  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
$\Sigma_{I, F}$, 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 $\phi$. 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 $\phi$ on the bounding
surfaces, is badly posed, being a boundary-value problem for a wave-like
(hyperbolic) set of equations. Following Feynman's $+i\epsilon$ prescription,
the problem is made well-posed by rotating the asymptotic time interval $T$
into the complex: $T\to{\mid} T{\mid}\exp(-i\theta)$, with $0<\theta\leq\pi/2$.
After calculating the amplitude for $\theta>0$, one takes the 'Lorentzian
limit' $\theta\to 0_+$ to obtain the Lorentzian quantum amplitude.
