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

Journal Article · Thu Jun 15 00:00:00 EDT 2006 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2005.11.011· OSTI ID:20767022

Farley, A N.St.J. ^[1]; D'Eath, P D ^[1]

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1. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)

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 {sigma}{sub F}. (For simplicity of exposition, we take the data on the initial surface {sigma}{sub I} to be exactly spherically symmetric.) The (large) Lorentzian proper-time interval between {sigma}{sub I} and {sigma}{sub F}, measured at spatial infinity, is denoted by T. We then consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action S{sub class}{sup (2)}, on the assumption that the time interval T has been rotated into the complex: T-> vertical bar T vertical bar exp(-i{theta}), for 0<{theta}=<{pi}/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 ({theta}=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 {theta}->0{sub +} of the semi-classical amplitude exp(iS{sub class}{sup (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 {phi}{sub AB}={phi}{sub (AB)} and the Weyl spinor {psi}{sub ABCD}={psi}{sub (ABCD)}. The magnetic boundary conditions are related to each other and to the natural s=12 boundary conditions by supersymmetry.

• OSTI ID: 20767022
• Journal Information: Annals of Physics (New York), Vol. 321, Issue 6; Other Information: DOI: 10.1016/j.aop.2005.11.011; PII: S0003-4916(05)00260-5; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
AMPLITUDES
BLACK HOLES
BOUNDARY CONDITIONS
BOUNDARY-VALUE PROBLEMS
COORDINATES
FIELD EQUATIONS
GAUGE INVARIANCE
GRAVITATIONAL COLLAPSE
GRAVITATIONAL WAVES
MAGNETIC FIELDS
PARITY
PERTURBATION THEORY
QUANTUM MECHANICS
SCALAR FIELDS
SCALARS
SPIN
SUPERSYMMETRY
TENSORS
TRANSFORMATIONS