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 small material bodies ${\phi^\epsilon}$ governed by the nonlinear Klein–Gordon equations to approximate the test particle. Given a vacuum spacetime ${([0, T]\times\mathbb{R}^3, h)}$ , we consider the initial value problem for the Einstein-scalar field system. For all sufficiently small ε and δ ≤ ε^q, q > 1, where δ, ε are the amplitude and size of the particle, we show the existence of the solution ${([0, T]\times\mathbb{R}^3, g, \phi^\epsilon)}$ to the Einstein-scalar field system 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 the given vacuum metric h. These results generalize those obtained by Stuart in (Ann Sci École Norm Sup (4) 37(2):312–362, 2004, J Math Pures Appl (9) 83(5):541–587, 2004).