论文标题
一般相对论中的高频极限和无尘埃壳解决方案
High-frequency limits and null dust shell solutions in general relativity
论文作者
论文摘要
考虑没有任何对称假设的爱因斯坦真空方程的特征初始值问题。在两个相交的无效Hypersurfaces上施加一系列数据，每个数据都由Spacelike $ 2 $ -SPHERES散发。假设数据序列是使沿空方向的指标的衍生物仅在$ l^2 $中均匀地界限，但是指标的衍生物沿着切向$ 2 $ -spheres的方向均匀地遵守较高的规律性均匀地界限。通过[J. Luk和I. Rodnianski，真空爱因斯坦方程的脉冲引力波的非线性相互作用，Camb。 J. Math。 5（4），2017年]，以下是特征初始值问题的顺序产生了一系列真空飞机$（\ Mathcal m，g_n）$中的一系列$ \ MATHCAL M $。由于存在定理仅需要非常低的规律性，因此溶液的序列可能表现出振荡和浓度，并且极限不必真空。尽管如此，我们证明，在传递到子序列后，指标以$ c^0 $收敛，而在$ w^{1,2} $中微弱地收敛到Einstein-Null Dust System的解决方案，其中有两个（潜在的测量值）null Dust。 此外，我们还表明，所有足够的定期解决方案对爱因斯坦无效的灰尘系统（具有潜在测量值的无效灰尘）适应了双重零坐标系，作为当地出现的，作为对爱因斯坦真空系统的弱限制，以上述方式。结果，我们还为Einstein-Null粉尘系统的解决方案提供了第一个一般的局部存在和独特性，无效的灰尘仅是措施。特别是作为一种特殊案例解决方案，具有繁殖和相互作用的无效灰尘壳。
Consider the characteristic initial value problem for the Einstein vacuum equations without any symmetry assumptions. Impose a sequence of data on two intersecting null hypersurfaces, each of which is foliated by spacelike $2$-spheres. Assume that the sequence of data is such that the derivatives of the metrics along null directions are only uniformly bounded in $L^2$ but the derivatives of the metrics along the directions tangential to the $2$-spheres obey higher regularity bounds uniformly. By the results in [J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math. 5(4), 2017], it follows that the sequence of characteristic initial value problems gives rise to a sequence of vacuum spacetimes $(\mathcal M, g_n)$ in a fixed double-null domain $\mathcal M$. Since the existence theorem requires only very low regularity, the sequence of solutions may exhibit both oscillations and concentrations, and the limit need not be vacuum. We prove nonetheless that, after passing to a subsequence, the metrics converge in $C^0$ and weakly in $W^{1,2}$ to a solution of the Einstein-null dust system with two families of (potentially measure-valued) null dust. We show moreover that all sufficiently regular solutions to the Einstein-null dust system (with potentially measure-valued null dust) adapted to a double null coordinate system arise locally as weak limits of solutions to the Einstein vacuum system in the manner described above. As a consequence, we also give the first general local existence and uniqueness result for solutions to the Einstein-null dust system for which the null dusts are only measures. This in particular includes as a special case solutions featuring propagating and interacting shells of null dust.
