The two-body problem: analytical results in a toy-model of relativistic gravity

TL;DR

This paper analyzes the two-body problem in Nordström's scalar gravity theory, deriving exact equations of motion, revealing the absence of an innermost circular orbit in the exact theory, and developing an iterative solution method that captures non-perturbative effects.

Contribution

The paper provides exact solutions and an iterative non-PN approach to the two-body problem in a relativistic gravity toy-model, highlighting differences from post-Newtonian approximations.

Findings

No innermost circular orbit in the exact theory.

Iterative solution converges rapidly and includes non-perturbative effects.

First iteration yields 4.5 PN radiation reaction.

Abstract

The two body problem in a scalar theory of gravity is investigated. We focus on the closest theory to General Relativity (GR), namely Nordstr\"om's theory of gravity (1913). The gravitational field can be exactly solved for any configuration of point-particles. We then derive the exact equations of motion of two inspiraling bodies including the exact self-forces terms. We prove that there is no innermost circular orbit (ICO) in the exact theory whereas we find (order-dependent) ICOs if post-Newtonian (PN) truncations are used. We construct a solution of the two body problem in an iterative (non-PN) way, which can be viewed as a series in powers of $(v/c)^{5}$. Besides this rapid convergence, each order also provides non-perturbative information. Starting from a circular Newtonian-like orbit, the first iteration already yields the 4.5 PN radiation reaction. These results not only shed light on some non-perturbative effects of relativistic gravity, but may also be useful to test numerical codes.