Geometric Optics Approximation for the Einstein Vacuum Equations

We show the stability of the geometric optics approximation in general relativity by constructing a family ( g λ ) λ ∈ ( 0 , 1 ] of high-frequency metrics solutions to the Einstein vacuum equations in 3 + 1 dimensions without any symmetry assumptions. In the limit λ → 0 this family approaches a fixe...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-09, Vol.402 (3), p.3109-3200
Main Author: Touati, Arthur
Format: Article
Language:English
Subjects:
Approximation
Classical and Quantum Gravitation
Complex Systems
Geometrical optics
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity
Relativity Theory
Tensors
Theoretical
Wave equations
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Summary:We show the stability of the geometric optics approximation in general relativity by constructing a family ( g λ ) λ ∈ ( 0 , 1 ] of high-frequency metrics solutions to the Einstein vacuum equations in 3 + 1 dimensions without any symmetry assumptions. In the limit λ → 0 this family approaches a fixed background g 0 solution of the Einstein-null dust system, illustrating the backreaction phenomenon. We introduce a precise second order high-frequency ansatz and identify a generalised wave gauge as well as polarization conditions at each order. The validity of our ansatz is ensured by a weak polarized null condition satisfied by the quadratic non-linearity in the Ricci tensor. The Einstein vacuum equations for g λ are recast as a hierarchy of transport and wave equations, and their coupling induces a loss of derivatives. In order to solve it, we take advantage of a null foliation associated to g 0 as well as a Fourier cut-off adapted to our ansatz. The construction of high-frequency initial data solving the constraint equations is the content of our companion paper (Touati in Commun Math Phys, 2023).
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-023-04790-x