The Newtonian limit of spacetimes describing uniformly accelerated particles

TL;DR

This paper investigates the conditions under which boost-rotation symmetric spacetimes have a Newtonian limit, emphasizing the importance of asymptotic flatness and illustrating results with specific examples.

Contribution

It establishes criteria for the existence of a Newtonian limit in boost-rotation symmetric spacetimes and clarifies the role of asymptotic flatness in this context.

Findings

Asymptotic flatness is essential for the Newtonian limit.

The Newtonian limit exists only at time symmetry and is given by a Poisson integral.

Generalized boost-rotation spacetimes describing particles in uniform fields lack a Newtonian limit.

Abstract

We discuss the Newtonian limit of boost-rotation symmetric spacetimes by means of the Ehler's frame theory. Conditions for the existence of such a limit are given and, in particular, we show that asymptotic flatness is an essential requirement for the existence of such a limit. Consequently, generalized boost-rotation symmetric spacetimes describing particles moving in uniform fields will not possess a Newtonian limit. In the cases where the boost-rotation symmetric spacetime is asymptotically flat and its Newtonian limit exists, then it is non-zero only for the instant of time symmetry and its value is given by a Poisson integral. The relation of this result with the (Newtonian) gravitational potential suggested by the weak field approximation is discussed. We illustrate our analysis through some examples: the two monopoles solution, the Curzon-Chazy particle solution, the generalized Bonnor-Swaminarayan solution, and the C metric.