On the geodesic hypothesis in general relativity

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 $ϵ$ and $\delta \le {ϵ}^q$, $q>1$, where $\delta$, $ϵ$ 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 ${\varphi }^{ϵ}$ is concentrated along a timelike geodesic. Moreover, the gravitational field produced by ${\varphi }^{ϵ}$ 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.