19,057 research outputs found
The hyperboloidal foliation method
The Hyperboloidal Foliation Method presented in this monograph is based on a
(3+1)-foliation of Minkowski spacetime by hyperboloidal hypersurfaces. It
allows us to establish global-in-time existence results for systems of
nonlinear wave equations posed on a curved spacetime and to derive uniform
energy bounds and optimal rates of decay in time. We are also able to encompass
the wave equation and the Klein-Gordon equation in a unified framework and to
establish a well-posedness theory for nonlinear wave-Klein-Gordon systems and a
large class of nonlinear interactions. The hyperboloidal foliation of Minkowski
spacetime we rely upon in this book has the advantage of being geometric in
nature and, especially, invariant under Lorentz transformations. As stated, our
theory applies to many systems arising in mathematical physics and involving a
massive scalar field, such as the Dirac-Klein-Gordon system. As it provides
uniform energy bounds and optimal rates of decay in time, our method appears to
be very robust and should extend to even more general systems.Comment: 160 page
Light speed variation from gamma ray bursts: criteria for low energy photons
We examine a method to detect the light speed variation from gamma ray burst
data observed by the Fermi Gamma-ray Space Telescope (FGST). We suggest new
criteria to determine the characteristic time for low energy photons by the
energy curve and the average energy curve, and obtain similar results compared
with those from the light curve. We offer a new criterion with both the light
curve and the average energy curve to determine the characteristic time for low
energy photons. We then apply the new criteria to the GBM NaI data, the GBM BGO
data, and the LAT LLE data, and obtain consistent results for three different
sets of low energy photons from different FERMI detectors.Comment: 26 latex pages, 23 figures, final version for publicatio
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
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