Horizon in Hawking radiation and in Random Matrix Theory
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Transcript of Horizon in Hawking radiation and in Random Matrix Theory
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)