New boundary conditions on the time-like conformal infinity of the Anti-de Sitter universe

Alain Bachelot

doi:10.1016/j.crma.2012.04.001

Partial Differential Equations/Mathematical Physics

New boundary conditions on the time-like conformal infinity of the Anti-de Sitter universe

Presented by: Yvonne Choquet-Bruhat

Alain Bachelot ¹

¹ Université de Bordeaux, institut de mathématiques, UMR CNRS 5251, 33405 Talence cedex, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 359-364.

Abstract
Résumé

We investigate the possible propagations of the gravitational waves in the 5-dimensional Anti-de Sitter universe. We construct a large family of unitary dynamics with respect to some high order energies that are conserved and positive. These dynamics are associated with asymptotic conditions on the conformal time-like boundary of the universe. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in $L^{2}$ . The proof needs the study of the Klein–Gordon equation on $R^{1 + 6}$ with a super-singular perturbation of $δ_{0}$ -type.

Nous étudions diverses propagations possibles des ondes gravitationnelles dans lʼunivers Anti-de Sitter pentadimensionnel. Nous construisons une grande famille de dynamiques unitaires par rapport à des énergies conservées dʼordre élevé et positives. Ces dynamiques sont associées à des conditions asymptotiques au bord conforme de genre temps de lʼunivers. Le point clef est lʼintroduction dʼun nouveau cadre fonctionnel hilbertien qui contient le graviton sans masse, lequel nʼest pas normalisable dans $L^{2}$ . La preuve sʼappuie sur lʼétude de lʼéquation de Klein–Gordon dans $R^{1 + 6}$ avec une perturbation super-singulière de type Dirac à lʼorigine.

Received: 2012-03-26
Accepted: 2012-04-04
Published online: 2012-04-01

DOI: 10.1016/j.crma.2012.04.001

Author's affiliations:

Alain Bachelot ¹

¹ Université de Bordeaux, institut de mathématiques, UMR CNRS 5251, 33405 Talence cedex, France

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     title = {New boundary conditions on the time-like conformal infinity of the {Anti-de} {Sitter} universe},
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Alain Bachelot. New boundary conditions on the time-like conformal infinity of the Anti-de Sitter universe. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 359-364. doi : 10.1016/j.crma.2012.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.001/

References
Cited by

[1] A. Bachelot The Dirac system on the Anti-de Sitter universe, Comm. Math. Phys., Volume 283 (2008), pp. 127-167

[2] A. Bachelot The Klein–Gordon equation in Anti-de Sitter cosmology, J. Math. Pures Appl. (9), Volume 96 (2011) no. 6, pp. 527-554

[3] A. Bachelot New dynamics in the Anti-de Sitter universe ${AdS}^{5}$ | arXiv

[4] A. Ishibashi; R.M. Wald Dynamics in non-globally-hyperbolic, static space-times: III. Anti-de Sitter space–time, Class. Quantum Grav., Volume 21 (2004), pp. 2981-3013

[5] P. Kurasov Triplet extensions I: Semibounded operators in the scale of Hilbert spaces, J. Anal. Math., Volume 107 (2009), pp. 251-286

[6] A. Vasy, The wave equation on asymptotically Anti-de Sitter spaces, Analysis and PDE, in press, . | arXiv

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