On the Local Existence for the Characteristic Initial Value Problem in General Relativity

Details 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

2011-07-05

arxiv.org/abs/1107.0898v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Scientific paper

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.

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