Vaidya Space-Time in Black-Hole Evaporation

Abstract

This paper continues earlier work on the quantum evaporation of black holes. This work has been concerned with the calculation and understanding of quantum amplitudes for final data perturbed slightly away from spherical symmetry on a space-like hypersurface Σ F at a late Lorentzian time T . For initial data, we take, for simplicity, sphericallysymmetric asymptotically-flat data for Einstein gravity with a massless scalar field on an initial surface Σ I at time t = 0 . Together, such boundary data give a quantum analogue of classical Einstein/scalar gravitational collapse to a black hole, perhaps starting from a diffuse, early-time configuration. Quantum amplitudes are calculated following Feynman's approach, by first rotating: T → |T | exp(−iθ) into the complex, where 0 < θ ≤ π/2 , then solving the corresponding complex classical boundary-value problem, which is expected to be well-posed provided θ > 0 , and computing its classical Lorentzian action S class and corresponding semi-classical quantum amplitude, proportional to exp(iS class ). For a locally-supersymmetric Lagrangian, describing supergravity coupled to supermatter, any loop corrections will be negligible, provided that the frequencies involved in the boundary data are well below the Planck scale. Finally, the Lorentzian amplitude is recovered by taking the limit θ → 0 + of the semi-classical amplitude. In the black-hole case, by studying the linearised spin-0 or spin-2 classical solutions in the above (slightly complexified) case, for the corresponding classical boundary-value problem with the given perturbative data on Σ F , one can compute an effective energy-momentum tensor < T µν > EF F , which has been averaged over several wavelengths of the radiation, and which describes the averaged extra energy-momentum contribution in the Einstein field equations, due to the perturbations. In general, this averaged extra contribution will be spherically symmetric, being of the form of a null fluid, describing the radiation (of quantum origin) streaming radially outwards. The corresponding space-time metric, in this region containing radially outgoing radiation, is of the Vaidya form. This, in turn, justifies the treatment of the adiabatic radial mode equations, for spins s = 0 and s = 2 , which is used elsewhere in this work.

Quantum amplitudes in black-hole evaporation: coherent and squeezed states

“…Finally, then, following Feynman's + iε prescription in the present context of black-hole quantum evaporation, one recovers the Lorentzian quantum amplitude (again, not just the probability density) for the quantum state including (say) created particles present at late times, by taking the limit of the semi-classical amplitude exp iS (2) class as δ → 0 + . As seen in [4,6,8], the black-hole radiation has the usual thermal spectrum, at the temperature 1/8πM I . Unless otherwise stated, we employ Planckian units, taking:…”

“…And W KB investigation of wave equations such as Eqs. (2.4,5) is also at the base of the present approach [4,6,8]. One might say that detailed knowledge concerning the (Lorentzian) event horizon is 'imbedded' in the relevant spin-s wave equations, such as Eqs.…”

“…As seen in [2,6], the spherically-symmetric background metric in this region can be represented to high accuracy by a Vaidya metric, which describes the outflow of massless matter, which is spherically symmetric, on the average. The principal condition for the validity of the adiabatic expansion is [6] that…”

“…The mode equation (2.12) does not depend on the quantum number m , whence we may choose ξ kℓm (r) = ξ kℓ (r) , independently of m . The boundary condition of regularity at the spatial origin {r = 0} [4,6] implies that…”

Surrounded Vaidya solution by cosmological fields

“…Afterwards, some models to describe the classical essence of this radiation in a language which is free from the usual quantum field theoretic tools and is more familiar to the astrophysicists and relativists, have been introduced. For instance, the Vaidya solution [8][9][10] has provided a simple classical model for the black hole radiation and has been vastly investigated in this regard [11][12][13][14][15][16][17], see also [18][19][20][21][22][23][24][25][26][27][28][29] for more studies. In fact, the Vaidya solution is one of the non-static solutions of the Einstein field equations and can be regarded as a generalization of the static Schwarzschild black hole solution.…”