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

Abstract

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.

49 pages, 0 figures

References in corpus (2)

• 3+1 Formalism and Bases of Numerical Relativity
• Local Metric with Parameterized Evolution

Cited by in corpus (2)

• Cosmic-time quantum mechanics and the passage-of-time problem
• An Evolving Spacetime Metric Induced by a `Static' Source