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. time Quantum effects lead to radiation with temperature T H ~10 K for black holes resulting from gravitational collapse with M>M Chandra= 3M 0 -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 to semiclassical 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” One can measure the correlation function: Prediction: -xx X’ Anti-correlation not only at x’=x but also at x’=-x (“Ghost” peak) Entangled pairs of phonons

Numerics - --

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 -2

The origin of the ghost peak Black holeRandom Matrix Theory Exponential redshift: 1/E Exponential unfolding: E x In both cases the sinh and cosh behavior arises from the flat-space behavior -2 Valid only for weak confinement

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)

Level statistics as a Luttinger liquid T=0 for WD RMT T=  for critical RMT Flat space- time 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

Temperature in the ground state as spontaneous symmetry breaking 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)