Self-force on point particles in orbit around a Schwarzschild black hole
We examine the motion of a point scalar or electromagnetic charge in orbit around a Schwarzschild black hole. As the particle moves it emits radiation and loses energy and angular momentum to the radiation field. A small part of this radiation backscatters from the curvature of spacetime and returns to the location of the particle. The interaction of the particle with this radiation gives rise to a self-force acting on the particle. Initially this self-force appears to be divergent at the position of the particle. Similar to the situation in quantum field theory, the field close to the particle requires renormalization, separating a finite physical contribution from the infinite renormalizable part. One way of handling the divergence in Schwarzschild spacetime is the mode-sum scheme introduced by Barack and Ori [1, 2]. We apply their scheme as well as the singular-regular decomposition of Detweiler and Whiting  to the problem at hand. In doing so we calculate what are commonly called the regularization parameters 'A, B, C' and ' D', extending previous work that only included the 'A, B' and 'C 'terms. In the scalar, electromagnetic and gravitational cases we calculate the regularization parameters for tetrad components of the field gradient, using only manifestly scalar quantities in the regularization. We also implement a numerical scheme to calculate the modes of the full retarded field for the scalar and electromagnetic cases. The gravitational case is left for future work, but could employ the same methods. To this end we use the characteristic grid evolution scheme of Price and Lousto [4, 5]. In the scalar case we implement a fourth order finite-difference scheme to calculate the retarded field. Our code can handle both circular and highly eccentric orbits around the black hole. In the electromagnetic case we only implement a second order accurate scheme to avoid the technical complexities of a fourth-order accurate code. We examine the influence of the conservative part of the self-force on the constants of motion along the orbit.