A new analytical method for self-force regularization I. scalar charged particle in Schwarzschild spacetime

TL;DR

This paper introduces a new analytical method using frequency domain Green function decomposition and post-Newtonian expansion to systematically compute the regularized self-force on a scalar charged particle in Schwarzschild spacetime, improving upon previous methods.

Contribution

It presents a novel analytical approach for self-force regularization that is systematic and can be extended to higher orders, applicable to scalar particles in Schwarzschild spacetime.

Findings

Developed a frequency domain Green function decomposition method.

Demonstrated the method for a scalar charged particle in Schwarzschild spacetime.

The approach allows high-order post-Newtonian expansions for self-force calculation.

Abstract

We formulate a new analytical method for regularizing the self-force acting on a particle of small mass $\mu$ orbiting a black hole of mass $M$, where $\mu\ll M$. At first order in $\mu$, 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 the 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 given. 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. 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. The generalization to the gravitational case is straightforward, except for some subtle issues related with the choice of gauge (which exists irrespective of regularization methods).