Wave asymptotics at a cosmological time-singularity: classical and quantum scalar fields

We investigate the propagation of the scalar waves in the FLRW universes beginning with a Big Bang and ending with a Big Crunch, a Big Rip, a Big Brake or a Sudden Singularity. We obtain the sharp description of the asymptotics for the solutions of the linear Klein-Gordon equation, and similar results for the semilinear equation with a subcritical exponent. We prove that the number of cosmological particle creation is finite under general assumptions on the initial Big Bang and the final Big Crunch or Big Brake.Comment: 40 p., 5 figure

New Dynamics in the Anti-De Sitter Universe AdS^5

This paper deals with the propagation of the gravitational waves in the Poincar\'e 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 L2 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 L2. Then the hamiltonian is not essentially self-adjoint in this new space and admits a lot of positive self-adjoint extensions

Wave Computation on the Hyperbolic Double Doughnut

We compute the waves propagating on the compact surface of constant negative curvature and genus 2. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. A spectral analysis of the wave allows to compute the first eigenvalues of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping and the ergodicity of the geodesic flow.Comment: 13 pages, 6 figure

The wave equation on the Schwarzschild metric II: Local decay for the spin 2 Regge Wheeler equation

Odd-type spin 2 perturbations of Einstein's equation can be reduced to the scalar Regge-Wheeler equation. We show that the weighted norms of solutions are in L^2 of time and space. This result uses commutator methods and applies uniformly to all relevant spherical harmonics.Comment: AMS-LaTeX, 8 pages with 1 figure. There is an errata to this paper at gr-qc/060807

Conformal scattering for a nonlinear wave equation on a curved background

The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild spacetime and a method used by H\"ormander for the Goursat problem. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity is then obtained. This requires a control of the nonlinearity uniform in time which comes from an estimates of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator

Global Waves with Non-Positive Energy in General Relativity

2000 Mathematics Subject Classification: 35Lxx, 35Pxx, 81Uxx, 83Cxx.The theory of the waves equations has a long history since M. Riesz and J. Hadamard. It is impossible to cite all the important results in the area, but we mention the authors related with our work: J. Leray [34] and Y. Choquet-Bruhat [9] (Cauchy problem), P. Lax and R. Phillips [33] (scattering theory for a compactly supported perturbation), L. H¨ ormander [27] and J-M. Bony [7] (microlocal analysis). In all these domains, V. Petkov has made fundamental contributions, mainly in microlocal analysis, scattering theory, dynamical zeta functions (see in particular the monography [42]). In this paper we present a survey of some recent results on the global existence and the asymptotic behaviour of waves, when the conserved energy is not definite positive. This unusual situation arises in important cosmological models of the General Relativity where the gravitational curvature is very strong. We consider the case of the closed time-like curves (violation of the causality) [1], and the charged black-holes (superradiance) [3

Lost in translation? Lo que los trabajos etnográficos nos dicen de los partidos políticos: una revision crítica de la literatura francesa

En la extensa literatura sobre los partidos políticos, el método etnográfico parece ser escasamente utilizado. Desde el trabajo pionero de Michels, que en cierta medida se basaba (aunque implícitamente) en la «observación participante», este enfoque sufrió un largo eclipse en la literatura anglosajona. Hoy en día, la investigación internacional favorece la confrontación de grandes conjuntos de datos, ya sean relativos a la afiliación, los líderes, las características de las reformas organizativas o los contenidos programáticos. El reciente florecimiento de una «etnografía política» (Auyero, Joseph y Mahler 200

Propagation of Massive Scalar Fields in Pre-Big Bang Cosmologies

We investigate the linear and semilinear massive Klein-Gordon equations in geometrical frameworks of type "Conformal Cyclic Cosmology" of R. Penrose, or "Singular Bouncing Scenario" as well. We give sufficient conditions on the decay of the mass to the fields be able to propagate across the Big-Bang

Scattering of massive Dirac fields on the Schwarzschild black hole spacetime

With a generally covariant equation of Dirac fields outside a black hole, we develop a scattering theory for massive Dirac fields. The existence of modified wave operators at infinity is shown by implementing a time-dependent logarithmic phase shift from the free dynamics to offset a long-range mass term. The phase shift we obtain is a matrix operator due to the existence of both positive and negative energy wave components.Comment: LaTex, 17 page