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 $[n+1]$-dimensional (with $n\geq 3$) 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 $n+1$ 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 $F_{\alpha\beta}$. 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