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

Abstract

Quantum amplitudes for $s=1$ at 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 a previous treatment of $s=0$ scalar-field perturbations of gravitational collapse at late times. In both the $s=1$ and $s=2$ cases, we isolate suitable 'co-ordinate' variables which can be taken as boundary data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the data on an initial pre-collapse surface $\Sigma_I$ to be exactly spherically symmetric. The (large) Lorentzian proper-time interval between $\Sigma_{I}, \Sigma_{F}$, measured at spatial infinity, is denoted by $T$. The complexified classical boundary-value problem is expected to be well-posed, provide that the time interval $T$ has been rotated into the complex: $T\to{\mid}T{\mid}\exp(-i\theta)$, for $0<\theta\leq{\pi}/2$. We calculate the second-variation classical Lorenztian action $S ^{(2)}_{\rm class}$. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as $\theta\to 0_+$ of the semi-classical amplitude $\exp(iS^{(2)}_{\rm class})$. The boundary data for $ s=1$ involve the Maxwell magnetic field; the data for $s=2$ involve the magnetic part of the Weyl curvature tensor. The magnetic boundary conditions are related to each other and to the natural $s={1 \over 2}$ boundary conditions by supersymmetry

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

“…The coupled Lorentzian-signature Einstein/scalar field equations for this spherically-symmetric configuration are given by the analytic continuation of the Riemannian field equations Eqs. (1.9-13) of [7], on making the replacement…”

“…where τ is the 'Riemannian time-coordinate' of [7], and where the real number ϑ should be rotated precisely from 0 to π/2 . The small anisotropic perturbations in the boundary data on the final late-time hypersurface Σ F consist, in the language of Sec.1, of the perturbed part λ h (1) ij of the intrinsic 3-dimensional spatial metric h ijF on Σ F , together with the perturbation λ φ (1) of the scalar field φ F on Σ F .…”

“…The first correction S (2) class is the secondvariation classical action and is bilinear in the linear-order λ h (1) ij , λ φ (1) corrections to the boundary data on Σ F . As in [7], one can indeed evaluate S (2) class jointly as a functional of the linearised boundary data λ h (1) ij , λ φ (1) F and a function of the complex variable T . One then computes the corresponding semi-classical quantum amplitude, proportional to exp iS (2) class , and one can also include loop corrections, if appropriate.…”

“…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.…”