Helically Symmetric $N$-Particle Solutions in Scalar Gravity

Within a scalar model theory of gravity, where the interaction between particles is given by the half-retarded plus half-advanced solution of the scalar wave equation, we consider an $N$-body problem: We investigate configurations of $N$ particles which form an equilateral $N$ angle and are in helical motion about their common center. We prove that there exists a unique equilibrium configuration and compute the equilibrium radius explicitly in a post-Newtonian expansion.

Figure 1

$N$ point particles of equal mass $m$ in uniform circular motion—the mutual interaction is given by the symmetric potential (half-retarded plus half-advanced potential). We prove that there exists a unique radius ${\overline{r}}_e$ such that the configuration is in equilibrium.