Entropy bounds Introduction Black hole entropy Entropy bounds Holography

Microscopic states What is entropy? S=k·ln( N ) S=-k  p i ln(p i ) S=-k  r(  ln  Ex: (microcanonical)  k  p i ln(p i ) =k  N ln N =k  ln N  =  p i |  > ii <  | S=k ln 3 1/3 S=-k 3 1/3 ln 1/3 = k ln 3 |☺☺O> |☺O☺> | O ☺☺> Macroscopic state

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

Entropy bounds What is the maximum of S? Extremum problem: S[p i ]=-k  p i ln p i subject to  p i =1 p i =1/ N S max =k ln dim H

S max =k 100ln2 dim H = Example 1: maximum entropy of 100 spin 1/2 particles  = | , ,…,  > Entropy bounds Example 2: maximum entropy of free fermions in a box  = |n 0,0,0, , n 0,0,0, , n  ħ/L,0,0, ,n 0,  ħ/L, 0, ,…,n ,  > Number of modes = 2  k 1  L 3   k 2 dk  L 3  3 S max  L 3  3 dim H = 2 N  2 L 3  3 Momentum mode Spin Maximal momentum S max  available phase space N  available phase space Generalization: dim H = O N Spin up and momentum k x =  ħ/L

Entropy bounds Why is this interesting? Black hole entropy  Entropy bounds on matter S max  Available phase space in quantum gravity r p  What should a unified theory look like ?

Black holes A wrong derivation yielding correct results: If nothing can escape then: Yielding: R s =2GM/c 2 R≤ Scwartzschield radius Black hole condition

x y The event horizon Schwartzshield radius x y t Event Horizon formed Schwartzshield radius Singularity formed Singularity formed Event Horizon formed

Black hole entropy (Bekenstein 1972) S>0 S bh = 0 Assumption S T >0 S T =0 The area of a black hole always increases: A≥  S bh =A/4 Via Hawking radiation: S bh = 4  kR 2 c 3 /4Għ Generalized second law S bh  A ; S T =S bh +S m 

Bekenstein entropy bound (Bekenstein 1981) Adiabatic lowering Initial entropy: Final entropy: SmSm E r E’ Energy is red-shifted: E’=Erc 2 /4MG Mass of black hole increases: M  M+  M  M+E’/c 2

Problems with the Bekenstein bound h S m <2  krE/cħ  S m  2  khE/cħ ?

Susskind entropy bound (Susskind 1995) S m,M R Initial stage SmSm S shell,c 2 R/2G-M Shell S m + S shell After collapse S BH S m ≤S BH =4  kR 2 c 3 /4Għ=A/4

Problems with a space-like bound SmSm R S m ≤A/4 ?

Bousso bound (Bousso 1999) x y t Light cone x y t Light sheet V S m ≤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))d 4 x A fundamental theory of nature should have the ‘correct’ number of degrees of freedom ? Gravity restricts the number of degrees of freedom available GNGN

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 sheet of 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 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?