On the geodesic hypothesis in general relativity

TL;DR

This paper rigorously derives Einstein's geodesic hypothesis in general relativity by approximating test particles with stable solitons in nonlinear wave equations, showing particles follow geodesics in a coupled Einstein-scalar field system.

Contribution

It provides a rigorous mathematical derivation of the geodesic hypothesis using soliton approximations in Einstein's equations, extending previous results by D. Stuart.

Findings

Energy of the scalar field concentrates along a timelike geodesic.

The gravitational field of the particle is negligibly small in $C^1$ norm.

Existence of solutions with particles following geodesics under small amplitude and size conditions.

Abstract

In this paper, we give a rigorous derivation of Einstein's geodesic hypothesis in general relativity. We use scaling stable solitons for nonlinear wave equations to approximate the test particle. Given a vacuum spacetime $([0, T]\times\mathbb{R}^3, h)$, we consider the scalar field coupled Einstein equations. For all sufficiently small $\epsilon$ and $\delta\leq \epsilon^q$, $q>1$, where $\delta$, $\epsilon$ are the amplitude and size of the particle, we show the existence of solution $([0, T]\times\mathbb{R}^3, g, \phi^\epsilon)$ to the coupled Einstein equations with the property that the energy of the particle $\phi^\epsilon$ is concentrated along a timelike geodesic. Moreover, the gravitational field produced by $\phi^\epsilon$ is negligibly small in $C^1$, that is, the spacetime metric $g$ is $C^1$ close to $h$. These results generalize those obtained by D. Stuart.