Self-Force on a Scalar Charge in Circular Orbit around a Schwarzschild Black Hole

TL;DR

This paper analytically calculates the scalar self-force on a particle in circular orbit around a Schwarzschild black hole up to the third post-Newtonian order using a new regularization method, revealing component-specific order appearances.

Contribution

It introduces and applies the power expansion regularization method to evaluate the scalar self-force at 3PN order in a Schwarzschild background.

Findings

The r-component of the self-force appears at 3PN order.

The t- and φ-components due to radiation reaction appear at 2PN and 1.5PN orders.

The method provides a systematic way to compute self-force components in curved spacetime.

Abstract

In an accompanying paper, we have formulated two types of regulariz_ation methods to calculate the scalar self-force on a particle of charge $q$ moving around a black hole of mass $M$, one of which is called the ``power expansion regularization''. In this paper, we analytically evaluate the self-force (which we also call the reaction force) to the third post-Newtonian (3PN) order on the scalar particle in circular orbit around a Schwarzschild black hole by using the power expansion regularization. It is found that the $r$-component of the self-force arises at the 3PN order, whereas the $t$- and $\phi$-components, which are due to the radiation reaction, appear at the 2PN and 1.5PN orders, respectively.