  Quantum amplitudes for $s=2$ gravitational-wave perturbations of
Einstein/scalar collapse to a black hole are treated by analogy with $s=1$
Maxwell perturbations. The spin-2 perturbations split into parts with odd and
even parity. We use the Regge-Wheeler gauge; at a certain point we make a gauge
transformation to an asymptotically-flat gauge, such that the metric
perturbations have the expected falloff behaviour at large radii. By analogy
with $s=1$, for $s=2$ natural 'coordinate' variables are given by the magnetic
part $H_{ij} (i,j=1,2,3)$ of the Weyl tensor, which can be taken as boundary
data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the
data on the initial surface $\Sigma_I$ to be exactly spherically-symmetric. The
(large) Lorentzian proper-time interval between $\Sigma_I$ and $\Sigma_F$,
measured at spatial infinity, is denoted by $T$. We follow Feynman's
$+i\epsilon$ prescription and rotate $T$ into the complex: $T\to{\mid}T{\mid}
\exp(-i\theta)$, for $0<\theta\leq\pi/2$. The corresponding complexified {\it
classical} boundary-value problem is expected to be well-posed. The Lorentzian
quantum amplitude is recovered by taking the limit as $\theta\to 0_+$. For
boundary data well below the Planck scale, and for a locally supersymmetric
theory, this involves only the semi-classical amplitude $\exp(iS^{(2)}_{\rm
class}$, where $S^{(2)}_{\rm class}$ denotes the second-variation classical
action. The relations between the $s=1$ and $s=2$ natural boundary data,
involving supersymmetry, are investigated using 2-component spinor language in
terms of the Maxwell field strength $\phi_{AB}=\phi_{(AB)}$ and the Weyl spinor
$\Psi_{ABCD}=\Psi_{(ABCD)}$.
