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.