The wave function of a black hole and the dynamical origin of entropy

Abstract

Recently [1,2 ] i t w as proposed to explain the dynamical origin of the entropy of a black hole by identifying its dynamical degrees of freedom with states of quantum elds propagating in the black-hole's interior. The present paper contains the further development of this approach. The no-boundary proposal (analogous to the Hartle-Hawking no-boundary proposal in quantum cosmology) is put forward for dening the wave function of a black hole. This wave function is a functional on the conguration space of physical elds (including the gravitational one) on the three-dimensional space with the Einstein-Rosen bridge topology. I t i s s h o wn that in the limit of small perturbations on the Kruskal background geometry the no-boundary wave function coincides with the HartleHawking vacuum state. The invariant denition of inside and outside modes is proposed. The density matrix describing the internal state of a black hole is obtained by a v eraging over the outside modes. This density matrix is used to dene the entropy of a black hole, which i s t o b e d i v ergent. It is argued that the quantum uctuations of the horizon which are internally present in the proposed formalism may give the necessary cut-o and provide a black hole with the nite entropy.

Macroscopic effects of the quantum trace anomaly

“…In these circumstances the low energy effective action (3.15) becomes a good approximation to the exact effective action (3.14). However, as has been noted by several authors [33], spacetimes with a static Killing horizon are locally conformally related to flat spacetime in the vicinity of the horizon. Because of this near horizon conformal symmetry we expect the anomalous action (3.12) to generate a stress tensor (3.36) which is a good approximation to the quantum stress tensor near an arbitrary Killing horizon.…”

Black Hole Entropy and Physics at Planckian Scales

“…For large M (M m P ) this probability is a sharp peak with width ≈ m P located near the value of M = M β ≡ β ∞ /8π. For β ∞ = 8πM and r B → ∞ this wavefunction coincides with a no-boundary wavefunction obtained in [31].…”

“…The discussion of the relation of the thermodynamical and statistical-mechanical entropy of a black hole requires off-shell calculations. The reason for this is quite simple and can be explained, for example, by using the approach based on the no-boundary wavefunction of BFZ [31]. The matrix elements in the |ϕ − basis of the operatorρ lnρ which enters the definition of the statistical-mechanical entropy (1.4) can be obtained by partially differentiating (3.24) with respect to β.…”

“…Here we present a modified version of the no-boundary approach of Ref. [31] which is analogous to the 'no-boundary ansatz' in quantum cosmology [36]. This ansatz singles out a set of no-boundary wavefunctions which is convenient for our purpose.…”

“…It was proposed in [31] to introduce a wave function of a black hole as the functional over the space of deformations of the Einstein-Rosen bridge. In this approach the wave function of a black hole depends on data located on both parts of the Einstein-Rosen bridge: an external Σ + (external degrees of freedom) and an inner Σ − one (internal degrees of freedom).…”