4,077 research outputs found
On the evolutionary form of the constraints in electrodynamics
The constraint equations in Maxwell theory are investigated. In analogy with
some recent results on the constraints of general relativity it is shown,
regardless of the signature and dimension of the ambient space, that the
"divergence of a vector field" type constraints can always be put into linear
first order hyperbolic form for which global existence and uniqueness of
solutions to an initial-boundary value problem is guaranteed.Comment: 10 pages, 1 figure; The published version contains several updates of
former one. The introduction is extended, and new sections with an explicit
example and with concluding remarks had also been adde
Constraints as evolutionary systems
The constraint equations for smooth -dimensional (with )
Riemannian or Lorentzian spaces satisfying the Einstein field equations are
considered. It is shown, regardless of the signature of the primary space, that
the constraints can be put into the form of an evolutionary system comprised
either by a first order symmetric hyperbolic system and a parabolic equation
or, alternatively, by a symmetrizable hyperbolic system and a subsidiary
algebraic relation. In both cases the (local) existence and uniqueness of
solutions are also discussed.Comment: 18 pages; exposition improved concerning the algebraic hyperbolic
system; references added; to appear in CQ
On rigidity of spacetimes with a compact Cauchy horizon
Smooth spacetimes with a compact Cauchy horizon ruled by closed null
geodesics are considered. The compact Cauchy horizon is assumed to be
non-degenerate. Then, supporting the validity of Penrose's strong cosmic censor
hypothesis, the existence of a smooth Killing vector field in a neighbourhood
of the horizon on the Cauchy development side is shown.Comment: 2 pages, contribution to the 9th Marcel Grossmann meeting (MG9),
Rome, July 200
Is the Bianchi identity always hyperbolic?
We consider dimensional smooth Riemannian and Lorentzian spaces
satisfying Einstein's equations. The base manifold is assumed to be smoothly
foliated by a one-parameter family of hypersurfaces. In both cases---likewise
it is usually done in the Lorentzian case---Einstein's equations may be split
into `Hamiltonian' and `momentum' constraints and a `reduced' set of field
equations. It is shown that regardless whether the primary space is Riemannian
or Lorentzian whenever the foliating hypersurfaces are Riemannian the
`Hamiltonian' and `momentum' type expressions are subject to a subsidiary first
order symmetric hyperbolic system. Since this subsidiary system is linear and
homogeneous in the `Hamiltonian' and `momentum' type expressions the
hyperbolicity of the system implies that in both cases the solutions to the
`reduced' set of field equations are also solutions to the full set of
equations provided that the constraints hold on one of the hypersurfaces
foliating the base manifold.Comment: 14 pages, no figures, the published versio
A simple method of constructing binary black hole initial data
By applying a parabolic-hyperbolic formulation of the constraints and
superposing Kerr-Schild black holes, a simple method is introduced to
initialize time evolution of binary systems. As the input parameters are
essentially the same as those used in the post-Newtonian (PN) setup the
proposed method interrelates various physical expressions applied in PN and in
fully relativistic formulations. The global ADM charges are also determined by
the input parameters, and no use of boundary conditions in the strong field
regime is made.Comment: Substantial simplification of the main argument. Supplemental
material available at http://www.kfki.hu/~iracz/SM-BH-data.pd
Black hole initial data without elliptic equations
We explore whether a new method to solve the constraints of Einstein's
equations, which does not involve elliptic equations, can be applied to provide
initial data for black holes. We show that this method can be successfully
applied to a nonlinear perturbation of a Schwarzschild black hole by
establishing the well-posedness of the resulting constraint problem. We discuss
its possible generalization to the boosted, spinning multiple black hole
problem
Superradiance or total reflection?
Numerical evolution of massless scalar fields on Kerr background is studied.
The initial data specifications are chosen to have compact support separated
from the ergoregion and to yield nearly monochromatic incident wave packets.
The initial data is also tuned to maximize the effect of superradiance.
Evidences are shown indicating that instead of the anticipated energy
extraction from black hole the incident radiation fail to reach the ergoregion
rather it suffers a nearly perfect reflection.Comment: 8 pages, 6 figures, contribution to the proceedings of the conference
on Relativity and Gravitation: 100 Years after Einstein in Pragu
On the use of projection operators in electrodynamics
In classical electrodynamics all the measurable quantities can be derived
from the gauge invariant Faraday tensor . Nevertheless, it is
often advantageous to work with gauge dependent variables. In [4],[2] and [8],
and in the present note too, the transformation of the vector potential in
Lorenz gauge to that in Coulomb gauge is considered. This transformation can be
done by applying a projection operator that extracts the transverse part of
spatial vectors. In many circumstances the proper projection operator is
replaced by a simplified transverse one. It is widely held that such a
replacement does not affect the result in the radiation zone. In this paper the
action of the proper and simplified transverse projections will be compared by
making use of specific examples of a moving point charge. It will be
demonstrated that whenever the interminable spatial motion of the source is
unbounded with respect to the reference frame of the observer the replacement
of the proper projection operator by the simplified transverse one yields, even
in the radiation zone, an erroneous result with error which is of the same
order as the proper Coulomb gauge vector potential itself.Comment: 15 pages, no figures, matched to the published versio
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