Vaidya Space-Time in Black-Hole Evaporation

Abstract

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 $\Sigma_{I,F}$. The Lorentzian proper-time separation between the surfaces, as measured at spatial infinity, is denoted by $T$. Following Feynman's $+i\epsilon$ approach, we rotate $T$ into the complex: $T\to {\mid}T{\mid} \exp(-i\theta)$, where $0<\theta\leq\pi/2$. The corresponding {\it classical} complex boundary-value problem is expected to be well-posed for $\theta > 0$. The Lorentzian amplitude is found by taking the limit $\theta \to 0_+$ of the quantum amplitude, itself closely approximated by the semi-classical expression $\exp(iS_{\rm class})$, where $S_{\rm class}$ is the classical action. For given weak anisotropic spin-0 and spin-2 boundary data on $\Sigma_F$, 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.