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 $0<\theta \le \pi /2$. We calculate the second-variation classical Lorenztian action $S_{\mathrm{class}}^{(2)}$. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as $\theta \to 0_{+}$ of the semi-classical amplitude $\exp (i{S}_{\mathrm{class}}^{(2)})$. 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 $s=\frac{1}{2}$ boundary conditions by supersymmetry.

Cited by in corpus (3)

• Black hole evaporation in a spherically symmetric non-commutative space-time
• Quantum amplitudes in black-hole evaporation: coherent and squeezed states
• Relic Radiation from an Evaporating Black Hole