Vaidya Space-Time in Black-Hole Evaporation

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Recently we have studied, using a boundary-value approach, quantum amplitudes resulting from gravitational collapse to a black hole. Suitable boundary data for all fields present are posed on initial and final space-like asymptotically flat hypersurfaces Σᵢ,F. The Lorentzian proper-time separation between the surfaces, as measured at spatial infinity, is denoted by T. Following Feynman's +iε approach, we rotate T into the complex: T→ T (-iθ), where 0<θ≤π/2. The corresponding classical complex boundary-value problem is expected to be well-posed for θ> 0. The Lorentzian amplitude is found by taking the limit θ→ 0+ of the quantum amplitude, itself closely approximated by the semi-classical expression (iS class), where S class is the classical action. For given weak anisotropic spin-0 and spin-2 boundary data on ΣF, one can compute an effective classical energy-momentum tensor in the interior, which has been averaged over several wave-lengths of the radiation. This averaged extra contribution will be spherically symmetric, equivalent to a null fluid, and describing the radial outward streaming of the radiation (of quantum origin). The corresponding space-time metric, in this region containing radially-outgoing radiation, is of the Vaidya form. This, in turn, justifies the treatment of the adiabatic radial mode equations, for spins s=0 and s=2, which is used throughout this larger project.