Facets of Tunneling: Particle production in external fields

TL;DR

This paper reviews particle production in electric fields and Schwarzschild-like spacetimes, highlighting differences in tunneling interpretations and introducing a semi-classical method for deriving Hawking radiation.

Contribution

It introduces a new semi-classical approach to derive Hawking radiation and discusses the limitations of tunneling interpretations in Schwarzschild-like spacetimes.

Findings

Standard Bogoliubov coefficients are recovered in electric fields.

Tunneling interpretation is invalid for Schwarzschild-like spacetimes.

A regularisation method for the horizon is proposed.

Abstract

This paper presents a critical review of particle production in an uniform electric field and Schwarzchild-like spacetimes. Both problems can be reduced to solving an effective one-dimensional Schrodinger equation with a potential barrier. In the electric field case, the potential is that of an inverted oscillator -x^2 while in the case of Schwarchild-like spacetimes, the potential is of the form -1/x^2 near the horizon. The transmission and reflection coefficients can easily be obtained for both potentials. To describe particle production, these coefficients have to be suitably interpreted. In the case of the electric field, the standard Bogoliubov coefficients can be identified and the standard gauge invariant result is recovered. However, for Schwarzchild-like spacetimes, such a tunnelling interpretation appears to be invalid. The Bogoliubov coefficients cannot be determined by using an identification process similar to that invoked in the case of the electric field. The reason for such a discrepancy appears to be that, in the tunnelling method, the effective potential near the horizon is singular and symmetric. We also provide a new and simple semi-classical method of obtaining Hawking's result in the (t,r) co-ordinate system of the usual standard Schwarzchild metric. We give a prescription whereby the singularity at the horizon can be regularised with Hawking's result being recovered. This regularisation prescription contains a fundamental asymmetry that renders both sides of the horizon dissimilar. Finally, we attempt to interpret particle production by the electric field as a tunnelling process between the two sectors of the Rindler metric.