Download Entropy bounds

Entropy bounds
•Introduction
•Black hole entropy
•Entropy bounds
•Holography
What is entropy?
Macroscopic state
S=k·ln(N)
Microscopic states
|☺☺O>
1/3
Ex: (microcanonical)
- k S piln(pi) =k S1/NlnN
=k SlnN
|☺O☺>
1/3
S=-k S piln(pi)
r= S pi|F>ii<F|
| O ☺☺>
1/3
S=-k 3 S=k
1/3 ln
ln1/3
3 = k ln 3
S=-k Tr(rlnr)
1

3
r 0


0

0
1
3
0

0

0

1

3
Examples
• Free particle:
3

 N  2 2  2 
3
 
S ( N , V , T )  Nk - N ln  
2
 V  mkT  


• Black body radiation:
S ( N ,V , T ) 
 2 k 4VT 4
15(c)3
• Debye Model (low temperatures):
S ( N ,V , T ) 
 2Vk 3T 4
10 3c 3 N
Entropy bounds
What is the maximum of S?
Extremum problem:
S[pi]=-kSpiln pi
subject to Spi=1
pi =1/N
1
 N
r  0

0

0 

0 

0 1
N
0
Smax=k ln dim H
Entropy bounds
1: maximum entropy of free
100 spin
1/2 particles
Example
2:
fermions
in a box
Generalization:
|n0,0,0,
, nphase
, nħ/L,0,0,,n 0,ħ/L, 0,,…,nL,L, L,>
|y>
|,
,…,
>space
N =available
0,0,0,
L
3 k2dk  L3L3
100
N
Number
of==modes
=
2S
1

L
dimH
2
dimH
O
k up and
Spin
Spin
Momentum
mode
momentum kx=ħ/L
N phase
L3L3 space

available
SSdimH
=k
100ln2
=
2
2
max
max
Smax L3L3
Maximal
momentum
Entropy bounds
Why is this interesting?
Black hole
entropy

Entropy
bounds
on matter
Smax
Available
What should
phase space
a unified theory look
in quantum
like

 gravity
p
?
r
Black holes
A wrong derivation yielding
correct results:
2GM
vescape 
R
If nothing can escape then:
vescape  c
Black
hole condition
Scwartzschield
radius
Yielding:
2GM
2
1
R≤2Rs=2GM/c
c R
The event horizon
Singularity
formed
Singularity
formed
t
Event Horizon
formed
Event Horizon
Schwartzshield
formed
radius
y
y
x
x
Schwartzshield
radius
Black hole entropy
(Bekenstein 1972)
The
area of a black hole always increases:
Assumption
A≥
S =0
bh
S>0
Generalized second law
 SbhA ; ST=Sbh+Sm
Via Hawking radiation:
Sbh = 4kR2c3/4Għ
ST=0
>0
Sbh
=A/4
Bekenstein entropy bound
(Bekenstein 1981)
Adiabatic lowering
Energy is red-shifted: E’=Erc2/4MG
Mass of black hole increases:
M M+M M+E’/c2
Sm
rE
Initial
E’
4kR2 c 3
entropy:
 Sm
4G
Final entropy:
S m  S bh
4kR2 c 3
 S bh
4G
dS bh
2krE

M 
dM
c
Problems with the Bekenstein bound
Sm<2krE/cħ
?
Sm 2khE/cħ
h
Susskind entropy bound
(Susskind 1995)
Sshell,c2R/2G-M
Initial stage
c2R
M
2G
Sm
Shell
R
Sm,M
c2R
M 
2G
Sm+ Sshell
After collapse
c2R
M 
2G
SBH
Sm≤SBH=4kR2c3/4Għ=A/4
Problems with a space-like bound
?
Sm≤A/4
Sm
R
Bousso bound
(Bousso 1999)
t
t
y
x
V
y
x
Light cone
Light sheet
Sm≤A/4
Possible conclusions from an
entropy bound
Dim H  A
Gravity restricts the number of
degrees of freedom available
In general, field theory over-counts
the available degrees of freedom
GN
L=L(F(x),Y(x))d4x
A fundamental theory of nature should
have the ‘correct’ number of degrees of freedom
?
The Holographic principle
(‘t Hooft 93, Susskind 94)
N, the number of degrees of freedom
involved in the description of L(B), must
not exceed A(B)/4. (Bousso 1999)
Proposition
The light sheet
of the region B
The surface area
of B
A D dimensional
quantum
theory of gravity
in planck units
may be described by a D-1 dimensional
Quantum field theory.
A working example: AdS/CFT
Quantum gravity in
D+1 dimensional
Anti de-Sitter space.
(Conformal) Field theory in
D dimensional
flat space
Current research
•How does one generalize the AdS/CFT
correspondence to other space-times?
•What is the role of gravity in holography?
•Is string theory holographic?