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 xμτ (μ=0,1,2,3) is parameterized by an external time τ. Combining insights from SHP electrodynamics and the ADM formalism in general relativity, we generalize the notion of a 4D spacetime M to a formal manifold M5=M×R, representing an admixture of geometry (the diffeomorphism invariance of M) and dynamics (the system evolution of Mτ with the monotonic advance of τ∈R). Strategically breaking the formal 5D symmetry of a metric gαβ(x,τ) (α,β=0,1,2,3,5) posed on M5, we obtain ten unconstrained Einstein equations for the τ-evolution of the 4D metric γμν(x,τ) 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.
Highlights
The Arnowitt Deser Misner (ADM) formalism [1] in general relativity (GR) expresses the Einstein field equations in canonical form, permitting a solution of particular field/matter configurations formulated as initial value problems
The resulting theory differs from five-dimensional (5D) gravitation, much as SHP U(1) gauge theory differs from 5D electrodynamics
In summarizing Einstein gravity as “Spacetime tells matter how to move; matter tells spacetime how to curve,” Wheeler [10] touched on certain general issues in relativity known collectively as the problem of time In nonrelativistic mechanics, space is viewed as the “arena” of physical motion, a manifold with given background metric in some coordinate system, while time is an external parameter introduced to mark the coordinate evolution that characterizes the motion of objects in space
Summary
The Arnowitt Deser Misner (ADM) formalism [1] in general relativity (GR) expresses the Einstein field equations in canonical form, permitting a solution of particular field/matter configurations formulated as initial value problems. As a canonical Hamiltonian formulation that splits four-dimensional (4D) spacetime into three-dimensional (3D) space and a selected time direction, ADM provides insight into general features of relativity, but is not always the most convenient of the 3+1 formulations for computation, especially numerical simulation. The SHP framework is a covariant canonical approach to relativistic classical and quantum mechanics, in which 4D spacetime events are defined with respect to coordinates x μ (μ = 0, 1, 2, 3) and an external evolution parameter τ. We construct a purely formal 4+1 −→ 5D manifold as a guide to formulating field equations that under the 5D −→ 4+1 foliation describe a spacetime metric γμν ( x, τ ) evolving with τ and preserving the required spacetime symmetries
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
