The Cauchy Problem for the Einstein Equations

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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.