A 4+1 Formalism for the Evolving Stueckelberg-Horwitz-Piron Metric

Abstract

We propose a field theory for the local metric in Stueckelberg–Horwitz–Piron (SHP) general relativity, a framework in which the evolution of classical four-dimensional (4D) worldlines xμτ (μ=0,1,2,3) is parameterized by an external time τ. Combining insights from SHP electrodynamics and the ADM formalism in general relativity, we generalize the notion of a 4D spacetime M to a formal manifold M5=M×R, representing an admixture of geometry (the diffeomorphism invariance of M) and dynamics (the system evolution of Mτ with the monotonic advance of τ∈R). Strategically breaking the formal 5D symmetry of a metric gαβ(x,τ) (α,β=0,1,2,3,5) posed on M5, we obtain ten unconstrained Einstein equations for the τ-evolution of the 4D metric γμν(x,τ) and five constraints that are to be satisfied by the initial conditions. The resulting theory differs from five-dimensional (5D) gravitation, much as SHP U(1) gauge theory differs from 5D electrodynamics.