193 research outputs found
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
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
The initial singularity of ultrastiff perfect fluid spacetimes without symmetries
We consider the Einstein equations coupled to an ultrastiff perfect fluid and
prove the existence of a family of solutions with an initial singularity whose
structure is that of explicit isotropic models. This family of solutions is
`generic' in the sense that it depends on as many free functions as a general
solution, i.e., without imposing any symmetry assumptions, of the
Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.Comment: 16 pages, journal 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
On Static n-body Configurations in Relativity
The static n-body problem of General Relativity states that there are, under
a reasonable energy condition, no static -body configurations for ,
provided the configuration of the bodies satisfies a suitable separation
condition. In this paper we solve this problem in the case that there exists a
closed, noncompact, totally geodesic surface disjoint from the bodies. This
covers the situation where the configuration has a reflection symmetry across a
noncompact surface disjoint from the bodies.Comment: 10 pages; result generalized to allow for more than one
asymptotically flat en
Near-inertial wave scattering by random flows
The impact of a turbulent flow on wind-driven oceanic near-inertial waves is
examined using a linearised shallow-water model of the mixed layer. Modelling
the flow as a homogeneous and stationary random process with spatial scales
comparable to the wavelengths, we derive a transport (or kinetic) equation
governing wave-energy transfers in both physical and spectral spaces. This
equation describes the scattering of the waves by the flow which results in a
redistribution of energy between waves with the same frequency (or,
equivalently, with the same wavenumber) and, for isotropic flows, in the
isotropisation of the wave field. The time scales for the scattering and
isotropisation are obtained explicitly and found to be of the order of tens of
days for typical oceanic parameters. The predictions inferred from the
transport equation are confirmed by a series of numerical simulations.
Two situations in which near-inertial waves are strongly influenced by flow
scattering are investigated through dedicated nonlinear shallow-water
simulations. In the first, a wavepacket propagating equatorwards as a result
from the -effect is shown to be slowed down and dispersed both zonally
and meridionally by scattering. In the second, waves generated by moving
cyclones are shown to be strongly disturbed by scattering, leading again to an
increased dispersion.Comment: Accepted for publication in Phys. Rev. Fluid
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
Late-time behaviour of the Einstein-Vlasov system with Bianchi I symmetry
The late-time behaviour of the Einstein-dust system is well understood for
homogeneous spacetimes. For the case of Bianchi I we have been able to show
that the late-time behaviour of the Einstein-Vlasov system is well approximated
by the Einstein-dust system assuming that one is close to the unique stationary
solution which is the attractor of the Einstein-dust system.Comment: 4 pages, based on a talk given at the Spanish Relativity Meeting
2010, to appear in Journal of Physics: Conference Series (JPCS
- …