Horizon in Hawking radiation and in Random Matrix Theory

Vladimir Kravtsov Abdus Salam ICTP,Trieste,

Italy

Collaboration: Fabio Franchini, ICTP

July 6, 2009, Euler Institute, St.Petersburg

Black hole and the horizon

Is the black hole black? Quantum effects and Hawking radiation.

tim

e

Quantum effects lead to radiation with

temperature

GM

cTH

3

8

1

TH ~10 K for black holes resulting from gravitational

collapse with M>MChandra=3M0

-8

Sonic Black Hole

Exterior of “black hole” Interior of “black hole”

Can be realized in a flow of BEC of cold atoms by tuning the density and interaction by applying laser radiation (laser trap) and magnetic field (Feshbach resonance)

Equivalence of BEC+phonons tosemiclassical gravity

See also a book: G.E.Volovik

“The universe in a helium droplet”

Motion along null-geodesics

Phonon propagation is a motion along null-geodesics of the 1+1 spacetime

Horizon in a sonic black hole

Horizon for v=c(x) (time derivative vanishes)

An advantage of being a “super-observer”

)(1 xn )'(1 xn

One can measure the correlation function: )()( 11 sxnxn

Prediction: -x xX’

Anti-correlation not only at x’=x but also at x’=-x (“Ghost” peak)

Entangled pairs of phonons

Numerics

-

--

)2/)'((sinh)'(ˆ)(ˆ

02

2

11 vxx

Txnxn Hawking

The “Ghost” peak in level correlations in random matrix theory with log-confinement

C.M.Canali, V.E.Kravtsov,

PRE, 51, R5185 (1995)

The same sinh and cosh

behavior as for sonic BH

-1-2 -2

The origin of the ghost peak

Black hole Random Matrix Theory

Exponential redshift:

|]|exp[)sgn(1 txtxtx

1/EExponential unfolding: )(En

|]|exp[)( xxsignE

Ex

In both cases the sinh and cosh behavior arises from the flat-space

behavior

-2 -2

2)'(

1)'()(

EEEnEn

Valid only for weak

confinement HHV H 2ln)(

Conjecture

Can the RMT with log-normal weight be reformulated in terms of kinematics in the curved space with a horizon?

We believe – YES (upon a

proper a parametric extension to introduce time)

sincos)( 21 HHH T2

Level statistics as a Luttinger liquid

T=0 for WD RMT

T=for critical RMT

Flat space-time

12

12

21||

mn

H nm

Mirlin & Fyodorov, 1996, Kravtsov & Muttalib 1997.

Luttinger liquid in a curved space with the horizon: an alternative way to introduce temperature

Flat Minkowski space in terms of the bar-co-ordinates: vacuum state in the bar-co-ordinates seen as thermal state with

temperature T=in the co-ordinates (x,t)

Ground state correlations of such a Luttinger liquid reproduces the Hawking radiation correlations

with the “ghost” term

2222 )cosh()(sinh dxxdtxds

Temperature in the ground state as spontaneous symmetry breaking

-1 Invariant RMT with log-normal weight

+

Non-invariant critical RMT

Hawking =

Multifractal statistics of eigenvectors with d-1

the same translationally-invariant part of level density correlations as in the invariant

RMT, Equivalent to Calogero-Sutherland model (Luttinger

liquid) at a temperature T=

Hawking >0 is equivalent to spontaneously

emerging preferential basis?

Conclusions Sonic black hole in BEC Ghost peak as signature of sonic Hawking

radiation Ghost peak in random matrix theories with

log-normal weight Role of exponential red-shift and

exponential unfolding Level statistics as Luttinger liquid (finite

temperature in a flat spacetime vs. ground state in a spacetime with a horizon)