Entropy bounds

•Introduction

•Black hole entropy

•Entropy bounds

•Holography

Microscopic states

What is entropy?

S=k·ln(N)

S=-k piln(pi)

S=-k r(ln

Ex: (microcanonical)kpiln(pi) =k NlnN

=k lnN

= pi|>ii<| S=k ln 3

1/3

1/3

1/3

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

|☺☺O>

|☺O☺>

| O ☺☺>

3

100

03

10

003

1

Macroscopic state

Examples• Free particle:

• Black body radiation:

• Debye Model (low temperatures):

23

22ln

2

3),,(

mkTV

NNNkTVNS

3

442

)(15),,(

c

VTkTVNS

Nc

TVkTVNS

33

432

10),,(

Entropy boundsWhat is the maximum of S?

Extremum problem:

S[pi]=-kpiln pi

subject to pi=1

pi =1/N

Smax=k ln dim H

N

N

100

00

001

Smax=k 100ln2

dimH= 2100

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

= |, ,…, >

Entropy bounds

Example 2: maximum entropy of free fermions in a box

= |n0,0,0,, n0,0,0, , nħ/L,0,0,,n 0,ħ/L, 0,,…,n,>

Number of modes = 2k1 L3k2dk L33

Smax L33

dimH= 2N 2L33

Momentummode

Spin Maximalmomentum

Smax available phase space

N available phase space

Generalization:

dimH= ONSpin up and momentum kx=ħ/L

Entropy boundsWhy is this interesting?

Black hole entropy

Entropy boundson matter

Smax

Available phase spacein quantum gravity

r

p

What shoulda unified theory look like

?

Black holes

A wrong derivation yielding correct results:

R

GM2vescape

If nothing can escape then:

cescapev

Yielding:

12

2

Rc

GMRs=2GM/c2R≤

Scwartzschield radiusBlack hole condition

x

y

The event horizon

Schwartzshield radius

x

y

t

Event Horizon formed

Schwartzshield radius

Singularityformed

Singularityformed

Event Horizon formed

Black hole entropy(Bekenstein 1972)

S>0Sbh = 0

Assumption

ST>0ST=0

The area of a black hole always increases:A≥

Sbh =A/4

Via Hawking radiation: Sbh = 4kR2c3/4Għ

Generalized second lawSbhA ; ST=Sbh+Sm

Bekenstein entropy bound(Bekenstein 1981)

Adiabatic lowering

Initial entropy: mSG

ckR

4

4 32

Final entropy: bhSG

ckR

4

4 32

c

krEM

dM

dSSS bhbhm

2

SmEr

E’

Energy is red-shifted: E’=Erc2/4MG

Mass of black hole increases: M M+M M+E’/c2

Problems with the Bekenstein bound

h

Sm<2krE/cħ

Sm 2khE/cħ?

Susskind entropy bound(Susskind 1995)

Sm,M

R

G

RcM

2

2

Initial stage

Sm

Sshell,c2R/2G-M

Shell

G

RcM

2

2

Sm+ Sshell

After collapse

G

RcM

2

2

SBH

Sm≤SBH=4kR2c3/4Għ=A/4

Problems with a space-like bound

Sm

R

Sm≤A/4?

Bousso bound(Bousso 1999)

x

y

t

Light cone

x

y

t

Light sheet

V

Sm≤A/4

Possible conclusions from an entropy bound

Dim H A

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

L=L((x),(x))d4x

A fundamental theory of nature shouldhave the ‘correct’ number of degrees of freedom

?

Gravity restricts the number of degrees of freedom available GN

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)

The light sheetof the region B

The surface area of B in planck units

A D dimensional quantum theory of gravity may be described by a D-1 dimensional Quantum field theory.

Proposition

A working example: AdS/CFT

Quantum gravity inD+1 dimensionalAnti de-Sitter space.

(Conformal) Field theory in D dimensionalflat 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?