Black hole information paradox:Analysis in a condensed matter analog

Theo M. Nieuwenhuizen Institute for Theoretical Physics

University of Amsterdam

Igor V. Volovich,

Steklov Mathematical Institute Moscow

SPhT, CEA SaclayDecember 16, 2004

   

Thermodynamics of black holes

• BH radiates at Hawking temperature• Absorbs 3K CMB radiation

• Thermodynamics applies:

Two temperature approach, as in glasses

Th.M. N. PRL 1998

Stephen Hawking’s website, since summer 2004

Press Release:One of the most intriguing problems in theoretical physics has been solved by Professor Stephen Hawking of the University of Cambridge. He presented his findings at GR17, an International Conference in Dublin, on Wednesday 21 July.

Black holes are often thought of as being regions of space into which matter and energy can fall, and disappear forever. In 1974, Stephen Hawking discovered that when one fused the ideas of quantum mechanics with those of general relativity, it was no longer true that black holes were completely black. They emitted radiation, now known as Hawking radiation. This radiation carried energy away from the black hole which meant that the black hole would gradually shrink and then disappear in a final explosive outburst.

These ideas led to a fundamental difficulty, the information paradox, the resolution of which is to be revealed in Dublin. The basic problem is that black holes, as well as eating matter, also appear to eat quantum mechanical information.

Yet the most fundamental laws of physics demand that this information be preserved as the universe evolves. The information paradox was explored and formalised by Hawking in 1975. Since then, many have tried to find a solution.

Whilst most physicists think that there must be a resolution of the paradox, nobody has really produced a believable explanation. In fact, seven years ago the issue prompted Hawking, together with Kip Thorne of Caltech, to make awager against John Preskill also of Caltech, that the information swallowed byblack holes could never be recovered.On Wednesday, Hawking conceded that he has lost the bet.

The way his new calculations work is to show that the event horizon, which is the surface of the black hole, has quantum fluctuations in it. These are the same uncertainties in position that were made famous by Heisenberg's uncertainty principle and are central to quantum mechanics. The fluctuations gradually allow all the information inside the black hole to leak out, thus allowing us to form a consistent picture. The information paradox is now unravelled.A complete description of this work will be published in professional journals and on the web in due course.

Setup

Introduction to the BH information paradoxProposed solutions

Toy system: Caldeira-Leggett model harmonic oscillator+phonon bath in ground state are coupled

and decoupled

Dynamical solutionState of central oscillatorWorkOccupation of bath modesEntropies

Lessons for black holes?

Summary

Introduction to BH information paradox

Schwartzschild metric

Bekenstein-Hawking Entropy

Hawking temperature

Radiation: photons, gravitons

Different modes uncorrelated: true thermal spectrum

Unitarity Paradox: also when starting from pure state, end up in mixed state

Proposed solutions

• Info comes out with Hawking radiation• Info comes out “at the end”• Info is retained in small stable remnant• Info escapes to “baby universes”• Quantum hair• Pure states don’t wear black• New physics is needed

Hawking ’75; Stephens+ ‘t Hooft+ Whiting 1994; Preskill 1992; Page 1993; Unruh+Wald, 1995; Anglin+Laflamme+Zurek+Paz, 1995; Myers 1997

Frolov+Novikov Black Hole Physics, Basic Concepts+New Developments, 1998

Toy system: Caldeira-Leggett model Harmonic oscillator in ground state Phonon bath in ground state

harm osc. & bath slowly coupled

BH formation energy/work in

slowly decoupled BH evaporation energy out

phonons remain photons remain small or no remnant

Ohmic spectrumU. Weiss Quantum Dissipative Systems, 1998N+Allahverdyan, PRE 2002: Perp. Mobile ...

)(tCoupling “constant”

Bath modes:unperturbed + reaction to central particle

Dynamics

Central particle:

noise+damping

T=0 noise

Ansatz

)(t /Near to adiabatic regime:

change smaller than damping

- short times: always possible

- long times: slower than 1 / t )(t

- Initial conditions washed out.

- Expand:

Laplace transforms of the f’s are rational in z

State of central particle

Quick decay to quasi-equilibrium at instantaneous )(t

Entropy:

Finally to zero

Particle back to ground state

Work done

When state of system does not changedt

Work done on system

Ends up as phonons running in the T=0 bath

Occupation of bath modes

+ total derivatives

Entropies

• Von Neumann entropy of total system: S(t)=S(0)=0

even though work is added: Unitary dynamics

• S(bath, t) = S(particle,t) because starting from pure state

Anglin, Zurek et all, PRD, 1995

• S(bath,t) starts from 0, goes to maximum, then back to zero

Even though work is added, converted into phonons

Supports that state is pure at all times

• Vanishing of S(bath): because of correlations

as large as occupation numbers

Coarse grained entropy: neglect correlations

jijiji pppxxx ,,

What to expect for Black Holes?

• There must be correlations: all radiation from same hole• Back reaction not yet taken into account: static BH • Static BH: where does energy of radiation come from ?

Fabbri, Navarro, Navarro-Salas, Olmo hep-th/0208004:

Extremal BH’s: back reaction brings large correlations

could explain paradox

Summary

In condensed matter no unitarity paradox

Pure state remains pure

Correlations: same order of magnitude as occupation numbers explain why entropy can vanish

Radiation vanishes in static regimes

In BH: similar setup expected static approximation is inconsistent coupling to some degrees of freedom inside BH ? (‘t Hooft, 1975-76; Page 1976)