6.1 Scalar wave equation in Kerr–Newman
Let us first discuss probe scalar fields on the Kerr–Newman black hole. The Klein–Gordon equation for a
charged massive spin 0 field of mass
and charge
was analyzed in Section 5.2. Expanding in
eigenmodes and using the fact that the equation is separable, we have
The equations for the functions
and
were written in (158) and (159). Substituting
, the equations can be written in a convenient way as
where
. The function
is defined as
and is evaluated either at
or
and
These equations can be solved by Heun functions, which are not among the usual special functions. A
solution can be found only numerically.