Entropy Bounds, Holography & 2nd Law ( in Cosmology )

אוניברסיטת בן-גוריון

Ram Brustein

Entropy bounds, holography, Causal Entropy Bound (CEB)

Quantum & Geometric entropies

GSL

•gr-qc/9904061=PRL 84 (00)•hep-th/9907032=PLB 471 (00)with S. Foffa & R. Sturani•hep-th/9912055=PRL 8? (00)with G. Veneziano

S,E

ERS

R

BEB :Bekenstein ‘81

Too much entropy/ too little energy ==> GSL

(For systems of limited gravity R>Rg =2 E GN )

Apply BEB to the universe ?! U is not a system of limited gravity …

Bekenstein ‘89

3422

31

,,

,

NTsNTHM

REHR

P

N

M

M

TNH

N

MT

P

P

P

2

4

43

NT

MNTNTRs P

Upper bound on curvature

2

22,

Pl

R

NG

R

NG

RSEERS

2

2

3

3

3PP l

R

l

R

l

V

UV

S

BEB is not compatible with QFT!

QFT:

Holographic principle:Any physical system can be completely specified by data stored on its boundary, without exceeding a density of one bit per Planck area.(adapted from Bousso hep-th/9911002 )

‘tHooft ‘93Susskind ‘95

Holographic entropy bound

2Pl

AHOL

SS But what is S ?

Bousso: use light-sheets=2+1D collections of light-rays orthogonal to surfaceFS: past ingoing light-sheet- wrong!B: 1. light-sheet of decreasing area = “inside” with converging geodesics 2. Stop when caustic ~ singularityneed “space-like projection”

TgTTgxd

RgRRgxdS

P

P

l

lCEB

21

0

41

21

0

41

Max

Max2

0

CEBSS 0CEB :

EVSVE

VS

R.B. & Veneziano ‘00

TgTTs

Pl 211 Max

ppsPl

,Max 31

Local form of CEB :

pppdiagT ,,, in cosmology:

CEB A bound on curvature(for RD FRW)

43 NTMNT PM

NPTH ,

Derivation of CEB :

( i ) Entropy is maximized by the largest stable BH (s) that can fit in a region

( i i) The largest stable BH is determined by causality: BH horizon < RCC

R3CC

R3CC

R3CC

V22

2

3

2

2

3,

PP

HH

P

HH

CC

CC

CC

CC

CC

R

VR

R

VSnS

RS

R

Vn

Find RCC : use cosmological perturbations

BEBCEBHOL

HOLCEBBEB

SSS

SSS

Comparison between entropy bounds

HOLBEBP

CEB SSRER

EVSP

2

211

Bousso is o.k.

limited gravity 2P

ER

not limited gravity 2P

ER

Quantum entropy: Entropy of quantum fluctuations

Modes “freeze”/ “thaw” “exit” / “reenter”

02 2 kkk kH

max

min )(

ln3

k

k kfknkdsquantum

Hquantum nS

NQuantum entropy is real !So what about 2nd law ?

PMk maxConstant !

R3CC

R3CC

R3CC

V

Causal boundaryhas geometric entropy

Proposed resolution: R.B., PRL 84 ‘00

Proof in progress

NG

RH CCGS

2

Entropy bounds: Geometric entropy dominates

0

HH

HH

H

QuantumClassical

NdndSnSdn

dSdSdS

02)()33( HHH

HHHH

H SnNSnH

N

MH P

Generalized second law

In cosmology: 2

2

31

31 ,

)(,

P

HHCC l

HS

H

anHR

2,0 HHH

R.B., PRL 84 ‘00

Conclusions

Holography modified by causality א

Singularity thms. modified by entropy בbounds

Hint: shortest length scale N ג

M P