A new approach to the evolving 4+1 spacetime metric

Abstract

We recently proposed [1, 2] field equations that prescribe a metric gαβ (x, τ) that is local in the spacetime coordinates x and evolves with the external “worldtime” τ of the Stueckelberg Horwitz Piron (SHP) framework. As in SHP electrodynamics, these field equations exhibit a formal 5D symmetry (α,β = 0, 1, 2, 3, 5), that is strategically broken to 4+1 representations of the Lorentz group. The resulting canonical formalism for this metric embodies a natural foliation of a 5D pseudo-manifold (encompassing both geometry and dynamics) into the τ-parameterized 4D spacetime posed in SHP theory. In this paper, we consider the linearized equations for weak gravitation in this 4+1 formalism, leading to a more straightforward and intuitive derivation of the coupled first-order evolution equations for the metric.