The Cauchy Problem for the Einstein Equations

摘要奖：DMS-0407732主要研究者：Lars Anderson该提案旨在研究爱因斯坦方程的柯西问题。将研究几个方面，包括宇宙时空的非线性稳定性，以及具有一个空间Killing场的各种版本的爱因斯坦时空方程的非线性稳定性。将进行时空奇点附近渐近行为的研究，特别关注BKL建议和渐近沉默的各个方面。这里将使用数值和分析方法相结合. 将研究爱因斯坦方程的椭圆规范公式。爱因斯坦方程描述了时空的几何形状，而时空的几何形状又决定了空间中物体（行星、恒星、星系）的运动，以及通过各自的场论确定的场和粒子的动力学。爱因斯坦方程所提供的宇宙几何观，导致了我们对宇宙的描述中一些最基本和最矛盾的特征，包括初始时空奇点或大爆炸和黑洞。对极端条件下的时空的数学研究，如大爆炸奇点附近或黑洞奇点附近，导致了一个遥远的猜想，称为“宇宙审查”，这意味着违反因果关系直觉概念的现象必须对观察者隐藏（审查）。在这个项目中，与宇宙审查有关的问题将使用基于计算机的技术和分析技术相结合来研究，特别关注几何从初始条件的演化。

AbstractAward: DMS-0407732Principal Investigator: Lars AnderssonThis proposal aims to investigate the Cauchy problem for theEinstein equations. Several aspects will be studied, includingnonlinear stability of cosmological spacetimes, as well as thenonlinear stability of various versions of the Einstein equationsfor spacetimes with one spatial Killing field. Studies of theasymptotic behavior near spacetime singularities will be carriedout, with particular attention to aspects of the BKL proposal andasymptotic silence. Here a combination of numerical andanalytical methods will be used. Elliptic gauge formulations forthe Einstein equations will be investigated. These are ofinterest for large scale numerical computations, as well as forthe Cauchy problem for rough initial data.The Einstein equations describe the geometry of spacetime, whichin turn determines the motion of bodies (planets, stars,galaxies) in space, as well as the dynamics of fields andparticles via their respective field theories. The geometric viewof the universe provided by the Einstein equations has led tosome of the most fundamental and paradoxical features of ourpicture of the universe, including the initial spacetimesingularity or Big Bang and black holes. The mathematical studyof spacetimes in extreme conditions such as near the Big Bangsingularity or near the singularity in black holes has led to afar reaching conjecture called ``Cosmic Censorship'', whichimplies that phenomena which violate the intuitive notions ofcausality must be hidden (censored) from observers. In thisproject, problems related to cosmic censorship are studied usinga combination of computer based and analytical techniques, with aparticular focus on the evolution of geometry from initialconditions.