19,062 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
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
Contact-eutectic-lens fabrication technique
Method enables use of crystal or semiconductor materials with selective spectral-response characteristics (ultraviolet, visible, or infrared wavelengths) in fabrication of contact lenses, reading glasses, and photographic processing equipment
Atmospheric Backscatter Model Development for CO Sub 2 Wavelengths
The results of investigations into the problems of modeling atmospheric backscatter from aerosols, in the lowest 20 km of the atmosphere, at CO2 wavelengths are presented, along with a summary of the relevant aerosol characteristics and their variability, and a discussion of the measurement techniques and errors involved. The different methods of calculating the aerosol backscattering function, both from measured aerosol characteristics and from optical measurements made at other wavelengths, are discussed in detail, and limits are placed on the accuracy of these methods. The effects of changing atmospheric humidity and temperature on the backscatter are analyzed and related to the actual atmosphere. Finally, the results of modeling CO2 backscatter in the atmosphere are presented and the variation with height and geographic location discussed, and limits placed on the magnitude of the backscattering function. Conclusions regarding modeling techniques and modeled atmospheric backscatter values are presented in tabular form
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