921 research outputs found
Stochastic integration based on simple, symmetric random walks
A new approach to stochastic integration is described, which is based on an
a.s. pathwise approximation of the integrator by simple, symmetric random
walks. Hopefully, this method is didactically more advantageous, more
transparent, and technically less demanding than other existing ones. In a
large part of the theory one has a.s. uniform convergence on compacts. In
particular, it gives a.s. convergence for the stochastic integral of a finite
variation function of the integrator, which is not c\`adl\`ag in general.Comment: 16 pages, some typos correcte
On a class of 2-surface observables in general relativity
The boundary conditions for canonical vacuum general relativity is
investigated at the quasi-local level. It is shown that fixing the area element
on the 2- surface S (rather than the induced 2-metric) is enough to have a well
defined constraint algebra, and a well defined Poisson algebra of basic
Hamiltonians parameterized by shifts that are tangent to and divergence-free on
$. The evolution equations preserve these boundary conditions and the value of
the basic Hamiltonian gives 2+2 covariant, gauge-invariant 2-surface
observables. The meaning of these observables is also discussed.Comment: 11 pages, a discussion of the observables in stationary spacetimes is
included, new references are added, typos correcte
Stationary untrapped boundary conditions in general relativity
A class of boundary conditions for canonical general relativity are proposed
and studied at the quasi-local level. It is shown that for untrapped or
marginal surfaces, fixing the area element on the 2-surface (rather than the
induced 2-metric) and the angular momentum surface density is enough to have a
functionally differentiable Hamiltonian, thus providing definition of conserved
quantities for the quasi-local regions. If on the boundary the evolution vector
normal to the 2-surface is chosen to be proportional to the dual expansion
vector, we obtain a generalization of the Hawking energy associated with a
generalized Kodama vector. This vector plays the role for the stationary
untrapped boundary conditions which the stationary Killing vector plays for
stationary black holes. When the dual expansion vector is null, the boundary
conditions reduce to the ones given by the non-expanding horizons and the null
trapping horizons.Comment: 11 pages, improved discussion section, a reference added, accepted
for publication in Classical and Quantum Gravit
On certain quasi-local spin-angular momentum expressions for small spheres
The Ludvigsen-Vickers and two recently suggested quasi-local spin-angular
momentum expressions, based on holomorphic and anti-holomorphic spinor fields,
are calculated for small spheres of radius about a point . It is shown
that, apart from the sign in the case of anti-holomorphic spinors in
non-vacuum, the leading terms of all these expressions coincide. In non-vacuum
spacetimes this common leading term is of order , and it is the product of
the contraction of the energy-momentum tensor and an average of the approximate
boost-rotation Killing vector that vanishes at and of the 3-volume of the
ball of radius . In vacuum spacetimes the leading term is of order ,
and the factor of proportionality is the contraction of the Bel-Robinson tensor
and an other average of the same approximate boost-rotation Killing vector.Comment: 16 pages, Plain Te
Self-intersection local time of planar Brownian motion based on a strong approximation by random walks
The main purpose of this work is to define planar self-intersection local
time by an alternative approach which is based on an almost sure pathwise
approximation of planar Brownian motion by simple, symmetric random walks. As a
result, Brownian self-intersection local time is obtained as an almost sure
limit of local averages of simple random walk self-intersection local times. An
important tool is a discrete version of the Tanaka--Rosen--Yor formula; the
continuous version of the formula is obtained as an almost sure limit of the
discrete version. The author hopes that this approach to self-intersection
local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has
been added and several inaccuracies have been corrected. To appear in Journal
of Theoretical Probabilit
Total angular momentum from Dirac eigenspinors
The eigenvalue problem for Dirac operators, constructed from two connections
on the spinor bundle over closed spacelike 2-surfaces, is investigated. A class
of divergence free vector fields, built from the eigenspinors, are found,
which, for the lowest eigenvalue, reproduce the rotation Killing vectors of
metric spheres, and provide rotation BMS vector fields at future null infinity.
This makes it possible to introduce a well defined, gauge invariant spatial
angular momentum at null infinity, which reduces to the standard expression in
stationary spacetimes. The general formula for the angular momentum flux
carried away be the gravitational radiation is also derived.Comment: 34 pages, typos corrected, four references added, appearing in Class.
Quantum Gra
Quasi-local energy-momentum and two-surface characterization of the pp-wave spacetimes
In the present paper the determination of the {\it pp}-wave metric form the
geometry of certain spacelike two-surfaces is considered. It has been shown
that the vanishing of the Dougan--Mason quasi-local mass , associated
with the smooth boundary of a spacelike
hypersurface , is equivalent to the statement that the Cauchy
development is of a {\it pp}-wave type geometry with pure
radiation, provided the ingoing null normals are not diverging on and the
dominant energy condition holds on . The metric on
itself, however, has not been determined. Here, assuming that the matter is a
zero-rest-mass-field, it is shown that both the matter field and the {\it
pp}-wave metric of are completely determined by the value of the
zero-rest-mass-field on and the two dimensional Sen--geometry of
provided a convexity condition, slightly stronger than above, holds. Thus the
{\it pp}-waves can be characterized not only by the usual Cauchy data on a {\it
three} dimensional but by data on its {\it two} dimensional boundary
too. In addition, it is shown that the Ludvigsen--Vickers quasi-local
angular momentum of axially symmetric {\it pp}-wave geometries has the familiar
properties known for pure (matter) radiation.Comment: 15 pages, Plain Tex, no figure
Gravitational energy and cosmic acceleration
Cosmic acceleration is explained quantitatively, as an apparent effect due to
gravitational energy differences that arise in the decoupling of bound systems
from the global expansion of the universe. "Dark energy" is a misidentification
of those aspects of gravitational energy which by virtue of the equivalence
principle cannot be localised, namely gradients in the energy due to the
expansion of space and spatial curvature variations in an inhomogeneous
universe. A new scheme for cosmological averaging is proposed which solves the
Sandage-de Vaucouleurs paradox. Concordance parameters fit supernovae
luminosity distances, the angular scale of the sound horizon in the CMB
anisotropies, and the effective comoving baryon acoustic oscillation scale seen
in galaxy clustering statistics. Key observational anomalies are potentially
resolved, and unique predictions made, including a quantifiable variance in the
Hubble flow below the scale of apparent homogeneity.Comment: 9 pages, 2 figures. An essay which received Honorable Mention in the
2007 GRF Essay Competition. To appear in a special issue of Int. J. Mod.
Phys.
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