A quantum analog model for a scalar particle interacting with a Kerr–de Sitter black hole

Abstract

We study the thin-layer quantization of a Schrödinger particle constrained to move along an oblate spheroidal surface embedded in Euclidean space as an analog model to a particle in the background of a black hole and of a wormhole. The Schrödinger equation for the superficial wavefunction leads to an ordinary differential equation with five singularities — four regular and one irregular. We manage to transform this equation into a generalized Heun’s equation, with three regular singular points and one irregular singularity at infinity. This equation was studied by Schäfke and Schmidt back in 1980. By means of an external potential, we are able to reproduce the equations of motion of a particle close both to a black hole and a wormhole, thus creating an effective analog model. Additionally, we present correct and exact solutions for the radial part of the Klein–Gordon equation in the Kerr–de Sitter metric.