15 research outputs found
Hertz potentials and asymptotic properties of massless fields
In this paper we analyze Hertz potentials for free massless spin-s fields on
the Minkowski spacetime, with data in weighted Sobolev spaces. We prove
existence and pointwise estimates for the Hertz potentials using a weighted
estimate for the wave equation. This is then applied to give weighted estimates
for the solutions of the spin-s field equations, for arbitrary half-integer s.
In particular, the peeling properties of the free massless spin-s fields are
analyzed for initial data in weighted Sobolev spaces with arbitrary,
non-integer weights.Comment: Regularity assumptions corrected. Orthogonality condition eliminate
A vector field method for relativistic transport equations with applications
We adapt the vector field method of Klainerman to the study of relativistic
transport equations. First, we prove robust decay estimates for velocity
averages of solutions to the relativistic massive and massless transport
equations, without any compact support requirements (in or ) for the
distribution functions. In the second part of this article, we apply our method
to the study of the massive and massless Vlasov-Nordstr\"om systems. In the
massive case, we prove global existence and (almost) optimal decay estimates
for solutions in dimensions under some smallness assumptions. In the
massless case, the system decouples and we prove optimal decay estimates for
the solutions in dimensions for arbitrarily large data, and in
dimension under some smallness assumptions, exploiting a certain form of
the null condition satisfied by the equations. The -dimensional massive case
requires an extension of our method and will be treated in future work.Comment: 72 pages, 3 figure
The Stability of the Minkowski space for the Einstein-Vlasov system
We prove the global stability of the Minkowski space viewed as the trivial
solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use
the vector field and modified vector field techniques developed in [FJS15;
FJS17]. In particular, the initial support in the velocity variable does not
need to be compact. To control the effect of the large velocities, we identify
and exploit several structural properties of the Vlasov equation to prove that
the worst non-linear terms in the Vlasov equation either enjoy a form of the
null condition or can be controlled using the wave coordinate gauge. The basic
propagation estimates for the Vlasov field are then obtained using only weak
interior decay for the metric components. Since some of the error terms are not
time-integrable, several hierarchies in the commuted equations are exploited to
close the top order estimates. For the Einstein equations, we use wave
coordinates and the main new difficulty arises from the commutation of the
energy-momentum tensor, which needs to be rewritten using the modified vector
fields.Comment: 139 page
The Conformal Einstein Field Equations with Massless Vlasov Matter
We prove the stability of de Sitter space-time as a solution to the
Einstein-Vlasov system with massless particles. The semi-global stability of
Minkowski space-time is also addressed. The proof relies on conformal
techniques, namely Friedrich's conformal Einstein field equations. We exploit
the conformal invariance of the massless Vlasov equation on the cotangent
bundle and adapt Kato's local existence theorem for symmetric hyperbolic
systems to prove a long enough time of existence for solutions of the evolution
system implied by the Vlasov equation and the conformal Einstein field
equations.Comment: 27 pages. To appear in Ann. Inst. Fourie
Propagation of polarized gravitational waves
The propagation of high-frequency gravitational waves can be analyzed using
the geometrical optics approximation. In the case of large but finite
frequencies, the geometrical optics approximation is no longer accurate, and
polarization-dependent corrections at first order in wavelength modify the
propagation of gravitational waves via a spin-orbit coupling mechanism. We
present a covariant derivation from first principles of effective ray equations
describing the propagation of polarized gravitational waves, up to first-order
terms in wavelength, on arbitrary spacetime backgrounds. The effective ray
equations describe a gravitational spin Hall effect for gravitational waves and
are of the same form as those describing the gravitational spin Hall effect of
light, derived from Maxwell's equations
Asymptotic Stability of Minkowski Space-Time with Non-compactly Supported Massless Vlasov Matter.
We prove the global asymptotic stability of the Minkowski space for the massless Einstein-Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein-Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components
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
Problème de Cauchy caractéristique et scattering conforme en relativité générale
This work presents two aspects of the characteristic Cauchy problem in general relativity. On the one hand, an integral formula for the characteristic Cauchy problem for the Dirac equation on a curved space-time is derived. This generalizes the work of Penrose in the 60's. The functional framework is adapted, so that the algebraic structures on spinors can be brought to distributions on spinors. This gives an integral formula which is simplified using the Geroch-Held-Penrose formalism. Penrose's formula on the Minkowski space-time is recovered for arbitrary spin. On the other hand, a conformal scattering theory for a conformally invariant nonlinear wave equation is established. Using a conformal rescaling, the space-time is completed with two null hypersurfaces representing respectively the past and future endpoints of null geodesics. The asymptotic behaviour of fields is then obtained by considering the traces of solutions of the rescaled equations on these hypersurfaces. The inversibility of these trace operators is obtained by solving a characteristic Cauchy problem and the conformal scattering operator is obtained by composing these trace operators.L'étude présentée dans ce travail de thèse aborde deux aspects du problème de Cauchy caractéristique en relativité générale. D'une part, une formule intégrale pour le problème de Cauchy caractéristique pour l'équation de Dirac est établie, généralisant les travaux de Penrose en espace-temps courbe. Ayant adapté le cadre fonctionnel pour obtenir une théorie des distributions adaptée à la structure algébriques des spineurs, le formalisme Geroch-Held-Penrose est utilisé pour décrire de la manière la plus précise possible la formule intégrale. La formule de Penrose en spin arbitraire sur l'espace-temps de Minkowski est retrouvée. D'autre part, une théorie de scattering conforme pour une équation des ondes non linéaire conformément invariante sur un espace asymptotiquement simple est construite. En effectuant un rééchelonnement conforme, l'espace-temps est complété en lui ajoutant une frontière constituée de deux hypersurfaces caractéristiques représentant respectivement les extrémités passées et futures des géodésiques de type lumière. Le comportement asymptotique des champs s'obtient alors en considérant les traces des solutions de l'équation conforme sur ces bords. L'inversibilité des opérateurs de trace s'obtient alors en résolvant un problème de Cauchy caractéristique sur ce bord et l'opérateur de scattering conforme par composition de ces opérateurs de trace