20 research outputs found
Charged black holes in compactified spacetimes
We construct and investigate a compactified version of the four-dimensional
Reissner-Nordstrom-NUT solution, generalizing the compactified Schwarzschild
black hole that has been previously studied by several workers. Our approach to
compactification is based on dimensional reduction with respect to the
stationary Killing vector, resulting in three-dimensional gravity coupled to a
nonlinear sigma model. Using that the original non-compactified solution
corresponds to a target space geodesic, the problem can be linearized much in
the same way as in the case of no electric nor NUT charge. An interesting
feature of the solution family is that for nonzero electric charge but
vanishing NUT charge, the solution has a curvature singularity on a torus that
surrounds the event horizon, but this singularity is removed when the NUT
charge is switched on. We also treat the Schwarzschild case in a more complete
way than has been done previously. In particular, the asymptotic solution (the
Levi-Civita solution with the height coordinate made periodic) has to our
knowledge only been calculated up to a determination of the mass parameter. The
periodic Levi-Civita solution contains three essential parameters, however, and
the remaining two are explicitly calculated here.Comment: 20 pages, 3 figures. v2: Typo corrected, reference adde
A unified treatment of cubic invariants at fixed and arbitrary energy
Cubic invariants for two-dimensional Hamiltonian systems are investigated
using the Jacobi geometrization procedure. This approach allows for a unified
treatment of invariants at both fixed and arbitrary energy. In the geometric
picture the invariant generally corresponds to a third rank Killing tensor,
whose existence at a fixed energy value forces the metric to satisfy a
nonlinear integrability condition expressed in terms of a Kahler potential.
Further conditions, leading to a system of equations which is overdetermined
except for singular cases, are added when the energy is arbitrary. As solutions
to these equations we obtain several new superintegrable cases in addition to
the previously known cases. We also discover a superintegrable case where the
cubic invariant is of a new type which can be represented by an energy
dependent linear invariant. A complete list of all known systems which admit a
cubic invariant at arbitrary energy is given.Comment: 16 pages, LaTeX2e, slightly revised version. To appear in J. Math.
Phys. vol 41, pp 370-384 (2000
Elastic Stars in General Relativity: II. Radial perturbations
We study radial perturbations of general relativistic stars with elastic
matter sources. We find that these perturbations are governed by a second order
differential equation which, along with the boundary conditions, defines a
Sturm-Liouville type problem that determines the eigenfrequencies. Although
some complications arise compared to the perfect fluid case, leading us to
consider a generalisation of the standard form of the Sturm-Liouville equation,
the main results of Sturm-Liouville theory remain unaltered. As an important
consequence we conclude that the mass-radius curve for a one-parameter sequence
of regular equilibrium models belonging to some particular equation of state
can be used in the same well-known way as in the perfect fluid case, at least
if the energy density and the tangential pressure of the background solutions
are continuous. In particular we find that the fundamental mode frequency has a
zero for the maximum mass stars of the models with solid crusts considered in
Paper I of this series.Comment: 22 pages, no figures, final version accepted for publication in
Class. Quantum Grav. The treatment of the junction conditions has been
improve
Lax pair tensors in arbitrary dimensions
A recipe is presented for obtaining Lax tensors for any n-dimensional
Hamiltonian system admitting a Lax representation of dimension n. Our approach
is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a
geometric Lax formulation. We also exploit the results to construct integrable
spacetimes, satisfying the weak energy condition.Comment: 8 pages, uses IOP style files. Minor correction. Submitted to J. Phys
Third rank Killing tensors in general relativity. The (1+1)-dimensional case
Third rank Killing tensors in (1+1)-dimensional geometries are investigated
and classified. It is found that a necessary and sufficient condition for such
a geometry to admit a third rank Killing tensor can always be formulated as a
quadratic PDE, of order three or lower, in a Kahler type potential for the
metric. This is in contrast to the case of first and second rank Killing
tensors for which the integrability condition is a linear PDE. The motivation
for studying higher rank Killing tensors in (1+1)-geometries, is the fact that
exact solutions of the Einstein equations are often associated with a first or
second rank Killing tensor symmetry in the geodesic flow formulation of the
dynamics. This is in particular true for the many models of interest for which
this formulation is (1+1)-dimensional, where just one additional constant of
motion suffices for complete integrability. We show that new exact solutions
can be found by classifying geometries admitting higher rank Killing tensors.Comment: 16 pages, LaTe
Axial perturbations of general spherically symmetric spacetimes
The aim of this paper is to present a governing equation for first order
axial metric perturbations of general, not necessarily static, spherically
symmetric spacetimes. Under the non-restrictive assumption of axisymmetric
perturbations, the governing equation is shown to be a two-dimensional wave
equation where the wave function serves as a twist potential for the
axisymmetry generating Killing vector. This wave equation can be written in a
form which is formally a very simple generalization of the Regge-Wheeler
equation governing the axial perturbations of a Schwarzschild black hole, but
in general the equation is accompanied by a source term related to matter
perturbations. The case of a viscous fluid is studied in particular detail.Comment: 16 pages, no figures, minor correction