Abstract
This paper deals with the propagation of the gravitational waves in the Poincaré patch of 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. This result does not contradict the statement of Breitenlohner-Freedman that the hamiltonian is essentially self-adjoint in L 2 and thus accordingly the dynamics is uniquely determined. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in L 2. Then the hamiltonian is not essentially self-adjoint in this new space and possesses a lot of different positive self-adjoint extensions. These dynamics satisfy a holographic principle: there exists a renormalized boundary value which completely characterizes the whole field in the bulk.
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Communicated by P. T. Chruściel
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Bachelot, A. New Dynamics in the Anti-de Sitter Universe AdS 5 . Commun. Math. Phys. 320, 723–759 (2013). https://doi.org/10.1007/s00220-013-1690-5
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DOI: https://doi.org/10.1007/s00220-013-1690-5