puzzle was a ‘non-puzzle’ to start with: one would say that Hawking did an approximate computation, and doing the computation more carefully would resolve the problem ...

Kip Thorne

John Preskill

Stephen Hawking

In 2004, Stephen Hawking surrendered his bet to John Preskill,based on such an argument of ‘small corrections’ ...

But Kip Thorne did not agree to surrender the bet ...

Theorem: Small corrections to Hawking’s leading order computation do NOT resolve the problem

(SDM arXiv: 09091038)

A

B CD

Basic tool : Strong Subadditivity (Lieb + Ruskai ’73)

E = hν =hc

λ(38)

�ψ�|H|ψ� ≈ �ψ�|Hs.c.|ψ�+O(�) (39)

lp � λ � Rs (40)

2� (41)

|ξ1� = |0�|0�+ |1�|1� (42)

|ξ2� = |0�|0� − |1�|1� (43)

SN+1 = SN + ln 2 (44)

S = Ntotal ln 2 (45)

|ψ� → |ψ1�|ξ1�+ |ψ2�|ξ2� (46)

||ψ2|| < � (47)

SN+1 < SN (48)

SN+1 > SN + ln 2− 2� (49)

S(A) = −Tr[ρA ln ρA] (50)

bN+1 cN+1 {b} {c} p = {cN+1 bN+1} (51)

S({b}+ p) > SN − � (52)

S(p) < � (53)

S(cN+1) > ln 2− � (54)

S({b}+ bN+1) + S(p) > S({b}) + S(cN+1) (55)

S(A+B) + S(B + C) ≥ S(A) + S(C) (56)

S({b}+ bN+1) > S) + ln 2− 2� (57)

S(A+B) ≥ |S(A)− S(B)| (58)

3

So, who is right ?

So if small corrections, do not work, what is the solution ?

If the corrections are of order , then the fraction of informationthat can be recovered is at most

�2�

Solving the black hole puzzles with string theory

String theory

It looks like a strange theory at first, but whatlooks like undesirable features turn out to bejust the ones that solve the puzzles of black holes

(a) The theory lives in 10 dimensions, so if we make a blackhole in 4 dimensions, then there will be 6 compact dimensions

6 compact circles

3+1 noncompact spacetime

(b) There are extended objects like strings, branes etc.

(c) The coupling constant g is a field, so its value can be made small or large

g

The Hawking radiation problem

(A) Weak coupling, take a string with some energy, compute radiation rate

(B) Strong coupling, the same energy makes a black hole ... compute Hawking radiation rate

Get an exact agreement of radiation rates !!

Thus the dynamics of string theory seems to know something about the physics of black holes ...

(Das+SDM 96)

But we have not solved Hawking’s puzzle ...

�

Different materials make the same mass hole, and then the radiation is the same ... so the radiation does not carry the information of the initial matter (information loss)

Weak coupling: we get the same rate as Hawking radiation rate from black holes, but this time different states of the string radiate differently ...

Problem is that the radiation is pulled out of the vacuum, and the vacuum has no information....

Could we put some ‘stuff ’ at the horizon so that we dont have the vacuum there ?

Difficulty: Any stuff put near the horizon just falls in, and we are back to the vacuum

Fuzzballs

(A) Weak coupling: Take some strings and branes, and join them to make a bound state in string theory

|n�total = (J−,total−(2n−2))

n1n5(J−,total−(2n−4))

n1n5 . . . (J−,total−2 )n1n5 |1�total (5)

A

4G= S = 2π

√n1n2n3

∆E =1

nR+

1

nR=

2

nR

∆E =2

nR

S = ln(1) = 0 (6)

S = 2√

2π√

n1n2 (7)

S = 2π√

n1n2n3 (8)

S = 2π√

n1n2n3n4 (9)

n1 ∼ n5 ∼ n

∼ n14 lp

∼ n12 lp

∼ n lp

M9,1 → M4,1 ×K3× S1

A

4G∼

�n1n5 − J ∼ S

A

4G∼√

n1n5 ∼ S

e2π√

2√

n1np

1 +Q1

r2

1 +Qp

r2

e2π√

2√

n1n5

w = e−i(t+y)−ikz w̃(r, θ, φ) (10)

B(2)MN = e−i(t+y)−ikz B̃(2)

MN(r, θ, φ) , (11)

2

We expect that the size of this bound state will be order planck length 10−33 cm

(B) At strong coupling we expect that a black hole horizon develops around this bound state

Then we have the usual Hawking pair production problem ...

(B’) But there is a surprise ... something else happens at strong coupling ...

The strings and branes swell up as we increase the coupling, and become as big as the expected size of the horizon ...

So a horizon never forms ...

So we never get the Hawking pair production process ...

Different states radiate differently ... so the radiation carries the information of what makes the hole !!

(SDM 97)

This solves Hawking’s black hole information paradox, so let us analyze in more detail what happened ...

Why do the strings and branes not just get sucked into the center ?

The detailed structure of this fuzzball became clearer in the next few years (2000-2004)

Fuzzball

Recall that in string theory we have extra dimensions ....

‘Inside of hole’

Let us draw just one space direction for simplicity

‘Inside of hole’

Now suppose there was an extra dimension (e.g., string theory has 6 extra dimensions)

People have thought of extra dimensions for a long time, but they seemed to have no particular significance for the black hole problem

‘Inside of hole’

But there is a completely different structure possible with compact dimensions ...

Contrast this with

The mass is captured by the energy in the curved manifold

‘Inside of hole’

1-dimension

Many dimensions

“Fuzzball”

The stuff at the horizon does not fall in because the stuff is actually a topology that provides to a smooth end to space

not part of spacetime

Hawking 2014: There is no horizon in a black hole ....

But he gave no mechanism to avoid the formation of a horizon ...

At present it seems that the only mechanism to get what he is saying is the fuzzball mechanism of string theory ...

Summary

Hawking 1974:

General relativity predicts black holes

Quantum mechanics around black holes is INCONSISTENT

String theory: Quantum gravity effects alter the structure of the black hole radically Fuzzballs

NO !!

When we try to crush matter into a black hole, then quantum measureterms start becoming important, even for macroscopic systems

Where else can we expect such a crushing of macroscopic amounts of matter ?

λ ∼ R (15)

∆t ∼ R

c(16)

tevap ∼G2M3

�c2(17)

T =1

8πGM=

dE

dS(18)

tevap ∼ 1063 years (19)

eS (20)

S = ln(# states) (21)

101077(22)

S =c3

�A

4G→ A

4G(c = � = 1) (23)

S = ln 1 = 0 (24)

Ψ ρ (25)

A ∼ e−Sgrav , Sgrav ∼1

16πG

�Rd4x ∼ GM2 (26)

# states ∼ eS , S ∼ GM2 (27)

A = 0 Sbek =A

4G= 0 Smicro = ln(1) = 0 (28)

�F (y − ct) (29)

n1 np e4π√

n1np (30)

Smicro = 4π√n1np Sbek−wald = 4π√

n1np Smicro = Sbek (31)

Ψf = e−iHtΨi Ψi = eiHtΨf (32)

S ∼ E34 S ∼ E S ∼ E2 S ∼ E

92 (33)

2

λ ∼ R (15)

∆t ∼ R

c(16)

tevap ∼G2M3

�c2(17)

T =1

8πGM=

dE

dS(18)

tevap ∼ 1063 years (19)

eS (20)

S = ln(# states) (21)

101077(22)

S =c3

�A

4G→ A

4G(c = � = 1) (23)

S = ln 1 = 0 (24)

Ψ ρ (25)

A ∼ e−Sgrav , Sgrav ∼1

16πG

�Rd4x ∼ GM2 (26)

# states ∼ eS , S ∼ GM2 (27)

A = 0 Sbek =A

4G= 0 Smicro = ln(1) = 0 (28)

�F (y − ct) (29)

n1 np e4π√

n1np (30)

Smicro = 4π√n1np Sbek−wald = 4π√

n1np Smicro = Sbek (31)

Ψf = e−iHtΨi Ψi = eiHtΨf (32)

S ∼ E34 S ∼ E S ∼ E2 S ∼ E

92 (33)

2

Radiation

String gas/brane gas ??

What happens here ?

THANK YOU !!