On the geodesic hypothesis in general relativity

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 $\delta \le {\epsilon }^q$, $q>1$, where $\delta$, $\epsilon$ are the amplitude and size of the particle, we show the existence of solution $([0,T]\times {\mathbb{R}}^3,g,{\varphi }^{\epsilon })$ to the coupled Einstein equations with the property that the energy of the particle ${\varphi }^{\epsilon }$ is concentrated along a timelike geodesic. Moreover, the gravitational field produced by ${\varphi }^{\epsilon }$ 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.

55 pages

References in corpus (3)

• On the quasilinear wave equations in time dependent inhomogeneous media
• Effective dynamics for solitons in the nonlinear Klein Gordon Maxwell system and the Lorentz force law
• Global solutions of nonlinear wave equations in time dependent inhomogeneous media