Spin-½ amplitudes in black-hole evaporation

Abstract

In recent papers, we have studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time. In this recent work, we have been concerned with evaluating quantum amplitudes (not just probabilities) for transitions from initial to final states. In a general asymptotically flat context, one may specify a quantum amplitude by posing boundary data on (say) an initial space-like hypersurface ΣI and a final space-like hypersurface ΣF. To complete the specification, one must also give the Lorentzian proper-time interval between the two boundary surfaces, as measured near spatial infinity. We have assumed that the Lagrangian contains Einstein gravity coupled to a massless scalar field ϕ, plus possible additional fields; there is taken to be a ‘background’ spherically symmetric solution (γμν, Φ) of the classical Einstein/scalar field equations. For bosonic fields, the gravitational and scalar boundary data can be taken to be gij and ϕ on the two hypersurfaces, where gij (i, j = 1, 2, 3) gives the intrinsic 3-metric on the boundary, and the 4-metric is gμν (μ, ν = 0, 1, 2, 3), the boundary being taken locally in the form {x0 = const}. The classical boundary value problem, corresponding to the calculation of this quantum amplitude, is badly posed, being a boundary value problem for a wave-like (hyperbolic) set of equations. Following Feynman's +iϵ prescription, one makes the problem well-posed by rotating the asymptotic time interval T into the complex: T → ∣T ∣ exp(−iθ), with 0 < θ ⩽ π/2. After calculating the amplitude for θ > 0, one then takes the ‘Lorentzian limit’ θ → 0+. Such quantum amplitudes have been calculated for weak s = 0 (scalar), s = 1 (photon) and s = 2 (graviton) anisotropic final data, propagating on the approximately Vaidya-like background geometry, in the region containing radially outgoing black-hole radiation. In this paper, we treat quantum amplitudes for the case of fermionic massless spin-½ (neutrino) final boundary data. Making use of boundary conditions originally developed for local supersymmetry, we find that this fermionic case can be treated in a way which parallels the bosonic case. In particular, we calculate the classical action as a functional of the fermionic data on the late-time surface ΣF; the quantum amplitude follows straightforwardly from this.