667 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
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
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
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
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
On quasi-local Hamiltonians in General Relativity
We analyse the definition of quasi-local energy in GR based on a Hamiltonian
analysis of the Einstein-Hilbert action initiated by Brown-York. The role of
the constraint equations, in particular the Hamiltonian constraint on the
timelike boundary, neglected in previous studies, is emphasized here. We argue
that a consistent definition of quasi-local energy in GR requires, at a
minimum, a framework based on the (currently unknown) geometric well-posedness
of the initial boundary value problem for the Einstein equations.Comment: 9 page
On quasi-local charges and Newman--Penrose type quantities in Yang--Mills theories
We generalize the notion of quasi-local charges, introduced by P. Tod for
Yang--Mills fields with unitary groups, to non-Abelian gauge theories with
arbitrary gauge group, and calculate its small sphere and large sphere limits
both at spatial and null infinity. We show that for semisimple gauge groups no
reasonable definition yield conserved total charges and Newman--Penrose (NP)
type quantities at null infinity in generic, radiative configurations. The
conditions of their conservation, both in terms of the field configurations and
the structure of the gauge group, are clarified. We also calculate the NP
quantities for stationary, asymptotic solutions of the field equations with
vanishing magnetic charges, and illustrate these by explicit solutions with
various gauge groups.Comment: 22 pages, typos corrected, appearing in Classical and Quantum Gravit
On the roots of the Poincare structure of asymptotically flat spacetimes
The analysis of vacuum general relativity by R. Beig and N. O Murchadha (Ann.
Phys. vol 174, 463 (1987)) is extended in numerous ways. The weakest possible
power-type fall-off conditions for the energy-momentum tensor, the metric, the
extrinsic curvature, the lapse and the shift are determined, which, together
with the parity conditions, are preserved by the energy-momentum conservation
and the evolution equations. The algebra of the asymptotic Killing vectors,
defined with respect to a foliation of the spacetime, is shown to be the
Lorentz Lie algebra for slow fall-off of the metric, but it is the Poincare
algebra for 1/r or faster fall-off. It is shown that the applicability of the
symplectic formalism already requires the 1/r (or faster) fall-off of the
metric. The connection between the Poisson algebra of the Beig-O Murchadha
Hamiltonians and the asymptotic Killing vectors is clarified. The value H[K^a]
of their Hamiltonian is shown to be conserved in time if K^a is an asymptotic
Killing vector defined with respect to the constant time slices. The angular
momentum and centre-of-mass, defined by the value of H[K^a] for asymptotic
rotation-boost Killing vectors K^a, are shown to be finite only for 1/r or
faster fall-off of the metric. Our center-of-mass expression is the difference
of that of Beig and O Murchadha and the spatial momentum times the coordinate
time. The spatial angular momentum and this centre-of-mass form a Lorentz
tensor, which transforms in the correct way under Poincare transformations.Comment: 34 pages, plain TEX, misleading notations changed, discussion
improved and corrected, appearing in Class. Quantum Gra
Witten spinors on maximal, conformally flat hypersurfaces
The boundary conditions that exclude zeros of the solutions of the Witten
equation (and hence guarantee the existence of a 3-frame satisfying the
so-called special orthonormal frame gauge conditions) are investigated. We
determine the general form of the conformally invariant boundary conditions for
the Witten equation, and find the boundary conditions that characterize the
constant and the conformally constant spinor fields among the solutions of the
Witten equations on compact domains in extrinsically and intrinsically flat,
and on maximal, intrinsically globally conformally flat spacelike
hypersurfaces, respectively. We also provide a number of exact solutions of the
Witten equation with various boundary conditions (both at infinity and on inner
or outer boundaries) that single out nowhere vanishing spinor fields on the
flat, non-extreme Reissner--Nordstr\"om and Brill--Lindquist data sets. Our
examples show that there is an interplay between the boundary conditions, the
global topology of the hypersurface and the existence/non-existence of zeros of
the solutions of the Witten equation.Comment: 23 pages, typos corrected, final version, accepted in Class. Quantum
Gra
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