AI 中文总结
该研究建立弗雷歇流形上的变分演算，将其应用于爱因斯坦演化方程的拉格朗日形式化，推导约束传播恒等式，证明相关定理并揭示受约束爱因斯坦演化的测地线性质。
AI 中文摘要
我们针对弗雷歇流形上的曲线建立变分演算，并将其应用于紧致流形的黎曼度量流形$\boldsymbol{\text{Mm}}$上爱因斯坦演化方程的拉格朗日形式化。对于与弱伪黎曼度量相关的力学拉格朗日量，我们建立了欧拉-拉格朗日方程、能量守恒以及无穷小诺特定理，该定理不要求对称向量场生成流。将此框架应用于DeWitt拉格朗日量，可将动量约束识别为诺特动量映射的消失，并直接证明其传播。我们还推导了逐点传输恒等式$\boldsymbol{\bigl(\text{Ham}_\boldsymbol{\boldsymbol{\text{λ}}}\boldsymbol{\boldsymbol{\text{dv}}}_\boldsymbol{\boldsymbol{g}}\boldsymbol{\bigr)}} = \boldsymbol{2α}\boldsymbol{\boldsymbol{\text{δ}}}_\boldsymbol{\boldsymbol{g}}\boldsymbol{\boldsymbol{\text{δ}}}_\boldsymbol{\boldsymbol{g}}^\boldsymbol{-}\boldsymbol{k}\boldsymbol{\boldsymbol{\text{dv}}}_\boldsymbol{\boldsymbol{g}}$，该恒等式蕴含哈密顿约束的传播。最后，我们证明了弗雷歇流形上弱伪黎曼度量的莫佩尔蒂-雅可比定理，并表明在总标量曲率势的零集之外，受约束的爱因斯坦演化是共形DeWitt度量$\boldsymbol{-4α S_\boldsymbol{\boldsymbol{λ}} G^\boldsymbol{-}}$的重新参数化单位速测地线。
英文摘要
We develop a variational calculus for curves on Fréchet manifolds and apply it to the Lagrangian formulation of the Einstein evolution equations on the manifold $\Mm$ of Riemannian metrics of a compact manifold. For mechanical Lagrangians associated with weak pseudo-Riemannian metrics, we establish the Euler--Lagrange equations, energy conservation, and an infinitesimal Noether theorem that does not require the symmetry vector field to generate a flow. Applied to the DeWitt Lagrangian, this framework identifies the momentum constraint with the vanishing of a Noether momentum map and gives a direct proof of its propagation. We also derive the pointwise transport identity \[ \partial_t\bigl(\Ham_λ\,\dv_g\bigr) = 2α\,δ_g(δ_g^-k)\,\dv_g\,, \] which implies propagation of the Hamiltonian constraint. Finally, we prove a Maupertuis-- Jacobi theorem for weak pseudo-Riemannian metrics on Fréchet manifolds and show that, away from the zero set of the total scalar-curvature potential, constrained Einstein evolutions are reparametrized unit-speed geodesics of the conformal DeWitt metric $-4αS_λG^-$.
Comments24 pages in amsart with 1in margins
