“正则奇点”与近似局部惯性系中重力波的散射
"Regularity Singularities" and the Scattering of Gravity Waves in Approximate Locally Inertial Frames
摘要：
爱因斯坦-欧拉方程的解是否足够光滑，可以在激波相互作用的点上采用局部惯性坐标，或者在这些点上是否存在“正则性奇点”，这是一个悬而未决的问题。术语{ \it正则性奇点}是由作者提出的，作为时空中的一个点，在那里引力度规张量是Lipschitz连续的（$C^{0,1}$），但在$C^{1,1}$图集的任何坐标系中都不平滑。对于球对称时空中的Einstein-Euler方程，已经证明了在允许任意相互作用的$C^{0,1}$中激波解的存在性理论，但是$C^{1,1}$是时空局部平坦所必需的光滑性。因此，正则性奇点的开放问题是关于在较大的$C^{1,1}$图谱内的激波处是否存在局部惯性坐标系的问题。为了澄清这个悬而未决的问题，我们确定了新的“科里奥利型”的影响，在几何的$C^{0,1}$冲击波度量和证明他们是必不可少的意义上说，他们永远不会消失的地图集内的{ \it光滑}坐标变换，通常假设在经典微分几何的地图集。因此，存在正则性奇点的问题等价于这些科里奥利型效应是否本质上是不可移除的和“真实的”，或者仅仅是可以移除的坐标效应（类似于经典的科里奥利力）的问题，通过去$C^{1,1}$变换的不太正则的图谱。如果基本上是不可移动的，它将有力地为广义相对论提供一个“真正的”新的物理效应，为正则性奇点的开放问题提供一个物理背景。
英文摘要：
It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term {\it regularity singularity} was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system of the $C^{1,1}$ atlas. An existence theory for shock wave solutions in $C^{0,1}$ admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but $C^{1,1}$ is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger $C^{1,1}$ atlas. To clarify this open problem, we identify new "Coriolis type" effects in the geometry of $C^{0,1}$ shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of {\it smooth} coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and `real', or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of $C^{1,1}$ transformations. If essentially non-removable, it would argue strongly for a `real' new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
