Alain Bachelot

Problème de Cauchy pour des systèmes hyperboliques semi‐linéaires

Resume On resout le probleme de Cauchy global pour des systemes d’equations de Schroedinger et de Klein-Gordon sans conservation d’energie et avec des donnees initiales peu regulieres. On obtient egalement des solutions locales pour des systemes de Dirac-Klein-Gordon avec non linearites quadratiques generalisant l’interaction de Yukawa.

Creation of Fermions at the Charged Black-Hole Horizon

We investigate the quantum state of the Dirac field at the horizon of a charged black-hole formed by a spherical gravitational collapse. We prove this state satisfies a KMS condition with the Hawking temperature and the chemical potential associated with the mass and the charge of the black-hole. Moreover, the fermions with charge of same sign to that of the black-hole are emitted more readily than those of opposite charge. It is a spontaneous loss of charge of the black-hole due to the quantum vacuum polarization.

Global properties of the wave equation on non-globally hyperbolic manifolds

Abstract We introduce a class of four-dimensional Lorentzian manifolds with closed curves of null type or timelike. We investigate some global problems for the wave equation: uniqueness of solution with data on a changing type hypersurface; existence of resonant states; scattering by a violation of the chronology; global Cauchy problem and asymptotic completeness of the wave operators for the chronological but non-causal metrics.

Scattering of scalar fields by spherical gravitational collapse

Abstract We study the scattering of a scalar field, massive or massless, by a collapsing spherical star. The main point of interest is the infinite Doppler effect measured by an observer, at rest in Schwarzschild coordinates. We construct the functional framework associated with this phenomenon, and we prove the existence and strong asymptotic completeness of the wave operators describing the scattering of the field by the space-time curvature and the asymptotically characteristic moving boundary of the star.

New Dynamics in the Anti-de Sitter Universe AdS5

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 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 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.

WAVE PROPAGATION AND SCATTERING FOR THE RS2 BRANE COSMOLOGY MODEL

We study the wave equation for the gravitational fluctuations in the Randall–Sundrum brane cosmology model. We solve the global Cauchy problem and we establish that the solutions are the sum of a slowly decaying massless wave localized near the brane, and a superposition of massive dispersive waves. We compute the kernel of the truncated resolvent. We prove some L1-L∞, L2-L∞ decay estimates and global Lp Strichartz type inequalities. We develop the complete scattering theory: existence and asymptotic completeness of the wave operators, computation of the scattering matrix, determination of the resonances on the logarithmic Riemann surface.

Resonances of Schwarzschild Black Holes

This paper is devoted to the theoretical and computational investigations of the scattering frequencies of scalar, electromagnetic, gravitationnal waves around a spherical Black Hole. We adopt a time dependent approach: construction of wave operators for the equation hyperbolic Regge-Wheeler equation; asymptotic completeness; outgoing and incoming spectral representations; meromorphic continuation of the Heisenberg matrix; approximation by dumping and cut-off of the potentials and interpretation of the semi group Z(t) in the framework of the Membrane Paradigme. We developp a new procedure for the computation of resonances by spectral analysis of the transient scattered wave, based on Prony’s algorithm.

Time dependent integral method for Maxwell's system with impedance boundary condition

We solve the problem of diffraction of an electromagnetic wave by a dissipative scatterer using a boundary integral method in time-domain directly. We prove the existence and uniqueness of the solution of this problem. We obtain the continuity and a relation of coercivity for the associated time-dependent formulation in this time functional framework. The discret approximation of the variational formulation leads to a stable marching-in-time scheme.

Waves in the Witten Bubble of Nothing and the Hawking Wormhole

We investigate the propagation of the scalar waves in the Witten space-time called “bubble of nothing” and in its remarkable sub-manifold, the Lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the functional framework associated with the energy. We perform a complete spectral analysis that allows us to get an explicit form of the solutions in terms of special functions. If the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In contrast, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. The quantized scattering operator leaves invariant the Fock vacuum: there is no creation of particles. The resonances can be defined in the massless case and they are purely imaginary.

La radiation Hawking à l’horizon d’un trou noir

Resume On met en evidence la polarisation du vide quantique vers un etat thermal de temperature Hawking, a l’horizon futur d’un trou noir cree par effondrement gravitationnel.