Spin-1 and spin-2 amplitudes in black-hole evaporation

In previous papers, we described work on real massless scalar (spin-0) perturbations propagating on the (approximately) spherically symmetric Vaidya-like background spacetime which remains after a black hole has evaporated completely. Here, we allow also for weak gravitational perturbations in the final data, corresponding to s = 2 (graviton) modes. We further allow for the possibility that the Lagrangian includes a contribution from a Maxwell field, and so include s = 1 (photon) modes. As with the previous spin-0 calculations, we rotate the (real) Lorentzian proper-time interval T between the initial and final hypersurfaces ΣI, ΣF, into the complex: T → vertical bar T vertical bar exp(-iθ), where 0 < θ ≤ π/2. The classical boundary-value problem becomes well posed. For example, the classical Maxwell action can be written as an explicit functional of the (suitably chosen) boundary data. Similarly for gravity (s = 2). By a process which parallels exactly the previous spin-0 calculation, one can obtain the quantum amplitude or wavefunctional for the final boundary data. The natural boundary data on ΣF for s = 1 are the (divergence-free) magnetic field Bi on ΣF; for s = 2 one takes the (symmetric, trace-free, divergence-free) magnetic part Hik of the Weyl tensor on ΣF; a similar expression holds for s = 1/2 (neutrinos). These relations are an aspect of local supersymmetry