Entropy bounds

1. Entropy bounds • Introduction • Black hole entropy • Entropy bounds • Holography

2. 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)

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

4. 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?

5. 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

6. 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?

7. A wrong derivation yielding correct results: If nothing can escape then: Yielding: Black holes Black hole condition Scwartzschield radius Rs=2GM/c2 R≤

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

9. 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

10. 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’

11. ? Sm 2pkhE/cħ h Problems with the Bekenstein bound Sm<2pkrE/cħ

12. 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

13. ? Sm≤A/4 Sm R Problems with a space-like bound

14. t t y y V x x Light cone Bousso bound(Bousso 1999) Light sheet Sm≤A/4

15. 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

16. 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.

17. A working example: AdS/CFT (Conformal) Field theory in D dimensional flat space Quantum gravity in D+1 dimensional Anti de-Sitter space.

18. 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?