2,978 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 the role of conformal three-geometries in the dynamics of General Relativity
It is shown that the Chern-Simons functional, built in the spinor
representation from the initial data on spacelike hypersurfaces, is invariant
with respect to infinitesimal conformal rescalings if and only if the vacuum
Einstein equations are satisfied. As a consequence, we show that in the phase
space the Hamiltonian constraint of vacuum general relativity is the Poisson
bracket of the imaginary part of this Chern-Simons functional and Misner's time
(essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the
condition on the canonical variables that the imaginary part of the Chern-
Simons functional be constant along the volume flow. The vacuum momentum
constraint can also be reformulated in a similar way as a (more complicated)
condition on the change of the imaginary part of the Chern-Simons functional
along the flow of York's time.Comment: 15 pages, plain Te
On the total mass of closed universes
The total mass, the Witten type gauge conditions and the spectral properties
of the Sen-Witten and the 3-surface twistor operators in closed universes are
investigated. It has been proven that a recently suggested expression
for the total mass density of closed universes is vanishing if and only if the
spacetime is flat with toroidal spatial topology; it coincides with the first
eigenvalue of the Sen-Witten operator; and it is vanishing if and only if
Witten's gauge condition admits a non-trivial solution.
Here we generalize slightly the result above on the zero-mass configurations:
if and only if the spacetime is holonomically trivial with toroidal
spatial topology. Also, we show that the multiplicity of the eigenvalues of the
(square of the) Sen-Witten operator is at least two, and a potentially viable
gauge condition is suggested. The monotonicity properties of through
the examples of closed Bianchi I and IX cosmological spacetimes are also
discussed. A potential spectral characterization of these cosmological
spacetimes, in terms of the spectrum of the Riemannian Dirac operator and the
Sen-Witten and the 3-surface twistor operators, is also indicated.Comment: 14 pages, plenary talk at the `Spanish Relativity Meeting in Portugal
2012', Guimar\~aes 3-7 September; Final version, appearing in General
Relativity and Gravitatio
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