We find a quasi-local expansion for the tail term of the Green's function for a particle with scalar charge moving outside the event horizon of a black hole of mass M. To do that we use a WKB-like ansatz for the mode functions and we solve the resulted differential equation by iteration. We then sum the mode contributions using Plana sum rule. The fact that we find the tail term as an analytic expression is important. We then use our expressions to calculate the self-force exerted upon a particle of scalar charge that has been held at rest from infinite past to some time after which it moves on a general geodesic of the space-time. We perform this computation first for the radial path of a particle released from rest and then generalize the method for a particle launched on a general geodesic.

We then turn to computing the noise kernel. The problem we are primarily concerned with is that of a massless, conformally coupled scalar field in the optical Schwarzschild (the ultrastatic spacetime conformal to the Schwarzschild black hole).

In contrast to previous work done on this topic, we keep the two points separate, and as a result work with non-renormalized Wightman functions. We give an expression in terms of an expansion in coordinate separation and conclude with an outlook.