A massless scalar particle coupled to the Wahlquist metric

Abstract

AbstractWe 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$$Q_0$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>or$$a_1$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>to zero. When both the NUT related constant$$a_1$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>and$$Q_0$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>are zero, the singly confluent Heun function is the solution. When we also equate the constant$$\nu _0$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ν</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>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.