128,246 research outputs found
Symmetries of the Einstein Equations
Generalized symmetries of the Einstein equations are infinitesimal
transformations of the spacetime metric that formally map solutions of the
Einstein equations to other solutions. The infinitesimal generators of these
symmetries are assumed to be local, \ie at a given spacetime point they are
functions of the metric and an arbitrary but finite number of derivatives of
the metric at the point. We classify all generalized symmetries of the vacuum
Einstein equations in four spacetime dimensions and find that the only
generalized symmetry transformations consist of: (i) constant scalings of the
metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms.
Our results rule out a large class of possible ``observables'' for the
gravitational field, and suggest that the vacuum Einstein equations are not
integrable.Comment: 15 pages, FTG-114-USU, Plain Te
New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory
DifferentialGeometry is a Maple software package which symbolically performs
fundamental operations of calculus on manifolds, differential geometry, tensor
calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the
variational calculus. These capabilities, combined with dramatic recent
improvements in symbolic approaches to solving algebraic and differential
equations, have allowed for development of powerful new tools for solving
research problems in gravitation and field theory. The purpose of this paper is
to describe some of these new tools and present some advanced applications
involving: Killing vector fields and isometry groups, Killing tensors and other
tensorial invariants, algebraic classification of curvature, and symmetry
reduction of field equations.Comment: 42 page
Handicapping currency design: counterfeit deterrence and visual accessibility in the United States and abroad
Despite the increasing use of electronic payments, currency retains an important role in the payments system of every country. Two aspects of currency usage drive currency design worldwide: deterring counterfeiting and making paper currency accessible to the visually impaired. Further, among the world's currencies, only U.S. banknotes are widely owned and used in transactions outside their country of issue (although the euro also has some external circulation). In this article, we compare and contrast major currencies and their design features. We conclude that the designs of the two most widely used currencies in the world-the U.S. dollar and the euro-have successfully deterred counterfeiting; data on other currencies are not public. We also conclude that, among the world's major currencies, U.S. banknotes have the fewest features to assist the visually impaired.Paper money design ; Coinage ; Counterfeits and counterfeiting
The role of phase dynamics in a stochastic model of a passively advected scalar
Collective synchronous motion of the phases is introduced in a model for the
stochastic passive advection-diffusion of a scalar with external forcing. The
model for the phase coupling dynamics follows the well known Kuramoto model
paradigm of limit-cycle oscillators. The natural frequencies in the Kuramoto
model are assumed to obey a given scale dependence through a dispersion
relation of the drift-wave form , where is a
constant representing the typical strength of the gradient. The present aim is
to study the importance of collective phase dynamics on the characteristic time
evolution of the fluctuation energy and the formation of coherent structures.
Our results show that the assumption of a fully stochastic phase state of
turbulence is more relevant for high values of , where we find that the
energy spectrum follows a scaling. Whereas for lower there
is a significant difference between a-synchronised and synchronised phase
states, and one could expect the formation of coherent modulations in the
latter case.Comment: Accepted for publication in Physics of Plasma
Equilibrium properties of the Skylab CMG rotation law
The equilibrium properties of the control moment gyroscopes of the Skylab are discussed. A rotation law is developed to produce gimbal rates which distribute the angular momentum contributions among the control moment gyroscopes to avoid gimbal stop encounters. The implications for gimbal angle management under various angular momentum situations are described. Conditions were obtained for the existence of equilibria and corresponding stability properties
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