Analytical calculation of self-force effects on a scalar particle in an eccentric orbit around a Schwarzschild black hole

TL;DR

This paper provides an analytical calculation of the scalar self-force effects on a particle in a slightly eccentric orbit around a Schwarzschild black hole, including explicit expressions up to 6PN order and $e^4$ eccentricity.

Contribution

It introduces a novel analytical method combining post-Newtonian expansion and small-eccentricity approximation to compute self-force effects in curved spacetime.

Findings

Derived explicit expressions for self-force components at 6PN and $e^4$ order.

Validated results by comparing fluxes with scalar-tensor theory calculations.

Found perfect agreement with previous scalar-tensor flux results after fixing scalar field value.

Abstract

In this work, we analytically investigate the effects of the scalar self-force exerted by a massless scalar field on a particle in a slightly eccentric orbit around a Schwarzschild black hole. By solving the Klein-Gordon equation in the curved spacetime background, using a combination of post-Newtonian (PN) expansion, and small-eccentricity approximation, we derive explicit expressions for the self-force components at the particle location, as well as for the associated energy and angular momentum fluxes. Our results are valid up to sixth post-Newtonian (6PN) order and fourth order in eccentricity ($e^4$). We compare asymptotic fluxes with those obtained in arXiv:2401.06844 for scalar-tensor (ST) theories. Once the relation between the two approaches has been established, we find perfect agreement by fixing the asymptotic value of the scalar field in ST theory $\phi_0 = 1$.