746 research outputs found
Stability of exact force-free electrodynamic solutions and scattering from spacetime curvature
Recently, a family of exact force-free electrodynamic (FFE) solutions was
given by Brennan, Gralla and Jacobson, which generalizes earlier solutions by
Michel, Menon and Dermer, and other authors. These solutions have been proposed
as useful models for describing the outer magnetosphere of conducting stars. As
with any exact analytical solution that aspires to describe actual physical
systems, it is vitally important that the solution possess the necessary
stability. In this paper, we show via fully nonlinear numerical simulations
that the aforementioned FFE solutions, despite being highly special in their
properties, are nonetheless stable under small perturbations. Through this
study, we also introduce a three-dimensional pseudospectral relativistic FFE
code that achieves exponential convergence for smooth test cases, as well as
two additional well-posed FFE evolution systems in the appendix that have
desirable mathematical properties. Furthermore, we provide an explicit analysis
that demonstrates how propagation along degenerate principal null directions of
the spacetime curvature tensor simplifies scattering, thereby providing an
intuitive understanding of why these exact solutions are tractable, i.e. why
they are not backscattered by spacetime curvature.Comment: 33 pages, 21 figures; V2 updated to match published versio
Modeling Shallow Water Flows on General Terrains
A formulation of the shallow water equations adapted to general complex
terrains is proposed. Its derivation starts from the observation that the
typical approach of depth integrating the Navier-Stokes equations along the
direction of gravity forces is not exact in the general case of a tilted curved
bottom. We claim that an integration path that better adapts to the shallow
water hypotheses follows the "cross-flow" surface, i.e., a surface that is
normal to the velocity field at any point of the domain. Because of the
implicitness of this definition, we approximate this "cross-flow" path by
performing depth integration along a local direction normal to the bottom
surface, and propose a rigorous derivation of this approximation and its
numerical solution as an essential step for the future development of the full
"cross-flow" integration procedure. We start by defining a local coordinate
system, anchored on the bottom surface to derive a covariant form of the
Navier-Stokes equations. Depth integration along the local normals yields a
covariant version of the shallow water equations, which is characterized by
flux functions and source terms that vary in space because of the surface
metric coefficients and related derivatives. The proposed model is discretized
with a first order FORCE-type Godunov Finite Volume scheme that allows
implementation of spatially variable fluxes. We investigate the validity of our
SW model and the effects of the bottom geometry by means of three synthetic
test cases that exhibit non negligible slopes and surface curvatures. The
results show the importance of taking into consideration bottom geometry even
for relatively mild and slowly varying curvatures
Field-theoretical formulations of MOND-like gravity
Modified Newtonian dynamics (MOND) is a possible way to explain the flat
galaxy rotation curves without invoking the existence of dark matter. It is
however quite difficult to predict such a phenomenology in a consistent field
theory, free of instabilities and admitting a well-posed Cauchy problem. We
examine critically various proposals of the literature, and underline their
successes and failures both from the experimental and the field-theoretical
viewpoints. We exhibit new difficulties in both cases, and point out the hidden
fine tuning of some models. On the other hand, we show that several published
no-go theorems are based on hypotheses which may be unnecessary, so that the
space of possible models is a priori larger. We examine a new route to
reproduce the MOND physics, in which the field equations are particularly
simple outside matter. However, the analysis of the field equations within
matter (a crucial point which is often forgotten in the literature) exhibits a
deadly problem, namely that they do not remain always hyperbolic. Incidentally,
we prove that the same theoretical framework provides a stable and well-posed
model able to reproduce the Pioneer anomaly without spoiling any of the
precision tests of general relativity. Our conclusion is that all MOND-like
models proposed in the literature, including the new ones examined in this
paper, present serious difficulties: Not only they are unnaturally fine tuned,
but they also fail to reproduce some experimental facts or are unstable or
inconsistent as field theories. However, some frameworks, notably the
tensor-vector-scalar (TeVeS) one of Bekenstein and Sanders, seem more promising
than others, and our discussion underlines in which directions one should try
to improve them.Comment: 66 pages, 6 figures, RevTeX4 format, version reflecting the changes
in the published pape
Reminiscences about numerical schemes
This preprint appeared firstly in Russian in 1997. Some truncated versions of this preprint were published in English and French, here a fully translated version is presented. The translation in English was done by O. V. Feodoritova and V. Deledicque to whom I express my gratitude
Steady inviscid transonic flows over planar airfoils: A search for a simplified procedure
A finite difference procedure based upon a system of unsteady equations in proper conservation form with either exact or small disturbance steady terms is used to calculate the steady flows over several classes of airfoils. The airfoil condition is fulfilled on a slab whose upstream extremity is a semi-circle overlaying the airfoil leading edge circle. The limitations of the small disturbance equations are demonstrated in an extreme example of a blunt-nosed, aft-cambered airfoil. The necessity of using the equations in proper conservation form to capture the shock properly is stressed. Ability of the steady relaxation procedures to capture the shock is briefly examined
Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties
Adiabatic vacuum states are a well-known class of physical states for linear
quantum fields on Robertson-Walker spacetimes. We extend the definition of
adiabatic vacua to general spacetime manifolds by using the notion of the
Sobolev wavefront set. This definition is also applicable to interacting field
theories. Hadamard states form a special subclass of the adiabatic vacua. We
analyze physical properties of adiabatic vacuum representations of the
Klein-Gordon field on globally hyperbolic spacetime manifolds (factoriality,
quasiequivalence, local definiteness, Haag duality) and construct them
explicitly, if the manifold has a compact Cauchy surface.Comment: 68 pages, Latex, no figures, minor changes in the text, 2 references
adde
Mathematical Imaging and Surface Processing
Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images.
This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains
Transonic flow studies
Major emphasis was on the design of shock free airfoils with applications to general aviation. Unsteady flow, transonic flow, and shock wave formation were examined
Topology and the Cosmic Microwave Background
Nature abhors an infinity. The limits of general relativity are often
signaled by infinities: infinite curvature as in the center of a black hole,
the infinite energy of the singular big bang. We might be inclined to add an
infinite universe to the list of intolerable infinities. Theories that move
beyond general relativity naturally treat space as finite. In this review we
discuss the mathematics of finite spaces and our aspirations to observe the
finite extent of the universe in the cosmic background radiation.Comment: Hilarioulsy forgot to remove comments to myself in previous version.
Reference added. Submitted to Physics Report
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