Scalar--Field Amplitudes in Black--Hole Evaporation

TL;DR

This paper calculates quantum amplitudes for black hole decay into flat space and radiation, showing no information loss and using complexified time to solve boundary-value problems in a semi-classical framework.

Contribution

It introduces a method to evaluate quantum amplitudes for black hole evaporation using complex time and boundary data, extending previous approaches to include scalar fields and supersymmetric theories.

Findings

Quantum amplitudes are approximately exp(-I), with I being the classical Euclidean action.

The boundary-value problem becomes well-posed with complexified time T, enabling unique solutions.

The approach applies to weak scalar fields and confirms no information loss in black hole decay.

Abstract

We study the quantum-mechanical decay of a Schwarzschild-like black hole into almost-flat space and weak radiation at a very late time, evaluating quantum amplitudes (not just probabilities) for transitions from initial to final states. No information is lost. The model contains gravity and a massless scalar field. The quantum amplitude to go from given initial to final bosonic data in a slightly complexified time-interval $T={\tau}{\exp}(-i{\theta})$ at infinity is approximately $\exp(-I)$, where $I$ is the (complex) Euclidean action of the classical solution filling in between the boundary data. And in a locally supersymmetric (supergravity) theory, the amplitude const. exp(-I) is exact. Dirichlet boundary data for gravity and the scalar field are posed on an initial spacelike hypersurface extending to spatial infinity, just prior to collapse, and on a corresponding final spacelike surface, sufficiently far to the future of the initial surface to catch all the Hawking radiation. In an averaged sense this radiation has an approximately spherically-symmetric distribution. If the time-interval $T$ were exactly real, the resulting `hyperbolic Dirichlet boundary-value problem' would not be well posed. If instead (`Euclidean strategy'), one takes $T$ complex, as above ($0<\theta{\leq}{\pi}/2$), the field equations become strongly elliptic, with a unique solution to the classical boundary-value problem. Expanding the bosonic part of the action to quadratic order in perturbations about the classical solution gives the quantum amplitude for weak-field final configurations, up to normalization. Such amplitudes are calculated for weak final scalar fields.