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