论文标题
延续的爱因斯坦方程解决方案的标准
Continuation Criterion For Solutions To The Einstein Equations
论文作者
论文摘要
在恒定平均外曲率（CMC）仪表中，我们证明了3+1维真空重力的持续条件。更确切地说，我们获得了定量标准，根据这些标准，将来可以无限期地扩展物理时空，以解决给定常规初始数据的爱因斯坦方程问题的解决方案。特别是，我们表明，时空Riemann Curvature的量规不变$ H^{2} $ SOBOLEV NORM在未来的时间方向上保持界限，前提是，如果所谓的静态矢量字段的所谓变形张量正常与所选的CMC Hypersurface验证空间$ l^\ infty。为此，我们实施了一种新型技术，通过使用Friedlander的参数进行张量波方程来获得这种精致的估计，并在弯曲的时空和Moncrief随后的改进上进行张量。最后，我们通过提供对结果的物理解释以及与确定论和弱宇宙审查制度问题的关系。
We prove a continuation condition in the context of 3+1 dimensional vacuum Einstein gravity in Constant Mean extrinsic Curvature (CMC) gauge. More precisely, we obtain quantitative criteria under which the physical spacetime can be extended in the future indefinitely as a solution to the Cauchy problem of the Einstein equations given regular initial data. In particular, we show that a gauge-invariant $H^{2}$ Sobolev norm of the spacetime Riemann curvature remains bounded in the future time direction provided the so-called deformation tensor of the unit timelike vector field normal to the chosen CMC hypersurfaces verifies a spacetime $L^\infty$ bound. To this end, we implement a novel technique to obtain this refined estimate by using Friedlander's parametrix for tensor wave equations on curved spacetime and Moncrief's subsequent improvement. We conclude by providing a physical explanation of our result as well as its relation to the issues of determinism and weak cosmic censorship.
