Spin-2 Amplitudes in Black-Hole Evaporation

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 ${\mathrm{\Sigma }}_F$. For simplicity, we take the data on the initial surface ${\mathrm{\Sigma }}_I$ to be exactly spherically-symmetric. The (large) Lorentzian proper-time interval between ${\mathrm{\Sigma }}_I$ and ${\mathrm{\Sigma }}_F$, measured at spatial infinity, is denoted by $T$. We follow Feynman’s $+iϵ$ 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 ${\varphi }_{AB}={\varphi }_{(AB)}$ and the Weyl spinor ${\mathrm{\Psi }}_{ABCD}={\mathrm{\Psi }}_{(ABCD)}$.