141 research outputs found
General Relativity and Gravitation: A Centennial Perspective
To commemorate the 100th anniversary of general relativity, the International
Society on General Relativity and Gravitation (ISGRG) commissioned a Centennial
Volume, edited by the authors of this article. We jointly wrote introductions
to the four Parts of the Volume which are collected here. Our goal is to
provide a bird's eye view of the advances that have been made especially during
the last 35 years, i.e., since the publication of volumes commemorating
Einstein's 100th birthday. The article also serves as a brief preview of the 12
invited chapters that contain in-depth reviews of these advances. The volume
will be published by Cambridge University Press and released in June 2015 at a
Centennial conference sponsored by ISGRG and the Topical Group of Gravitation
of the American Physical Society.Comment: 37 page
Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow
We study constant mean curvature Lorentzian hypersurfaces of
from the point of view of its Cauchy problem. We
completely classify the spherically symmetric solutions, which include among
them a manifold isometric to the de Sitter space of general relativity. We show
that the spherically symmetric solutions exhibit one of three (future)
asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like
cylinder isometric to some and (iii) infinite
expansion to the future converging asymptotically to a time translation of the
de Sitter solution. For class (iii) we examine the future stability properties
of the solutions under arbitrary (not necessarily spherically symmetric)
perturbations. We show that the usual notions of asymptotic stability and
modulational stability cannot apply, and connect this to the presence of
cosmological horizons in these class (iii) solutions. We can nevertheless show
the global existence and future stability for small perturbations of class
(iii) solutions under a notion of stability that naturally takes into account
the presence of cosmological horizons. The proof is based on the vector field
method, but requires additional geometric insight. In particular we introduce
two new tools: an inverse-Gauss-map gauge to deal with the problem of
cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson
tensor to obtain natural energy quantities.Comment: Version 2: 60 pages, 1 figure. Changes mostly to fix typographical
errors, with the exception of Remark 1.2 and Section 9.1 which are new and
which explain the extrinsic geometry of the embedding in more detail in terms
of the stability result. Version 3: updated reference
Mathematical general relativity: a sampler
We provide an introduction to selected recent advances in the mathematical
understanding of Einstein's theory of gravitation.Comment: Some updates. A shortened version, to appear in the Bulletin of the
AMS, is available online at
http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2010-01304-
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
A guide to the Choquard equation
We survey old and recent results dealing with the existence and properties of
solutions to the Choquard type equations and some of its variants and extensions.Comment: 39 page
Nonlinear optical waves in disordered ferroelectrics
This thesis describes an experimental, numerical and theoretical investigation of nonlinear optical phenomena in disordered photorefractive ferroelectrics in proximity of their phase-transition temperature. The work addresses different physical issues that find in nonlinear optics a common fertile research arena and are closely related to each other in the considered systems. Nonlinear wave dynamics in the spatial domain, where self-interaction of propagating waves generally results into non-spreading localized wavepackets such as spatial solitons, is extended in photorefractive ferroelectrics to non-equilibrium regimes characterized by stochastic instabilities and large material fluctuations. We discover the emergence of rogue waves, localized perturbations of abnormal intensity, whose understanding is challenging in various physical contexts and resides in the general problem of long-tail statistical distributions in complex systems. We identify their origin in spatiotemporal soliton dynamics in a saturable nonlinearity which can support scale-invariant waveforms. Properties and predictability of the observed extreme events are investigated, and, in particular, we demonstrate their active control through the spatial incoherence scale of the optical field. Moreover, we report how their emergence is sustained by turbulent transitions to an incoherent and disordered optical state triggered by modulational instability. The onset of strong turbulence for propagating optical waves has remained unobserved up to now and our results demonstrate a new experimental setting for its study. When the functional form of the nonlinearity is turned into a nonlocal one due
to diffusive fields, this setting also exploits photonics to address fundamental physical problems and access to otherwise hidden phenomena. The natural spreading of waves during propagation, representing the wavelength-defined ultimate limit to spatial
resolution, can be eliminated and reversed leading to diffraction cancellation and anti-diffraction of light. Since these behaviors on modifying the nature of underlying Schrödinger equation, we are the first to demonstrate how nonlinearity can make the spatial light distribution behave as the wavefunction of a quantum particle with negative mass. All these findings have roots in the nonlinear optical response of critical disordered ferroelectric crystals, which are also extremely interesting from the condensed matter point of view. In fact, competition of different microscopic structural phases and the associated polar-domain dynamics at the nanoscale results into non-ergodic dipolar-glass behaviors giving giant responses such as giant polarization, piezoelectricity and electro-optic effect. Disordered ferroelectrics crystals are investigated electro-optically across their ferroelectric phase-transition, where we report the observation of an anomalous electro-optic effect compatible with ultracold dipolar reorientation. In compounds presenting spatial inhomogeneity in their chemical composition, we discover a new ferroelectric phase of matter in which polar domains spontaneously coordinate into a mesoscopic coherent polarization super-crystals. This phase mimics standard solid-state structures but on scales that are thousands of times larger and represent the first spontaneous three-dimensional photonic crystal
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