Facets of Tunneling: Particle production in external fields
  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. 
K. Srinivasan and T. Padmanabhan