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 of L. M. Burko and Y. T. Liu, Phys. Rev. D 64, 024006 (2001).

Figure 1

The retarded field at $x$ depends on the entire particle’s history from $-\infty <\tau \le {\tau }_{\mathrm{ret}}$. The advanced field at $x$ depends on the entire future of the particle from ${\tau }_{\mathrm{adv}}\le \tau <\infty$. The singular field depends only on the trajectory of the particle between the advanced and retarded points.

Figure 2

The spatial coordinates of the worldline of a static particle are fixed over coordinate time.