Entropy localization and distribution in the Hawking radiation

Horacio Casini

CONICET-Intituto Balseiro – Centro Atómico Bariloche

Introduction

a) Ambiguities in the localized entropy S(V) in relativistic quantum field theory

Reduced density matrix

Entanglement entropy

S(V) divergent due to vacuum fluctuations: ambiguities due to regularization

Standard calculation requires artificial boundary conditionsand taking the large volume limit. For a finite region ambiguities arise due to these boundary conditions (or bounding box). Gravity prevents the large volume limit of homogeneous mater: formation of horizons.

In general decoupling S(V) ~ c1 Volume + c2 Area +… is expected, but this is not the case for diluted gases:Hawking radiation

Radiation shell

Boundary contribution

Where is the entropy in the Hawking radiation? Is it related to the flux of energygiven by the energy momentum tensor?

b) Information loss paradox in black hole evaporation

Black hole formation and evaporation as a unitary process for external observers

Quasi-thermal radiation, butin a pure total state.

Page: the first bit of information is recovered after half the evaporation time

The semiclassical picture tells a very different story

Correlations are present for ever between the asymptotic region and the BH, which as a space-timeRegion is never destroyed. These correlations should Induce information loss for observers who only see the exterior part

How to quantify this information loss due to entanglement (correlation)?It is possible to have information recovery (for external observers) in the semiclassical picture?

c) Entropy bounds

Holographic bound

Bekenstein bound (not in this talk)

It is possible to state them in a regularization independent way?

Are they necessarily violated in QFT?

d) It is possible to understand the finiteness of black hole entropy in the semiclassical context?

Entanglement entropy and mutual information

The functions g are local and extensive on the boundary due to UV origin of divergences.To produce a physical quantity subtract

The divergences cancel in I(A,B) which is finite and well definedI(A,B) is called the mutual information between A and B

Properties of the mutual information

Positivity

Monotonously increasing withsize: smooth

Upper bound to correlations

Lower bound to entropy

For pure global states

Mutual information in the black hole evaporation

Monotonicity: the I(A,B) increases with B. Also S(B)>1/2 I(A,B)

Then it is not possible that correlations established during the whole evaporation period make the final state pure once somenon zero entropy has developed at the early stages.

Some concrete calculation can be made…

For a massless scalar field in the two dimensional approximation (r,t coordinates only in the spherically symmetric case, qualitatively good, quantitatively wrong)the problem is mapped to a 1+1 conformal case where the entanglement entropy is known

(for A in contact to the horizon)

Then I(A,B) is proportional to the area of the horizon in Planck units and greater than twice the BH entropy

1- I(A,B) is extensive in the radiation region, as should be for a measure of the radiation “entropy”

2- A can be exponentially small near the point where the horizon emerges without appreciably changing the results. This location of the entangled information inside the black hole (in a nearly flat region) leads to a large violation of the holographic bound

How stable is this violation?

Backscattering: it can be shown that for free fields

Argument for a conformal theory

Unitary operator which is a symmetry of I(A,B)It acts as a spacetime point transformation in region I if the theory is conformal, and in the late part of region II

If thermal equilibrium is achieved.

Can we understand the finiteness of the black hole entropy semiclassicaly?

S(A) cannot be finite in QFT

This limit diverges. But if there is a Hagedorn maximal temperature(for example with a number of fields exponentially increasing with the fieldmass )the mutual information is meaningless (diverges) for But it can be finite at the transition point

Conclusions

A physical notion of entropy requires the specification of a coarse graining. The information shared by two sets A and B, I(A,B), is naturally forced by QFT as a coarse graining when considering the localized entropy in a region B. It seems to be the right tool to deal with the entropy carried by the vacuum fluctuations.

Any entropy can be considered mutual information with a hidden sector in a global pure state. For the Hawking radiation the black hole is this hidden sector. The monotonicity of I(A,B) sharps the information paradox:It shows that information is monotonically lost during the evaporation in the semiclassical case, independently of the interactions present in the QFT theory

The holographic bound is violated by a large amount in a flat region by the correlations with a far away region

Quantum gravity can change this. But how?1-Fluctuations in the position of the horizon2-No tensor product structure of the Hilbert space of spatially separated regions

A finite BH entropy means a large reduction of degrees of freedom at high energies to regularize the entanglement entropy. But a large number of degrees of freedom at high energies giving place to a Hagedorn transition leads to a natural distance cutoff to the mutual information. This may suggest to interpret the finite black hole entropy as a limit on the information that can be shared between the inside and the outside

Extensivity properties of I(A,B) not well understood