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 μ and charge qe was analyzed in Section 5.2. Expanding in eigenmodes and using the fact that the equation is separable, we have
−iωt+imΟ• Φ (t,r,πœƒ,Ο• ) = e S (πœƒ)R (r). (197 )
The equations for the functions S(πœƒ) and R(r) were written in (158View Equation) and (159View Equation). Substituting A0 = K + a2(ω2 − μ2) aω,l,m l, the equations can be written in a convenient way as
[ 1 m2 ] ----∂ πœƒ(sin πœƒ∂πœƒ) − --2---+ a2(ω2 − μ2)cos2 πœƒ + Kl S (πœƒ) = 0, (198 ) [sinπœƒ sin πœƒ ] ------α(r+-)2------ -----α-(r− )2----- ∂r(Δ ∂r) + (r − r )(r − r ) − (r − r )(r − r ) − Kl + V (r) R (r) = 0, (199 ) + + − − + −
where Δ (r ) = (r − r+ )(r − r− ). The function α (r) is defined as
α(r) = (2M r − Q2 )ω − am − Qrqe, (200 )
and is evaluated either at r+ or r− and
V (r) = (ω2 − μ2)r2 + 2ω (M ω − q Q )r − ω2Q2 + (2M ω − q Q)2. (201 ) e e
These equations can be solved by Heun functions, which are not among the usual special functions. A solution can be found only numerically.
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