Spin-1 Amplitudes in Black-Hole Evaporation

TL;DR

This paper extends previous work on quantum amplitudes in black-hole evaporation to include spin-1 (photon) fields, deriving the classical action and quantum amplitude for electromagnetic boundary data in a gravitational collapse scenario.

Contribution

It introduces a method to compute quantum amplitudes for spin-1 fields in black-hole evaporation, linking magnetic boundary conditions to supersymmetry and gravitational wave data.

Findings

Derived the classical action for photon boundary data.

Established the relation between magnetic boundary conditions and gravitational data.

Provided the quantum amplitude expression for spin-1 fields in black-hole scenarios.

Abstract

Our earlier work on the quantum amplitude for a scalar field in black-hole evaporation, following gravitational collapse, is here extended to Maxwell theory. Boundary data are specified on initial and final space-like hypersurfaces $\Sigma_{I,F}$, separated by a large Lorentzian proper-time interval $T$, as measured at spatial infinity. The initial boundary data may be chosen (say) to be spherically symmetric, corresponding to a nearly-spherical configuration prior to gravitational collapse. The final data include the intrinsic 3-metric and scalar field, restricted to $\Sigma_F$, in addition to spin-1 data, naturally taken to be the magnetic field $B_i$ on $\Sigma_{I,F} (i=1,2,3)$. For a locally-supersymmetric theory, the quantum amplitude should be proportional to $\exp(iS_{\rm class})$, apart from corrections which are very small when the frequencies in the boundary data are small compared to the Planck scale. Here, $S_{\rm class}$ is the action of the classical solution. The Lorentzian amplitude is found by taking the limit $\theta\to 0_+$. By a method similar to that used in the spin-0 case, one obtains the quantum amplitude for photon data on $\Sigma_F$. The magnetic boundary conditions are related by supersymmetry to the natural spin-2 (gravitational-wave) boundary conditions, which involve fixing the magnetic part of the Weyl tensor.