演化Stueckelberg-Horwitz-Piron度规的4+1形式
A 4+1 Formalism for the Evolving Stueckelberg-Horwitz-Piron Metric
摘要：
我们提出了一个局部度规的场理论在Stueckelberg-Horwitz-Piron（SHP）广义相对论，一个框架中，经典的四维（4D）世界线$x^μ\left( τ\right)$（$μ= 0,1,2,3 $）的演变参数化的外部时间$τ$。结合SHP电动力学和广义相对论中的ADM形式主义的见解，我们将4D时空$\mathcal{M}$的概念推广到形式流形$\mathcal{M}_5 = \mathcal{M} \times R$，表示几何（$\mathcal{M}$的同胚不变性）和动力学（$\mathcal{M} \left( τ\right)$的系统演化与$τ\in R$的单调推进）的混合。策略性地打破了在$\mathcal{M}_5$上提出的度量$g_{αβ}(x,τ)$（$α,β= 0,1,2,3,5 $）的形式5D对称性，我们得到了10个用于$τ$ -4D度量$γ_{μν}(x,τ)$演化的无约束爱因斯坦方程和5个初始条件所要满足的约束。由此产生的理论不同于五维（5D）引力，就像SHP U（1）规范理论不同于五维电动力学一样。
英文摘要：
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 $x^μ\left( τ\right)$ ($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 $\mathcal{M}$ 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 $\mathcal{M} \left( τ\right)$ with the monotonic advance of $τ\in R$). Strategically breaking the formal 5D symmetry of a metric $g_{αβ}(x,τ)$ ($α,β= 0,1,2,3,5 $) posed on $\mathcal{M}_5$, we obtain ten unconstrained Einstein equations for the $τ$-evolution of the 4D metric $γ_{μν}(x,τ)$ 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.
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