Quantum Amplitudes in Black-Hole Evaporation I. Complex Approach

Abstract

Here we examine the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time, in order to evaluate quantum amplitudes (not just probabilities) for final states. No information is lost in collapse to a black hole. Boundary data are specified on initial and final hypersurfaces ${\Sigma }_{I,F}$, separated by a Lorentzian proper-time interval $T$, as measured at spatial infinity. For simplicity, consider Einstein gravity coupled minimally to a massless scalar field $\varphi$. In Lorentzian signature, the classical Dirichlet boundary-value problem, corresponding to specification of the intrinsic spatial metric $h_{ij}(i,j=1,2,3)$ and $\varphi$ on the bounding surfaces, is badly posed, being a boundary-value problem for a wave-like (hyperbolic) set of equations. Following Feynman's $+i\epsilon$ prescription, the problem is made well-posed by rotating the asymptotic time interval $T$ into the complex: $T\to \mid T\mid \exp (-i\theta )$, with $0<\theta \le \pi /2$. After calculating the amplitude for $\theta >0$, one takes the 'Lorentzian limit' $\theta \to 0_{+}$ to obtain the Lorentzian quantum amplitude.

References in corpus (1)

• What local supersymmetry can do for quantum cosmology

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• Quantum amplitudes in black-hole evaporation: Spins 1 and 2
• Quantum Amplitudes in Black-Hole Evaporation: Complex Approach and Spin-0 Amplitude