66 research outputs found
A model problem for the initial-boundary value formulation of Einstein's field equations
In many numerical implementations of the Cauchy formulation of Einstein's
field equations one encounters artificial boundaries which raises the issue of
specifying boundary conditions. Such conditions have to be chosen carefully. In
particular, they should be compatible with the constraints, yield a well posed
initial-boundary value formulation and incorporate some physically desirable
properties like, for instance, minimizing reflections of gravitational
radiation.
Motivated by the problem in General Relativity, we analyze a model problem,
consisting of a formulation of Maxwell's equations on a spatially compact
region of spacetime with timelike boundaries. The form in which the equations
are written is such that their structure is very similar to the
Einstein-Christoffel symmetric hyperbolic formulations of Einstein's field
equations. For this model problem, we specify a family of Sommerfeld-type
constraint-preserving boundary conditions and show that the resulting
initial-boundary value formulations are well posed. We expect that these
results can be generalized to the Einstein-Christoffel formulations of General
Relativity, at least in the case of linearizations about a stationary
background.Comment: 25 page
Radiation from accelerated black holes in an anti-de Sitter universe
We study gravitational and electromagnetic radiation generated by uniformly
accelerated charged black holes in anti-de Sitter spacetime. This is described
by the C-metric exact solution of the Einstein-Maxwell equations with a
negative cosmological constant Lambda. We explicitly find and interpret the
pattern of radiation that characterizes the dependence of the fields on a null
direction from which the (timelike) conformal infinity is approached. This
directional pattern exhibits specific properties which are more complicated if
compared with recent analogous results obtained for asymptotic behavior of
fields near a de Sitter-like infinity. In particular, for large acceleration
the anti-de Sitter-like infinity is divided by Killing horizons into several
distinct domains with a different structure of principal null directions, in
which the patterns of radiation differ.Comment: 19 pages, 11 colour figures, submitted to Phys. Rev. D [Low quality
figures are included in this version because of arXive size restrictions. The
version with the standard quality figures is available at
http://utf.mff.cuni.cz/~podolsky/jppubl.htm.
Twistor Bundles, Einstein Equations and Real Structures
We consider sphere bundles P and P' of totally null planes of maximal
dimension and opposite self-duality over a 4-dimensional manifold equipped with
a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be
appropriate for the encoding of both the selfdual and the Einstein-Weyl
equations for the 4-metric. This encoding is realized in terms of the
properties of certain well defined geometrical objects on PP'. The formulation
is suitable for both complex- and real-valued metrics. It unifies results for
all three possible real signatures. In the purely Riemannian positive definite
case it implies the existence of a natural almost hermitian structure on PP'
whose integrability conditions correspond to the self-dual Einstein equations
of the 4-metric. All Einstein equations for the 4-metric are also encoded in
the properties of this almost hermitian structure on PP'.Comment: Paper accepted in Classical and Quantum Gravity, Special issue in
honour of Professor Andrzej Trautma
The Raychaudhuri equations: a brief review
We present a brief review on the Raychaudhuri equations. Beginning with a
summary of the essential features of the original article by Raychaudhuri and
subsequent work of numerous authors, we move on to a discussion of the
equations in the context of alternate non--Riemannian spacetimes as well as
other theories of gravity, with a special mention on the equations in
spacetimes with torsion (Einstein--Cartan--Sciama--Kibble theory). Finally, we
give an overview of some recent applications of these equations in General
Relativity, Quantum Field Theory, String Theory and the theory of relativisitic
membranes. We conclude with a summary and provide our own perspectives on
directions of future research.Comment: 35 pages, two figures, to appear in the special issue of Pramana
dedicated to the memory of A. K. Raychaudhur
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