Static Kerr Green's Function in Closed Form and an Analytic Derivation   of the Self-Force for a Static Scalar Charge in Kerr Space-Time
  We derive a closed-form solution for the Green's function for the wave equation of a static (with respect to an undragged, static observer at infinity) scalar charge in the Kerr space-time. We employ our solution to obtain an analytic expression for the self-force on such a charge, comparing our results to those previously obtained using the mode-sum regularization prescription. 
Adrian C. Ottewill and Peter Taylor
gr-qc