Several solutions of the Klein-Gordon equation in Kerr-Newman spacetime and the BSW effect

TL;DR

This paper derives exact solutions to the charged Klein-Gordon equation in Kerr-Newman spacetime, analyzes collision intensities of field excitations, and demonstrates the quantum analogue of the BSW effect with unbounded collision intensities at the horizon.

Contribution

It provides explicit hypergeometric solutions and extends the BSW effect analysis to quantum field theory in Kerr-Newman spacetime.

Findings

Collision intensity is bounded in nonextremal black holes.

Unbounded collision intensity occurs at extremal horizons when conditions are met.

Quantum effects inherit the classical BSW effect, leading to infinite collision energies.

Abstract

We investigate the radial part of the charged massive Klein-Gordon equation in Kerr-Newman spacetime, and in several specific situations, obtain exact solutions by means of essentially hypergeometric functions or their confluent types. Using these global solutions and generally obtained local solutions, we calculate a sort of intensity of the collision of two field excitations, which is a slight generalization of the trace of the stress tensor. We find that when the black hole is nonextremal, the intensity of the collision of two ingoing modes is bounded. However, in the extremal limit, more precisely $\hbar \kappa_H \rightarrow 0$, the upper bound grows so that when the frequency of one of the two modes satisfies the critical relation, the intensity of the collision at the horizon becomes unboundedly large. Furthermore, the intensity of the collision of ingoing and outgoing modes is always unbounded, as well as in the classical particle theory. Our results suggest that the BSW effect is inherited by the quantum theory.