A massless scalar particle coupled to the Wahlquist metric

TL;DR

This paper analyzes the wave equation for a massless scalar field in the Wahlquist metric, revealing solutions in terms of Heun functions and generalizing previous results for type-D metrics.

Contribution

It demonstrates that all variations of the Wahlquist metric lead to solutions expressed as Heun functions, extending prior work on type-D metrics.

Findings

Solutions are expressed in terms of general Heun functions.

Special cases yield confluent and double confluent Heun solutions.

Generalizes previous results for type-D metrics to Wahlquist metric variations.

Abstract

We study the solutions of the wave equation where a massless scalar field is coupled to the Wahlquist metric, a type-D solution. We first take the full metric and then write simplifications of the metric by taking some of the constants in the metric null. When we do not equate any of the arbitrary constants in the metric to zero, we find the solution is given in terms of the general Heun function, apart from some simple functions multiplying this solution. This is also true, if we equate one of the constants $Q_0$ or $a_1$ to zero. When both the NUT-related constant $a_1$ and $Q_0$ are zero, the singly confluent Heun function is the solution. When we also equate the constant $\nu_0$ to zero, we get the double confluent Heun-type solution. In the latter two cases, we have an exponential and two monomials raised to powers multiplying the Heun type function. Thus, we generalize the Batic et al. result for type-D metrics for this metric and show that all variations of the Wahlquist metric give Heun type solutions.