2,590 research outputs found
Decay of the Maxwell field on the Schwarzschild manifold
We study solutions of the decoupled Maxwell equations in the exterior region
of a Schwarzschild black hole. In stationary regions, where the Schwarzschild
coordinate ranges over , we obtain a decay rate of
for all components of the Maxwell field. We use vector field methods
and do not require a spherical harmonic decomposition.
In outgoing regions, where the Regge-Wheeler tortoise coordinate is large,
, we obtain decay for the null components with rates of
, , and . Along the event horizon and in ingoing regions, where ,
and when , all components (normalized with respect to an ingoing null
basis) decay at a rate of C \uout^{-1} with \uout=t+r_* in the exterior
region.Comment: 37 pages, 5 figure
The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein-Gordon fields
We study the initial value problem for two fundamental theories of gravity,
that is, Einstein's field equations of general relativity and the
(fourth-order) field equations of f(R) modified gravity. For both of these
physical theories, we investigate the global dynamics of a self-gravitating
massive matter field when an initial data set is prescribed on an
asymptotically flat and spacelike hypersurface, provided these data are
sufficiently close to data in Minkowski spacetime. Under such conditions, we
thus establish the global nonlinear stability of Minkowski spacetime in
presence of massive matter. In addition, we provide a rigorous mathematical
validation of the f(R) theory based on analyzing a singular limit problem, when
the function f(R) arising in the generalized Hilbert-Einstein functional
approaches the scalar curvature function R of the standard Hilbert-Einstein
functional. In this limit we prove that f(R) Cauchy developments converge to
Einstein's Cauchy developments in the regime close to Minkowski space. Our
proofs rely on a new strategy, introduced here and referred to as the
Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of
the Hyperboloidal Foliation Method (HFM) which we used earlier for the
Einstein-massive field system but for a restricted class of initial data. Here,
the data are solely assumed to satisfy an asymptotic flatness condition and be
small in a weighted energy norm. These results for matter spacetimes provide a
significant extension to the existing stability theory for vacuum spacetimes,
developed by Christodoulou and Klainerman and revisited by Lindblad and
Rodnianski.Comment: 127 pages. Selected chapters from a boo
Global existence and modified scattering for the solutions to the Vlasov-Maxwell system with a small distribution function
The purpose of this paper is twofold. In the first part, we provide a new
proof of the global existence of the solutions to the Vlasov-Maxwell system
with a small initial distribution function. Our approach relies on vector field
methods, together with the Glassey-Strauss decomposition of the electromagnetic
field, and does not require any support restriction on the initial data or
smallness assumption on the Maxwell field. Contrary to previous works on Vlasov
systems in dimension , we do not modify the linear commutators and avoid
then many technical difficulties.
In the second part of this paper, we prove a modified scattering result for
these solutions. More precisely, we obtain that the electromagnetic field has a
radiation field along future null infinity and approaches, for large time, a
smooth solution to the vacuum Maxwell equations. As for the Vlasov-Poisson
system, in constrast, the distribution function converges to a new density
function along modifications of the characteristics of the free relativistic
transport equation. In order to define these logarithmic corrections, we
identify an effective asymptotic Lorentz force. By considering logarithmical
modifications of the linear commutators, defined in terms of derivatives of the
asymptotic Lorentz force, we finally prove higher order regularity results for
the limit distribution function.Comment: In this second version of the article, the electromagnetic field is
now allowed to be large. There are other improvements as well and typos have
been correcte
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