A Spacetime Metric for the 4 + 1 Formalism

In his foundational work on classical and quantum electrodynamics, Stueckelberg introduced an external evolution parameter, τ, in order to overcome difficulties associated with the problem of time in relativity. Stueckelberg particle trajectories are described by the evolution of spacetime events under the monotonic advance of τ, the basis for the Feynman–Stueckelberg interpretation of particle–antiparticle interactions. An event is a solution to τ-parameterized equations of motion, which, under simple conditions, including the elimination of pair processes, can be reparameterized by the proper time of motion. The 4+1 formalism in general relativity (GR) extends this framework to provide field equations for a τ-dependent local metric γμν(x,τ) induced by these Stueckelberg trajectories, leading to τ-parameterized geodesic equations in an evolving spacetime. As in standard GR, the linearized theory for weak fields leads to a wave equation for the local metric induced by a given matter source. While previous attempts to solve the wave equation have produced a metric with the expected features, the resulting geodesic equations for a test particle lead to unreasonable trajectories. In this paper, we discuss the difficulties associated with the wave equation and set up the more general ADM-like 4+1 evolution equations, providing an initial value problem for the metric induced by a given source. As in the familiar 3+1 formalism, the metric can be found as a perturbation to an exact solution for the metric induced by a known source. Here, we propose a metric, ansatz, with certain expected properties; obtain the source that induces this metric; and use them as the initial conditions in an initial value problem for a general metric posed as a perturbation to the ansatz. We show that the ansatz metric, its associated source, and the geodesic equations for a test particle behave as required for such a model, recovering Newtonian gravitation in the nonrelativistic limit. We then pose the initial value problem to obtain more general solutions as perturbations of the ansatz.