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General Relativity and the Einstein Equations$
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Yvonne Choquet-Bruhat

Print publication date: 2008

Print ISBN-13: 9780199230723

Published to Oxford Scholarship Online: May 2009

DOI: 10.1093/acprof:oso/9780199230723.001.0001

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Global Existence Theorems: Asymptotically Euclidean Data

Global Existence Theorems: Asymptotically Euclidean Data

(p.482) XV Global Existence Theorems: Asymptotically Euclidean Data
General Relativity and the Einstein Equations

Yvonne Choquet-Bruhat

Oxford University Press

This chapter shows how the Penrose transform can be used to prove global existence of solutions of various semilinear field equations. It outlines the foundation points of Friedrich's conformal system, and explains how a conformal transformation of a future causal cone in Minkowski spacetime of dimension greater than or equal to six on to another such light cone gives a global existence theorem of solutions of the vacuum Einstein equations with small data which are Schwarzschild outside of a compact set. The chapter indicates some of the arguments of the book Non-Linear Stability of Minkowski Space, and states some further properties proved in another book by Nicolo and Klainerman. Finally, it sketches the main steps of the proof by Lindblad and Rodnianski of the global existence in wave coordinates, for small initial data.

Keywords:   Penrose transformation, wave equations, Friedrich conformal system, Einstein equations, Christodoulou–Klainerman theorem, Klainerman–Nicolo theorem, Linblad– Rodnianski theorem

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