Analytic reductions of self-force calculations in curved spacetimes

Abstract

For the case of an electric or scalar charge travelling along a timelike geodesic in a curved, background spacetime, we apply a Green's theorem argument to transform the tail contribution to the particle's self-force at a point p along its trajectory to a form which involves only an integral over the past light cone from p. One potential advantage of this reformulation is that the tail contribution to the fundamental solution for a (self-adjoint) tensor wave equation in an arbitrary spacetime is (at least within so-called causal domains) explicitly computable on the past light cone of an arbitrary point through the integration of certain linear transport equations defined along the null generators of the cone. The conventional approach to computing the self-force requires this tail field throughout the interior of the past light cone and extracts it from the particle's (numerically generated) total field through an intricate mode-by-mode numerical decomposition. By contrast, our approach requires that the particle's total field be paired with the (explicitly computable) tail field on the light cone itself and then integrated over this cone. We thus avoid the need for the aforementioned mode-by-mode decomposition of the particle's total field into direct ⊕ tail contributions. We speculate that in some circumstances it may be sufficient to truncate the particle's total field, as needed in our calculation, to its (also explicitly computable) direct Liénard Wiechert approximation.