A new analytical method for self-force regularization. I - Charged scalar particles in schwarzschild spacetime

We formulate a new analytical method for regularizing the self-force acting on a particle of small mass μ orbiting a black hole of mass M, where μ ≪ M. At first order in μ, the geometry is perturbed and the motion of the particle is affected by its self-force. The self-force, however, diverges at the location of the particle, and hence should be regularized. It is known that a properly regularized self-force is given by the tail part (or the R-part) of the self-field, obtained by subtracting the direct part (or the S-part) from the full self-field. The most successful method of regularization proposed so far relies on the spherical harmonic decomposition of the self-force, the so-called mode-sum regularization or mode decomposition regularization. However, except for some special orbits, no systematic analytical method for computing the regularized self-force has been constructed. In this paper, utilizing a new decomposition of the retarded Green function in the frequency domain, we formulate a systematic method for the computation of the self-force in the time domain. Our method relies on the post-Newtonian (PN) expansion, but the order of the expansion can be arbitrarily high. To demonstrate the essence of our method, in this paper, we focus on a scalar charged particle on the Schwarzschild background. Generalization to the gravitational case is straightforward, except for some subtle issues related with the choice of gauge (which exists irrespective of regularization methods).