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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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| Published in: | Communications in mathematical physics 2023-09, Vol.402 (3), p.3109-3200 |
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| container_end_page | 3200 |
| container_issue | 3 |
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| container_title | Communications in mathematical physics |
| container_volume | 402 |
| creator | Touati, Arthur |
| description | 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). |
| doi_str_mv | 10.1007/s00220-023-04790-x |
| format | article |
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(
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).</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-023-04790-x</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>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</subject><ispartof>Communications in mathematical physics, 2023-09, Vol.402 (3), p.3109-3200</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-f583ae5d12a4409161febd7b2c7fdd9c3cf01c9a8fa8fe84241e881863e2cda73</citedby><cites>FETCH-LOGICAL-c319t-f583ae5d12a4409161febd7b2c7fdd9c3cf01c9a8fa8fe84241e881863e2cda73</cites><orcidid>0000-0002-0359-1665</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27900,27901</link.rule.ids></links><search><creatorcontrib>Touati, Arthur</creatorcontrib><title>Geometric Optics Approximation for the Einstein Vacuum Equations</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>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).</description><subject>Approximation</subject><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Geometrical optics</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Relativity</subject><subject>Relativity Theory</subject><subject>Tensors</subject><subject>Theoretical</subject><subject>Wave equations</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKt_wFPAc3Qm2WazN0upVSj0ol5Dmk10i93dJruw_nvTruBNGJjLe2_efITcItwjQP4QATgHBlwwyPIC2HBGJpgJzqBAeU4mAAhMSJSX5CrGHQAUXMoJeVy5Zu-6UFm6abvKRjpv29AM1d50VVNT3wTafTq6rOrYuaqm78b2_Z4uD_1JEK_JhTdf0d387il5e1q-Lp7ZerN6WczXzAosOuZnShg3K5GbLDt2Qu-2Zb7lNvdlWVhhPaAtjPJpnMp4hk4pVFI4bkuTiym5G3NTu0PvYqd3TR_qdFJzNRNCAXKZVHxU2dDEGJzXbUivhG-NoI-k9EhKJ1L6REoPySRGU0zi-sOFv-h_XD_UQGyJ</recordid><startdate>20230901</startdate><enddate>20230901</enddate><creator>Touati, Arthur</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-0359-1665</orcidid></search><sort><creationdate>20230901</creationdate><title>Geometric Optics Approximation for the Einstein Vacuum Equations</title><author>Touati, Arthur</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-f583ae5d12a4409161febd7b2c7fdd9c3cf01c9a8fa8fe84241e881863e2cda73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Approximation</topic><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Geometrical optics</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Relativity</topic><topic>Relativity Theory</topic><topic>Tensors</topic><topic>Theoretical</topic><topic>Wave equations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Touati, Arthur</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Touati, Arthur</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Geometric Optics Approximation for the Einstein Vacuum Equations</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2023-09-01</date><risdate>2023</risdate><volume>402</volume><issue>3</issue><spage>3109</spage><epage>3200</epage><pages>3109-3200</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>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).</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-023-04790-x</doi><tpages>92</tpages><orcidid>https://orcid.org/0000-0002-0359-1665</orcidid></addata></record> |
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| ispartof | Communications in mathematical physics, 2023-09, Vol.402 (3), p.3109-3200 |
| issn | 0010-3616 1432-0916 |
| language | eng |
| recordid | cdi_proquest_journals_2853380126 |
| source | Springer Nature |
| 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 |
| title | Geometric Optics Approximation for the Einstein Vacuum Equations |
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