Samir D. Mathur The Ohio State Universitynaglecture/style/nagpublic.pdfThe Hawking radiation problem...
Transcript of Samir D. Mathur The Ohio State Universitynaglecture/style/nagpublic.pdfThe Hawking radiation problem...
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 !!