On the Local Existence for the Characteristic Initial Value Problem in General Relativity
On the Local Existence for the Characteristic Initial Value Problem in General Relativity
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DOI：
10.1093/imrn/rnr201
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发表时间：
2011-07
期刊：
影响因子：
--
通讯作者：
J. Luk
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文献类型：
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作者：
J. Luk
给定一个被截断的入射零锥和一个被截断的出射零锥相交于具有光滑特征初值的两个球面$S$，Rendall的一个定理表明，真空爱因斯坦方程在$S$的未来可以在$S$的一个小邻域内求解。我们证明了真空爱因斯坦方程实际上可以在未来的锥的邻域内求解，只要约束方程在零锥上初始满足即可。这一证明是基于能量类型的估计，并在很大程度上依赖于双零面理中爱因斯坦方程的零结构。
Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.