Quantum amplitudes in black-hole evaporation: Spins 1 and 2

Quantum amplitudes for s = 1 Maxwell fields and for s = 2 linearised gravitational-wave perturbations of a spherically symmetric Einstein/massless scalar background, describing gravitational collapse to a black hole, are treated by analogy with the previous treatment of s = 0 scalar-field perturbations of gravitational collapse at late times. Both the spin-1 and the spin-2 perturbations split into parts with odd and even parity. Their detailed angular behaviour is analysed, as well as their behaviour under infinitesimal coordinate transformations and their linearised field equations. In general, we work in the Regge–Wheeler gauge, except that, at a certain point, it becomes necessary to make a gauge transformation to an asymptotically flat gauge, such that the metric perturbations have the expected fall-off behaviour at large radii. In both the s = 1 and s = 2 cases, we isolate suitable ‘coordinate’ variables which can be taken as boundary data on a final space-like hypersurface ΣF. (For simplicity of exposition, we take the data on the initial surface ΣI to be exactly spherically symmetric.) The (large) Lorentzian proper-time interval between ΣI and ΣF, measured at spatial infinity, is denoted by T  . We then consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action Sclass(2), on the assumption that the time interval T has been rotated into the complex: T → |T| exp (−iθ), for 0 < θ ⩽ π/2. This complexified classical boundary-value problem is expected to be well-posed, in contrast to the boundary-value problem in the Lorentzian-signature case (θ = 0), which is badly posed, since it refers to hyperbolic or wave-like field equations. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as θ → 0+ of the semi-classical amplitude exp(iSclass(2)). The boundary data for s = 1 involve the (Maxwell) magnetic field, while the data for s = 2 involve the magnetic part of the Weyl curvature tensor. These relations are also investigated, using 2-component spinor language, in terms of the Maxwell field strength ϕAB = ϕ(AB) and the Weyl spinor ΨABCD = Ψ(ABCD). The magnetic boundary conditions are related to each other and to the natural s=12 boundary conditions by supersymmetry.