Geometric optics approximation for the Einstein vacuum equations

Arthur Touati

doi:10.5802/slsedp.159

Research article

Geometric optics approximation for the Einstein vacuum equations

Arthur Touati ¹

¹ Institut des Hautes Études Scientifiques 91440 Bures-sur-Yvette, France

Séminaire Laurent Schwartz — EDP et applications (2022-2023), Talk no. 7, 13 p.

Abstract

We present recent works on the construction of high-frequency solutions to the Einstein vacuum equations in general relativity, based on [Tou22b, Tou23]. In these articles, the author shows the existence of a family of vacuum spacetimes ${(g_{λ})}_{λ \in (0, 1]}$ in generalised wave gauge oscillating at frequency $λ^{- 1}$ and defined by a geometric optics ansatz. In this survey, we first review the Cauchy theory for the Einstein vacuum equations in wave gauge, geometric optics and Burnett’s conjecture in general relativity. This will motivate our main result, for which we provide a sketch of proof high-lighting both quasi-linear and semi-linear challenges.

Published online: 2023-07-12

DOI: 10.5802/slsedp.159

Author's affiliations:

Arthur Touati ¹

¹ Institut des Hautes Études Scientifiques 91440 Bures-sur-Yvette, France

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     author = {Arthur Touati},
     title = {Geometric optics approximation for the {Einstein} vacuum equations},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:7},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2022-2023},
     doi = {10.5802/slsedp.159},
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     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.159/}
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Arthur Touati. Geometric optics approximation for the Einstein vacuum equations. Séminaire Laurent Schwartz — EDP et applications (2022-2023), Talk no. 7, 13 p. doi : 10.5802/slsedp.159. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.159/

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