Quantum Amplitudes in Black-Hole Evaporation I. Complex Approach

Abstract

Here we examine the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time, in order to evaluate quantum amplitudes (not just probabilities) for final states. No information is lost in collapse to a black hole. Boundary data are specified on initial and final hypersurfaces $Σ_{I, F}$, separated by a Lorentzian proper-time interval $T$, as measured at spatial infinity. For simplicity, consider Einstein gravity coupled minimally to a massless scalar field $ϕ$. In Lorentzian signature, the classical Dirichlet boundary-value problem, corresponding to specification of the intrinsic spatial metric $h_{ij} (i,j =1,2,3)$ and $ϕ$ on the bounding surfaces, is badly posed, being a boundary-value problem for a wave-like (hyperbolic) set of equations. Following Feynman&#39;s $+iε$ prescription, the problem is made well-posed by rotating the asymptotic time interval $T$ into the complex: $T\to{\mid} T{\mid}\exp(-iθ)$, with $0<θ\leqπ/2$. After calculating the amplitude for $θ>0$, one takes the &#39;Lorentzian limit&#39; $θ\to 0_+$ to obtain the Lorentzian quantum amplitude.