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

Abstract

Here, quantum amplitudes 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 = 1 linearised Maxwell-field perturbations. As with the spin-1 case, 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 vacuum Einstein) 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 asymptoticallyflat gauge, such that the metric perturbations have the expected falloff behaviour at large radii. As with the s = 1 treatment, so in this s = 2 case 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 secondvariation classical Lorentzian action S (2) class , 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(iS (2) class ) . The boundary data for s = 2 involve the magnetic part of the Weyl tensor, just as the data for s = 1 involve the usual (Maxwell) magnetic field. 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) .

Quantum amplitudes in black-hole evaporation: coherent and squeezed states

“…An analogous description holds for fields of all the spins 1 2 , 1 and 2 that have so far been checked [2,5,7]. When considering perturbative boundary data for a field of any spin, posed on Σ F in describing a final state resulting from black-hole evaporation, we denote by {a skℓmP } a set of analogous 'Fourier-like' coefficients, where s gives the particle spin, k the frequency, (ℓ, m) the angular quantum numbers, and P = ± 1 the parity (for s = 0 ).…”

“…We begin by reviewing this approach, which in [2][3][4][5][6][7][8] was described and applied to quantum amplitudes (not just probabilities) for particle production, following gravitational collapse to a black hole. In specifying either the classical boundary-value problem or the quantum amplitude to be computed, one takes an initial space-like ypersurface Σ I and a final space-like hypersurface Σ F .…”

“…In the present paper, we study quantum amplitudes found via Feynman's approach, as discussed and calculated in [2][3][4][5][6][7][8], but now in the context of coherent states [13], which resemble 'classical states', and of squeezed states [14], which are purely quantum-mechanical. As above, our motivation originated with the study of the final radiation which remains after a black hole has evaporated completely, but there are strong connections also with the relic Cosmic Microwave Background Radiation (CMBR) induced by inflationary cosmological perturbations.…”

Black hole evaporation in a spherically symmetric non-commutative spacetime

“…(ii) The related series of papers in Refs. [2,3,4,5,6,7,8,9,10], concerned with evaluating quantum amplitudes for transitions from initial to final states, in agreement with a picture where information is not lost, and the end state of black hole evaporation is a combination of outgoing radiation states.…”

“…We have been therefore led to study how non-commutativity would affect the analysis of quantum amplitudes in black hole evaporation performed in Refs. [2,3,4,5,6,7,8,9,10]. Following Ref.…”