Solutions of a charged scalar field in five-dimensional helicoid solution with electromagnetic field

TL;DR

This paper investigates the behavior of a charged scalar field in a five-dimensional helicoid spacetime with an electromagnetic field, deriving solutions to the Klein-Gordon equation and analyzing the effects of the electromagnetic parameter.

Contribution

It provides a detailed analysis of the Klein-Gordon equation in a complex five-dimensional background, including numerical and approximate solutions, highlighting the electromagnetic field's influence.

Findings

The angular part reduces to a double confluent Heun equation.

The radial equation is generally unsolvable in closed form but can be simplified in special cases.

The electromagnetic field acts as an effective cutoff on the radial coordinate.

Abstract

We study a charged and massive scalar field in the background of the Nutku-Ghezelbash-Kumar metric which is obtained by the addition of a time coordinate to the Nutku helicoid metric in a non-trivial way. The angular part of the Klein-Gordon equation can be written as a double confluent Heun equation. The radial equation cannot be solved in terms of a known function in its general form. However, in some special cases, the radial equation can also be written explicitly as a double confluent Heun equation. We study the full radial equation numerically and observe that the electromagnetic field parameter defines an effective cut-off on the range of the radial coordinate. Finally, we obtain a quasi-exact solution with an approximation.