Black hole tunnelling entropy and the spectrum of gravity

Abstract

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. ⋆ Supported in part by NATO CRG 890404 † Postal address: Université Libre de Bruxelles, Campus Plaine, C.P.225, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. c The curve c is the hypersurface Σ ′B.H. c which lay in the intersection of E B.H.and E A and tends to Σ B.H. c in the limit η → 0.

A primer for black hole quantum physics

“…A possible constant of integration in eq. (3.115) is a subject of much debate [23]. If there is a remnant at the end of evaporation then S BH (0) is its entropy.…”

Entropy generation by tunneling in 2+1 gravity

“…Classical considerations seem to indicate that this object has small or vanishing degeneracy in contrast to the infinite degenerate object found in the case of tunneling to the Schwarzschild black hole. [5] This should be verified explicitly…”

“…Classical considerations seem to indicate that this object has small or vanishing degeneracy in contrast to the infinite degenerate object found in the case of tunneling to the Schwarzschild black hole. [5] This should be verified explicitly by including also quantum fluctuation of the shell. Yet, if the tunneling approach for the eternal black hole does hint on the dimensionality of the Hilbert space of the collapsing black hole, and if indeed the initial configuration that tunnels to the black hole is not degenerate, then this indicates a significant difference between 3-D and 4-D black holes.…”

“…We first review the tunneling approach [1,5] and its relation to entropy. Consider a system of gravity coupled to source fields with the action S tot = S gravity + S matter .…”

“…A new way to obtain the entropy from a (semi-classical) theory of quantum gravity, which presumably may yield the value of the integration constant, was suggested recently in Ref. [1,5]. It is well known that in the absence of surface terms at infinity there is no "external" time parameter that can be used to define a unitary evolution, or a Schrödinger-like equation for the matter.…”