广义相对论Shock Wave相互作用点是时空非局部平坦的“正则奇点”
Points of General Relativisitic Shock Wave Interaction are "Regularity Singularities" where Spacetime is Not Locally Flat
摘要：
我们证明了在广义相对论中，在激波相互作用点附近的$C^{1,1}$坐标变换类中，球对称时空中引力度规张量的正则性不能从$C^{0,1}$提升到$C^{1,1}$，而不迫使度规张量的行列式在相互作用点处为零。这与Israel定理相反，Israel定理指出这样的坐标变换总是存在于光滑的单激波表面上的点的邻域中。因此，这些结果意味着，激波相互作用点代表了时空中演化的完美流体的一种新的奇异性，这种奇异性在物理上是完全有意义的，可以从光滑初始数据的演化中形成，但在任何坐标变换下，时空都不是局部闵可夫斯基的。特别是，在这样的奇点，δ函数源的引力度量张量的二阶导数存在于所有坐标系的$C^{1,1}$图集，但由于取消，曲率张量保持一致有界。
英文摘要：
We show that the regularity of the gravitational metric tensor in spherically symmetric spacetimes cannot be lifted from $C^{0,1}$ to $C^{1,1}$ within the class of $C^{1,1}$ coordinate transformations in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel's Theorem which states that such coordinate transformations always exist in a neighborhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of singularity for perfect fluids evolving in spacetime, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the spacetime is not locally Minkowskian under any coordinate transformation. In particular, at such singularities, delta function sources in the second derivatives of the gravitational metric tensor exist in all coordinate systems of the $C^{1,1}$ atlas, but due to cancelation, the curvature tensor remains uniformly bounded.
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