2005/10/07 by A. N. St. J. Farley, Farley, A. N. St. J., P. D. D'Eath +1
Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/0510028
arxiv created 2005/10/07 · arxiv updated 2009/12/01
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 ΣI, 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 ϕ. In Lorentzian signature, the classical Dirichlet boundary-value problem, corresponding to specification of the intrinsic spatial metric hij (i,j =1,2,3) and ϕ on the bounding surfaces, is badly posed, being a boundary-value problem for a wave-like (hyperbolic) set of equations. Following Feynman's +iε prescription, the problem is made 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 takes the 'Lorentzian limit' θ→ 0+ to obtain the Lorentzian quantum amplitude.