论文标题
一般相对论中通用壳的变分形式主义
Variational formalism for generic shells in general relativity
论文作者
论文摘要
我们在存在完全无约束的签名（通用壳）的薄壳的情况下研究重力场的变异原理。在分别针对时机和无效签名的外壳之前，已经给出了这种变分的配方，但到目前为止尚无统一治疗。我们将壳方程识别为沿着超出表面损坏的极端问题的自然边界条件，在该方面允许度量张量不可分割。由于爱因斯坦 - 希尔伯特作用的二阶性质使得与变异配方不确定的边界值问题需要引入正则化方案。我们研究了几种此类正规化方案，并证明了它们的等效性。我们表明，从该变异过程中得出的统一壳方程将通过小号和以色列通过分布理论获得的过去结果，用于固定因果类型的高空以及MARS和SENOVILLA的通用外壳。预计这些结果将提供有用的指南，以在修改后的重力理论中沿通用性高空的薄壳方程和连接条件制定。
We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equations derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.
