On “hyperboloidal” Cauchy data for vacuum einstein equations and obstructions to smoothness of Scri
On “hyperboloidal” Cauchy data for vacuum einstein equations and obstructions to smoothness of Scri
复制标题
真空爱因斯坦方程的“双曲面”柯西数据和 Scri 平滑性的障碍
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
P. Chruściel
中科院分区：
文献类型：
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作者：
L. Andersson;P. Chruściel
详细分析了真空爱因斯坦方程的“双曲面”柯西数据在初始数据表面共形边界处的几何性质与时空几何之间的关系。我们证明，存在平滑或多齐性 Scri（即，度量可根据 ofr−j logir 而不是根据 ofr−j 进行扩展的 Scri）的必要条件是初始数据表面的共形边界的剪切力消失。我们推导了“边界约束”，为了与弗里德里希的共形框架兼容，初始数据集必须满足该约束。我们表明，存在平滑 Scri（不一定是完整的）的充分条件是初始数据表面的共形边界的剪切力和共形重新缩放的初始数据的边界的平滑度消失。我们还表明，在约束方程解的共角边界处渐近展开中一些对数项的出现与共角边界处 Weyl 张量的不消失有关。
The relationship between the geometric properties of “hyperboloidal” Cauchy data for vacuum Einstein equations at the conformal boundary of the initial data surface and between the space-time geometry is analyzed in detail. We prove that a necessary condition for existence of a smooth or a polyhomogeneous Scri (i.e., a Scri around which the metric is expandable in terms ofr−j logir rather than in terms ofr−j) is the vanishing of the shear of the conformal boundary of the initial data surface. We derive the “boundary constraints” which have to be satisfied by an initial data set for compatibility with Friedrich's conformal framework. We show that a sufficient condition for existence of a smooth Scri (not necessarily complete) is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data. We also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non-vanishing of the Weyl tensor at the conformal boundary.