173 research outputs found
Perfect fluids and generic spacelike singularities
We present the conformally 1+3 Hubble-normalized field equations together
with the general total source equations, and then specialize to a source that
consists of perfect fluids with general barotropic equations of state.
Motivating, formulating, and assuming certain conjectures, we derive results
about how the properties of fluids (equations of state, momenta, angular
momenta) and generic spacelike singularities affect each other.Comment: Considerable changes have been made in presentation and arguments,
resulting in sharper conclusion
Monotonic functions in Bianchi models: Why they exist and how to find them
All rigorous and detailed dynamical results in Bianchi cosmology rest upon
the existence of a hierarchical structure of conserved quantities and monotonic
functions. In this paper we uncover the underlying general mechanism and derive
this hierarchical structure from the scale-automorphism group for an
illustrative example, vacuum and diagonal class A perfect fluid models. First,
kinematically, the scale-automorphism group leads to a reduced dynamical system
that consists of a hierarchy of scale-automorphism invariant sets. Second, we
show that, dynamically, the scale-automorphism group results in
scale-automorphism invariant monotone functions and conserved quantities that
restrict the flow of the reduced dynamical system.Comment: 26 pages, replaced to match published versio
A new proof of the Bianchi type IX attractor theorem
We consider the dynamics towards the initial singularity of Bianchi type IX
vacuum and orthogonal perfect fluid models with a linear equation of state. The
`Bianchi type IX attractor theorem' states that the past asymptotic behavior of
generic type IX solutions is governed by Bianchi type I and II vacuum states
(Mixmaster attractor). We give a comparatively short and self-contained new
proof of this theorem. The proof we give is interesting in itself, but more
importantly it illustrates and emphasizes that type IX is special, and to some
extent misleading when one considers the broader context of generic models
without symmetries.Comment: 26 pages, 5 figure
Dynamics of Bianchi type I elastic spacetimes
We study the global dynamical behavior of spatially homogeneous solutions of
the Einstein equations in Bianchi type I symmetry, where we use non-tilted
elastic matter as an anisotropic matter model that naturally generalizes
perfect fluids. Based on our dynamical systems formulation of the equations we
are able to prove that (i) toward the future all solutions isotropize; (ii)
toward the initial singularity all solutions display oscillatory behavior;
solutions do not converge to Kasner solutions but oscillate between different
Kasner states. This behavior is associated with energy condition violation as
the singularity is approached.Comment: 28 pages, 11 figure
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
Self-gravitating statinary spherically symmetric systems in relativistic galactic dynamics
We study equilibrium states in relativistic galactic dynamics which are described by solutions of the Einstein-Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses
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
Homoclinic chaos and energy condition violation
In this letter we discuss the connection between so-called homoclinic chaos
and the violation of energy conditions in locally rotationally symmetric
Bianchi type IX models, where the matter is assumed to be non-tilted dust and a
positive cosmological constant. We show that homoclinic chaos in these models
is an artifact of unphysical assumptions: it requires that there exist
solutions with positive matter energy density that evolve through the
singularity and beyond as solutions with negative matter energy density
. Homoclinic chaos is absent when it is assumed that the dust particles
always retain their positive mass.In addition, we discuss more general models:
for solutions that are not locally rotionally symmetric we demonstrate that the
construction of extensions through the singularity, which is required for
homoclinic chaos, is not possible in general.Comment: 4 pages, RevTe
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Sequential dependencies between trials in free choice tasks
In two previous experiments we investigated the neural precursors of subjects' "free" choices for one of two options (pressing one of two buttons, and choosing between adding and subtracting numbers). In these experiments the distribution of sequence lengths was taken as an approximate indicator of the randomness (or lack of sequential dependency) of the choice sequences. However, this method is limited in its ability to reveal sequential dependencies. Here we present a more detailed individual-subject analysis and conclude that despite of the presence of significant sequential dependencies the subjects' behavior still approximates randomness, as measured by an entropy rate (on pooled data) of 0.940 bit / trial and 0.965 bit / trial in the two experiments. We also provide the raw single-subject behavioral data
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