1. 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 Saclay December 16, 2004

2. 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

3. 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.

4. 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 a wager against John Preskill also of Caltech, that the information swallowed by black 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.

5. Setup Introduction to the BH information paradox Proposed solutions Toy system: Caldeira-Leggett model harmonic oscillator+phonon bath in ground state are coupled and decoupled Dynamical solution State of central oscillator Work Occupation of bath modes Entropies Lessons for black holes? Summary

6. 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

7. 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

8. Toy system: Caldeira-Leggett model Harmonic oscillator in ground state Phonon bath in ground state harm osc. & bath slowly coupled BH formationenergy/work in slowly decoupledBH evaporationenergy out phonons remainphotons remainsmall or no remnant Ohmic spectrum U. Weiss Quantum Dissipative Systems, 1998 N+Allahverdyan, PRE 2002: Perp. Mobile... Coupling “constant”

9. Bath modes: unperturbed + reaction to central particle Dynamics Central particle: noise+damping T=0 noise Ansatz

10. Near to adiabatic regime: change smaller than damping - short times: always possible - long times: slower than 1 / t - Initial conditions washed out. - Expand: Laplace transforms of the f’s are rational in z

11. State of central particle Quick decay to quasi-equilibrium at instantaneous Entropy: Finally to zero Particle back to ground state

12. Work done When state of system does not change Work done on system Ends up as phonons running in the T=0 bath

13. Occupation of bath modes + total derivatives

14. 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

15. 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

16. 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)