2002/09/30 by Andrew Farley, Andrew N. S. J. Farley, Farley, Andrew N. S. J.
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Quantum Electrodynamics and Casimir Effect #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/0209113
25 pages, 1 figure
arxiv created 2002/09/30 · arxiv updated 2009/11/30
We compute the semi-classical quantum amplitude to go from an initial spherically symmetric bosonic matter and gravitational field configuration to a final radiation configuration, corresponding to the relic Hawking radiation from a non-rotating, chargeless black hole which evaporates completely. This is obtained via the classical action integral which is solely a boundary term. On discretising the classical action, the quantum amplitude can be expressed in terms of generalised coherent states of the harmonic oscillator. A squeezed-state representation is obtained by complexifying the proper time separation T at spatial infinity between the initial and final space-like hypersurfaces. Such a procedure is deemed necessary as the two-surface problem for Dirichlet boundary data and wave-like perturbations is not well posed. We find that infinitesimal rotation into the lower complex T plane is equivalent to a highly-squeezed final state for the relic radiation, similar to the relic gravitational-wave background in cosmology. This final state is a pure state, and so the unpredictability associated with the final momentarily-naked singularity is avoided. The cosmological analogy is the tunnelling from an initial smooth Euclidean or timeless state to a classical universe. The high-squeezing limit corresponds to a final state of the Hawking flux which is indistinguishable from a stochastic collection of standing waves. The phases conjugate to the field amplitudes are squeezed to discrete values. We also discuss the entropy of the final radiation in the high-squeezing limit.