260 research outputs found
Spherically symmetric relativistic stellar structures
We investigate relativistic spherically symmetric static perfect fluid models
in the framework of the theory of dynamical systems. The field equations are
recast into a regular dynamical system on a 3-dimensional compact state space,
thereby avoiding the non-regularity problems associated with the
Tolman-Oppenheimer-Volkoff equation. The global picture of the solution space
thus obtained is used to derive qualitative features and to prove theorems
about mass-radius properties. The perfect fluids we discuss are described by
barotropic equations of state that are asymptotically polytropic at low
pressures and, for certain applications, asymptotically linear at high
pressures. We employ dimensionless variables that are asymptotically homology
invariant in the low pressure regime, and thus we generalize standard work on
Newtonian polytropes to a relativistic setting and to a much larger class of
equations of state. Our dynamical systems framework is particularly suited for
numerical computations, as illustrated by several numerical examples, e.g., the
ideal neutron gas and examples that involve phase transitions.Comment: 23 pages, 25 figures (compressed), LaTe
Dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations which are locally rotationally symmetric
The dynamics of a class of cosmological models with collisionless matter and
four Killing vectors is studied in detail and compared with that of
corresponding perfect fluid models. In many cases it is possible to identify
asymptotic states of the spacetimes near the singularity or in a phase of
unlimited expansion. Bianchi type II models show oscillatory behaviour near the
initial singularity which is, however, simpler than that of the mixmaster
model.Comment: 27 pages, 3 figures, LaTe
Matter and dynamics in closed cosmologies
To systematically analyze the dynamical implications of the matter content in
cosmology, we generalize earlier dynamical systems approaches so that perfect
fluids with a general barotropic equation of state can be treated. We focus on
locally rotationally symmetric Bianchi type IX and Kantowski-Sachs orthogonal
perfect fluid models, since such models exhibit a particularly rich dynamical
structure and also illustrate typical features of more general cases. For these
models, we recast Einstein's field equations into a regular system on a compact
state space, which is the basis for our analysis. We prove that models expand
from a singularity and recollapse to a singularity when the perfect fluid
satisfies the strong energy condition. When the matter source admits Einstein's
static model, we present a comprehensive dynamical description, which includes
asymptotic behavior, of models in the neighborhood of the Einstein model; these
results make earlier claims about ``homoclinic phenomena and chaos'' highly
questionable. We also discuss aspects of the global asymptotic dynamics, in
particular, we give criteria for the collapse to a singularity, and we describe
when models expand forever to a state of infinite dilution; possible initial
and final states are analyzed. Numerical investigations complement the
analytical results.Comment: 23 pages, 24 figures (compressed), LaTe
An almost isotropic cosmic microwave temperature does not imply an almost isotropic universe
In this letter we will show that, contrary to what is widely believed, an
almost isotropic cosmic microwave background (CMB) temperature does not imply
that the universe is ``close to a Friedmann-Lemaitre universe''. There are two
important manifestations of anisotropy in the geometry of the universe, (i) the
anisotropy in the overall expansion, and (ii) the intrinsic anisotropy of the
gravitational field, described by the Weyl curvature tensor, although the
former usually receives more attention than the latter in the astrophysical
literature. Here we consider a class of spatially homogeneous models for which
the anisotropy of the CMB temperature is within the current observational
limits but whose Weyl curvature is not negligible, i.e. these models are not
close to isotropy even though the CMB temperature is almost isotropic.Comment: 5 pages (AASTeX, aaspp4.sty), submitted to Astrophysical Journal
Letter
Spatially self-similar spherically symmetric perfect-fluid models
Einstein's field equations for spatially self-similar spherically symmetric
perfect-fluid models are investigated. The field equations are rewritten as a
first-order system of autonomous differential equations. Dimensionless
variables are chosen in such a way that the number of equations in the coupled
system is reduced as far as possible and so that the reduced phase space
becomes compact and regular. The system is subsequently analysed qualitatively
with the theory of dynamical systems.Comment: 21 pages, 6 eps-figure
Tilted two-fluid Bianchi type I models
In this paper we investigate expanding Bianchi type I models with two tilted
fluids with the same linear equation of state, characterized by the equation of
state parameter w. Individually the fluids have non-zero energy fluxes w.r.t.
the symmetry surfaces, but these cancel each other because of the Codazzi
constraint. We prove that when w=0 the model isotropizes to the future. Using
numerical simulations and a linear analysis we also find the asymptotic states
of models with w>0. We find that future isotropization occurs if and only if . The results are compared to similar models investigated previously
where the two fluids have different equation of state parameters.Comment: 14 pages, 3 figure
Theory of Newtonian self-gravitating stationary spherically symmetric systems
We investigate spherically symmetric equilibrium states of the Vlasov-Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure.Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support
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
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