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Entropy bounds. Introduction Black hole entropy Entropy bounds Holography. Macroscopic state. 1/3. 1/3. Microscopic states. Ex: (microcanonical) - k S p i ln(p i ) =k S1/ N ln N =k S ln N. 1/3. |☺☺O>. |☺O☺>. | O ☺☺>. r = S p i | F > ii < F |. What is entropy?.

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entropy bounds

Entropy bounds

  • Introduction
  • Black hole entropy
  • Entropy bounds
  • Holography
slide2

Macroscopic state

1/3

1/3

Microscopic states

Ex: (microcanonical)

- k S piln(pi) =k S1/NlnN

=k SlnN

1/3

|☺☺O>

|☺O☺>

| O ☺☺>

r= S pi|F>ii<F|

What is entropy?

S=k·ln(N)

S=-k S piln(pi)

S=-k 3 1/3 ln 1/3 = k ln 3

S=k ln 3

S=-k Tr(rlnr)

examples
Examples
  • Free particle:
  • Black body radiation:
  • Debye Model (low temperatures):
slide4

Extremum problem:

S[pi]=-kSpiln pi

subject to Spi=1

pi =1/N

Smax=k ln dim H

Entropy bounds

What is the maximum of S?

entropy bounds1

Example 1: maximum entropy of 100 spin 1/2 particles

Example 2: maximum entropy of free fermions in a box

|y> = |n0,0,0,, n0,0,0, , npħ/L,0,0,,n 0,pħ/L, 0,,…,nL,L, L,>

|y> = |, ,…, >

Spin

Maximal

momentum

Momentum

mode

Spin up and

momentum kx=pħ/L

Entropy bounds

Generalization:

N  available phase space

Number of modes = 2Sk1  L3Lk2dk  L3L3

dimH= 2100

dimH= ON

Smax available phase space

Smax=k 100ln2

dimH= 2N  2L3L3

Smax L3L3

entropy bounds2

Available

phase space

in quantum

gravity

What should

a unified theory look like

Entropy

bounds

on matter

Black hole

entropy

Smax

p

?

r

Entropy bounds

Why is this interesting?

black holes

A wrong derivation yielding correct results:

If nothing can escape then:

Yielding:

Black holes

Black hole condition

Scwartzschield radius

Rs=2GM/c2

R≤

the event horizon

Singularity

formed

Singularity

formed

t

Event Horizon formed

Event Horizon

formed

Schwartzshield radius

y

y

x

x

Schwartzshield

radius

The event horizon
black hole entropy bekenstein 1972

Assumption

Sbh = 0

Generalized second law

SbhA ; ST=Sbh+Sm

S>0

Black hole entropy(Bekenstein 1972)

The area of a black hole always increases:

A≥

Via Hawking radiation:

Sbh = 4pkR2c3/4Għ

Sbh =A/4

ST=0

ST>0

bekenstein entropy bound bekenstein 1981

Adiabatic lowering

Sm

E

r

Initial entropy:

Final entropy:

Bekenstein entropy bound(Bekenstein 1981)

Energy is red-shifted: E’=Erc2/4MG

Mass of black hole increases:

M M+dM

M+E’/c2

E’

problems with the bekenstein bound

?

Sm 2pkhE/cħ

h

Problems with the Bekenstein bound

Sm<2pkrE/cħ

susskind entropy bound susskind 1995

Initial stage

Sshell,c2R/2G-M

Sm

R

Shell

Sm,M

Sm+ Sshell

After collapse

SBH

Susskind entropy bound(Susskind 1995)

Sm≤SBH=4pkR2c3/4Għ=A/4

bousso bound bousso 1999

t

t

y

y

V

x

x

Light cone

Bousso bound(Bousso 1999)

Light sheet

Sm≤A/4

possible conclusions from an entropy bound

Gravity restricts the number of

degrees of freedom available

GN

In general, field theory over-counts the available degrees of freedom

L=L(F(x),Y(x))d4x

A fundamental theory of nature should

have the ‘correct’ number of degrees of freedom

?

Possible conclusions from an entropy bound

Dim H A

the holographic principle t hooft 93 susskind 94

The surface area of B

in planck units

The light sheet

of the region B

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

A D dimensional quantum theory of gravity

may be described by a D-1 dimensional

Quantum field theory.

a working example ads cft
A working example: AdS/CFT

(Conformal) Field theory in D dimensional

flat space

Quantum gravity in

D+1 dimensional

Anti de-Sitter space.

current research
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?