Bogoliubov transformations for amplitudes in black-hole evaporation

Abstract

In a previous Letter, we outlined an approach to the calculation of quantum amplitudes appropriate for studying the black-hole radiation which follows gravitational collapse. This formulation must be different from the familiar one (which is normally carried out by considering Bogoliubov transformations), since it yields quantum amplitudes relating to the final state, and not just the usual probabilities for outcomes at a late time and large radius. Our approach simply follows Feynman's +iǫ prescription. Suppose that, in specifying the quantum amplitude to be calculated, initial data for Einstein gravity and (say) a massless scalar field are specified on an asymptotically-flat space-like hypersurface Σ I , and final data similarly specified on a hypersurface Σ F , where both Σ I and Σ F are diffeomorphic to R 3 . Denote by T the (real) Lorentzian proper-time interval between Σ I and Σ F , as measured at spatial infinity. Then rotate: T → |T | exp(−iθ) , 0 < θ ≤ π/2 . The classical boundary-value problem is then expected to become well-posed on a region of topology I × R 3 , where I is the interval 0, |T | . For a locally-supersymmetric theory, the quantum amplitude is expected to be dominated by the semi-classical expression exp(iS class ), where S class is the classical action. Hence, one can find the Lorentzian quantum amplitude from consideration of the limit θ → 0 + . In the usual approach, the only possible such final surfaces Σ F are in the strong-field region shortly before the curvature singularity; that is, one cannot have a Bogoliubov transformation to a smooth surface 'after the singularity'. In our complex approach, however, one can put arbitrary smooth gravitational data on Σ F , provided that it has the correct mass M ; thus we do have Bogoliubov transformations to surfaces 'after the singularity in the Lorentzian-signature geometry' -the singularity is simply by-passed in the analytic continuation (see below). In this Letter, we consider Bogoliubov transformations in our approach, and their possible relation to the probability distribution and density matrix in the traditional approach. In particular, we find that our probability distribution for configurations of the final scalar field cannot be expressed in terms of the diagonal elements of some density-matrix distribution.

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.…”

“…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…”

“…When considering perturbative boundary data for a field of any spin, posed on Σ F in describing a final state resulting from black-hole evaporation, we denote by {a skℓmP } a set of analogous 'Fourier-like' coefficients, where s gives the particle spin, k the frequency, (ℓ, m) the angular quantum numbers, and P = ± 1 the parity (for s = 0 ). For simplicity, we study in this Paper only bosonic perturbations, of spins s = 0, 1, 2, as treated in [2][3][4][6][7][8]. In each case, we found that the quantum amplitude or wave functional is of the semi-classical form, being given by…”

“…For a complete specification of the problem (if indeed such a boundary-value problem is well-posed -see below), one further needs to give the Lorentzian propertime interval T which separates Σ I from Σ F , as measured at spatial infinity. In the papers [3,4], as in the present paper, we take the simplest possibility for the matter fields present, namely, a real massless scalar field φ . Then, suitable boundary data on Σ I and on Σ F are expected to be (h ij , φ) I,F .…”