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

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 $x^\mu \left( \tau \right)$ ($\mu = 0,1,2,3$) is parameterized by an external time $\tau$. Combining insights from SHP electrodynamics and the ADM formalism in general relativity, we generalize the notion of a 4D spacetime $\mathcal{M}$ to a formal manifold $\mathcal{M}_5 = \mathcal{M} \times R$, representing an admixture of geometry (the diffeomorphism invariance of $\mathcal{M}$) and dynamics (the system evolution of $\mathcal{M} \left( \tau \right)$ with the monotonic advance of $\tau \in R$). Strategically breaking the formal 5D symmetry of a metric $g_{\alpha\beta}(x,\tau)$ ($\alpha,\beta = 0,1,2,3,5$) posed on $\mathcal{M}_5$, we obtain ten unconstrained Einstein equations for the $\tau$-evolution of the 4D metric $\gamma_{\mu\nu}(x,\tau)$ 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.