Abstract
We propose a semiclassical method to calculate S-matrix elements for two-stage gravitational transitions involving matter collapse into a black hole and evaporation of the latter. The method consistently incorporates back-reaction of the collapsing and emitted quanta on the metric. We illustrate the method in several toy models describing spherical self-gravitating shells in asymptotically flat and AdS space-times. We find that electrically neutral shells reflect via the above collapse-evaporation process with probability exp(-B), where B is the Bekenstein-Hawking entropy of the intermediate black hole. This is consistent with interpretation of exp(B) as the number of black hole states. The same expression for the probability is obtained in the case of charged shells if one takes into account instability of the Cauchy horizon of the intermediate Reissner-Nordstrom black hole. Our semiclassical method opens a new systematic approach to the gravitational S-matrix in the non-perturbative regime.
Highlights
Ψf (a) collapsing matter (b) black hole (c) Hawking radiation reconciliation of the black hole evaporation with unitarity would require drastic departures from the classical geometry in the vicinity of an old black hole horizon
We propose a semiclassical method to calculate S-matrix elements for twostage gravitational transitions involving matter collapse into a black hole and evaporation of the latter
We find that electrically neutral shells reflect via the above collapse-evaporation process with probability exp(−B), where B is the Bekenstein-Hawking entropy of the intermediate black hole
Summary
2.1 Semiclassical S-matrix for gravitational scattering The S-matrix is defined as. where Uis the evolution operator; free evolution operators U0 on both sides transform from Schrodinger to the interaction picture. If P is small, the gravitational interaction is weak and the particles scatter trivially without forming a black hole In this regime the integral in eq (2.2) is saturated by the saddle-point configuration Φcl satisfying the classical field equations with boundary conditions related to the initial and final states [22]. We introduce a functional Tint[Φ] with the following properties: it is (i) diff-invariant; (ii) positive-definite if Φ is real; (iii) finite if Φ approaches flat space-time at t → ±∞; (iv) divergent for any configuration containing a black hole in the asymptotic future. We increase the value of P to P > P∗ assuming that continuously deformed saddle-point configurations Φ remain physical.7 In this way we obtain the modified solutions and the semiclassical amplitude at any P. This functional should satisfy the conditions (i)-(iv) listed before eq (2.3)
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