4+1 Gravitation in the SHP Formalism

The Stueckelberg-Horwitz-Piron (SHP) formalism describes particles and fields traced out as spacetime events functionally dependent on an external evolution parameter τ. This approach addresses a number of difficulties associated with the problem of time. In SHP general relativity, the state of the unconstrained phase space variables {xμ(τ),pν(τ)} specifies a 4D block spacetime M(τ) that evolves to an infinitesimally close 4D block spacetime M(τ+δτ) under a scalar Hamiltonian. As the configuration of matter and energy evolves with τ it induces changes in the spacetime metric γμν(x,τ), leading to τ-dependent geodesic equations for the phase space variables. The 4+1 approach in gravitation generalizes the 3+1 formalism of Arnowitt, Deser, and Misner (ADM) to construct τ-dependent Einstein field equations, a canonical Hamiltonian formalism, and an initial value problem for γμν(x,τ). To conform to known gravitational phenomenology, we must respect the 5D symmetries associated with the free fields-the geometrical constructs relevant to M(τ) as an embedded hypersurface-and the O(3,1) symmetries of 4D matter. The 4+1 formalism has been discussed in a series of publications. The goal of this paper is to provide a systematic review of the subject, make a few corrections and some significant additions, and present the theory in a concise and orderly fashion.