Vaidya space-time in black-hole evaporation

摘要

We take a boundary-value approach to quantum amplitudes arising in gravitational collapse to a black hole. Pose boundary data on initial and final space-like hypersurfaces Sigma(F,I) , separated at spatial infinity by a Lorentzian proper-time interval T. Quantum amplitudes are calculated following Feynman's approach; rotate: T -> vertical bar T vertical bar exp (-i theta) into the complex, where 0 < theta <= pi/2, and solve the corresponding well-posed complex classical boundary-value problem. We compute the classical Lorentzian action S-class and corresponding semi-classical quantum amplitude, proportional to exp(iS(class)). To recover the Lorentzian amplitude, take the limit theta -> 0(+) of the semi-classical amplitude. For the classical boundary-value problem with given perturbative boundary data, we compute an effective spherically-symmetric energy-momentum tensor < T-mu nu > (EFF), averaged over several wavelengths of the radiation, describing the averaged extra energy-momentum contribution in the Einstein field equations, due to the perturbations. This takes the form of a null fluid, describing the radiation (of quantum origin) streaming radially outwards. The classical space-time metric, in this region of the space time, is of Vaidya form, justifying the adiabatic radial mode equations, for spins s = 0 and s = 2.