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.

中文翻译：

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进化中的 Stueckelberg-Horwitz-Piron 度量的 4+1 形式主义

我们提出了 Stueckelberg-Horwitz-Piron (SHP) 广义相对论中局部度量的场论，该框架中经典四维 (4D) 世界线 xμτ (μ=0,1,2,3) 的演化被参数化由外部时间τ。结合 SHP 电动力学和广义相对论中的 ADM 形式主义的见解，我们将 4D 时空 M 的概念推广到形式流形 M5=M×R，代表几何（M 的微分同胚不变性）和动力学（系统演化）的混合Mτ 与 τ∈R 的单调推进）。策略性地打破了在 M5 上提出的度量 gαβ(x,τ) (α,β=0,1,2,3,5) 的形式 5D 对称性，我们获得了 10 个无约束爱因斯坦方程，用于 4D 度量 γμν 的 τ 演化(x,τ) 和初始条件要满足的五个约束条件。