On the black hole unitarity issue

I discuss features required for preserving unitarity in black hole decay and concepts underlying such a perspective. Unitarity requires that correlations extend on the scale of the horizon. I show, in a toy model inspired by string theories, that such correlations can indeed arise. The model suggests that, after a time of order 4M ln M following the onset of Hawking radiation, quantum effects could maintain throughout the decay a collapsing star within a Planck distance of its Schwarzschild radius. In this way information loss would be avoided. The concept of black hole ``complementarity'', which could reconcile these macroscopic departures from classical physics with the equivalence principle, is interpreted in terms of weak values of quantum operators.Comment: 14 pages, latex, no figures. Talk presented at the Puri workshop (December 1996

OPERATOR WEAK VALUES AND BLACK HOLE COMPLEMENTARITY

In conventional field theories, the emission of Hawking radiation in the background of a collapsing star requires transplanckian energy fluctuations. These fluctuations are encoded in the weak values of the energy-momentum operator constructed from matrix elements between both -in and -out states. It is argued that taming of these weak values by back-reaction may lead to geometrical backgrounds which are also build from weak values of the gravitational field operators. This leads to different causal histories of the black hole as reconstructed by observers crossing the horizon at different times but reduces, in accordance with the equivalence principle, to the classical description of the collapse for the proper history of the star as recorded by an observer comoving with it. For observers never crossing the horizon, the evaporation would be interpreted within a topologically trivial ``achronon geometry" void of horizon and singularity: after the initial ignition of the radiation from pair creation out of the vacuum of the collapsing star of mass M, as in the conventional theory, the source of the thermal radiation would shift gradually to the star itself in a time at least of order $4M\ln 2M$. The burning of the star could be consistent with a quantum unitary evolution along the lines suggested by 't Hooft. A provisional formal expression of general black hole complementarity is proposed and its possible relevance for testing features of a theory of quantum gravity is suggested.Comment: presented at the Oskar Klein Centenary Symposium (September 1994); 29 pages, phyzzx, no figure

Black Hole Tunneling Entropy and the Spectrum of Gravity

The tunneling approach for entropy generation in quantum gravity is applied to black holes. The area entropy is recovered and shown to count only a tiny fraction of the black hole degeneracy. The latter stems from the extension of the wave function outside the barrier. In fact the semi-classical analysis leads to infinite degeneracy. Evaporating black holes leave then infinitely degenerate "planckons" remnants which can neither decay into, nor be formed from, ordinary matter in a finite time. Quantum gravity opens up at the Planck scale into an infinite Hilbert space which is expected to provide the ultraviolet cutoff required to render the theory finite in the sector of large scale physics.Comment: 26 pages + 3 figures, phyzzx macropackage, figures available from Author

Statistical Entropy of Schwarzschild Black Holes

The entropy of a seven dimensional Schwarzschild black hole of arbitrary large radius is obtained by a mapping onto a near extremal self-dual three-brane whose partition function can be evaluated. The three-brane arises from duality after submitting a neutral blackbrane, from which the Schwarzschild black hole can be obtained by compactification, to an infinite boost in non compact eleven dimensional space-time and then to a Kaluza-Klein compactification. This limit can be defined in precise terms and yields the Bekenstein-Hawking value up to a factor of order one which can be set to be exactly one with the extra assumption of keeping only transverse brane excitations. The method can be generalized to five and four dimensional black holes.Comment: 11 pages, LaTex, no figures, corrected typ

G+++ Invariant Formulation of Gravity and M-Theories: Exact BPS Solutions

We present a tentative formulation of theories of gravity with suitable matter content, including in particular pure gravity in D dimensions, the bosonic effective actions of M-theory and of the bosonic string, in terms of actions invariant under very-extended Kac-Moody algebras G+++. We conjecture that they host additional degrees of freedom not contained in the conventional theories. The actions are constructed in a recursive way from a level expansion for all very-extended algebras G+++. They constitute non-linear realisations on cosets, a priori unrelated to space-time, obtained from a modified Chevalley involution. Exact solutions are found for all G+++. They describe the algebraic properties of BPS extremal branes, Kaluza-Klein waves and Kaluza-Klein monopoles. They illustrate the generalisation to all G+++ invariant theories of the well-known duality properties of string theories by expressing duality as Weyl invariance in G+++. Space-time is expected to be generated dynamically. In the level decomposition of E8+++ = E11, one may indeed select an A10 representation of generators Pa which appears to engender space-time translations by inducing infinite towers of fields interpretable as field derivatives in space and time.Comment: Latex 45 pages, 1 figure. Discussion on pages 19 and 20 altered. Appendix B amplified. 4 footnotes added. 2 references added. Acknowledgments updated. Additional minor correction

Representations of G+++ and the role of space-time

We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras G+++ with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of G+++. In particular, for E8+++, this applies to all its fundamental representations. However, there are some important examples, such as An+++, where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on G+++. Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of G+++ which is consistent with the interpretation of the former as charges associated with brane solutions.Comment: 45 pages, 9 figures, 9 tables, te

Kac-Moody Symmetries of Ten-dimensional Non-maximal Supergravity Theories

A description of the bosonic sector of ten-dimensional N=1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D_8. We also describe the extension of N=1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B_8. Finally, we discuss the related points for the N=1 supergravity coupled to an arbitrary number of abelian vector gauge fields

E10 and SO(9,9) invariant supergravity

We show that (massive) D=10 type IIA supergravity possesses a hidden rigid SO(9,9) symmetry and a hidden local SO(9) x SO(9) symmetry upon dimensional reduction to one (time-like) dimension. We explicitly construct the associated locally supersymmetric Lagrangian in one dimension, and show that its bosonic sector, including the mass term, can be equivalently described by a truncation of an E10/K(E10) non-linear sigma-model to the level \ell<=2 sector in a decomposition of E10 under its so(9,9) subalgebra. This decomposition is presented up to level 10, and the even and odd level sectors are identified tentatively with the Neveu--Schwarz and Ramond sectors, respectively. Further truncation to the level \ell=0 sector yields a model related to the reduction of D=10 type I supergravity. The hyperbolic Kac--Moody algebra DE10, associated to the latter, is shown to be a proper subalgebra of E10, in accord with the embedding of type I into type IIA supergravity. The corresponding decomposition of DE10 under so(9,9) is presented up to level 5.Comment: 1+39 pages LaTeX2e, 2 figures, 2 tables, extended tables obtainable by downloading sourc

On the Opening of Branes

We relate, in 10 and 11 dimensional supergravities, configurations of intersecting closed branes with vanishing binding energy to configurations where one of the branes opens and has its boundaries attached to the other. These boundaries are charged with respect to fields living on the closed brane. The latter hosts electric and magnetic charges stemming from dual pairs of open branes terminating on it. We show that charge conservation, gauge invariance and supersymmetry entirely determine these charges and these fields, which can be seen as Goldstone fields of broken supersymmetry. Open brane boundary charges can annihilate, restoring the zero binding energy configuration. This suggests emission of closed branes by branes, a generalization of closed string emission by D-branes. We comment on the relation of the Goldstone fields to matrix models approaches to M-theory.Comment: 13 pages, LaTeX, no figure