Local foliations and optimal regularity of Einstein spacetimes

JOURNAL OF GEOMETRY AND PHYSICS

卷:59

Article

Chen, Bing-Long³  LeFloch, Philippe G.^1,2 

[1] Univ Paris 06, Lab Jacques Louis Lions, F-75252 Paris, France

[2] Univ Paris 06, CNRS, F-75252 Paris, France

[3] Sun Yat Sen Univ, Dept Math, Guangzhou 510275, Guangdong, Peoples R China

关键词: Lorentzian geometry;    General relativity;    Einstein field equations;    Mean curvature foliation;    Harmonic coordinates;    Optimal regularity;   

DOI  :  10.1016/j.geomphys.2009.04.002

来源: Elsevier

【 摘 要 】

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CIVIC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CIVIC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations. (C) 2009 Elsevier B.V. All rights reserved.

【 授权许可】

Free