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

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 $φ_0 = 1$.

Figures (3)

• Figure 1: Regularized field $\psi_R(y, e; \chi)$ evaluated at $\chi = 0, \pi , 2\pi$, setting $M=1$.
• Figure 2: Here, we show the $\phi$-component of the SF. We selected the same value for both $\chi$ and $e$. The behaviour is rather similar to the one presented by the regularized field.
• Figure 3: We show here the energy and angular momentum flux in the first row, while in the second we show the difference with respect to the circular case. The fluxes do not differ significantly for the different values of eccentricity we chose.