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We propose a field theory for the local metric inStueckelberg--Horwitz--Piron (SHP) general relativity, a framework in which theevolution 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 generalrelativity, we generalize the notion of a 4D spacetime $\mathcal{M}$ to aformal manifold $\mathcal{M}_5 = \mathcal{M} \times R$, representing anadmixture of geometry (the diffeomorphism invariance of $\mathcal{M}$) anddynamics (the system evolution of $\mathcal{M} \left( \tau \right)$ with themonotonic advance of $\tau \in R$). Strategically breaking the formal 5Dsymmetry 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 forthe $\tau$-evolution of the 4D metric $\gamma_{\mu\nu}(x,\tau)$ and fiveconstraints that are to be satisfied by the initial conditions. The resultingtheory differs from five-dimensional (5D) gravitation, much as SHP U(1) gaugetheory differs from 5D electrodynamics.

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