35 research outputs found
Multi-body spherically symmetric steady states of Newtonian self-gravitating elastic matter
We study the problem of static, spherically symmetric, self-gravitating
elastic matter distributions in Newtonian gravity. To this purpose we first
introduce a new definition of homogeneous, spherically symmetric (hyper)elastic
body in Euler coordinates, i.e., in terms of matter fields defined on the
current physical state of the body. We show that our definition is equivalent
to the classical one existing in the literature and which is given in
Lagrangian coordinates, i.e., in terms of the deformation of the body from a
given reference state. After a number of well-known examples of constitutive
functions of elastic bodies are re-defined in our new formulation, a detailed
study of the Seth model is presented. For this type of material the existence
of single and multi-body solutions is established.Comment: 33 pages, 1 figure. v2 matches final published versio
On dynamical systems approaches and methods in cosmology
We discuss dynamical systems approaches and methods applied to flat
Robertson-Walker models in -gravity. We argue that a complete description
of the solution space of a model requires a global state space analysis that
motivates globally covering state space adapted variables. This is shown
explicitly by an illustrative example, , ,
for which we introduce new regular dynamical systems on global compactly
extended state spaces for the Jordan and Einstein frames. This example also
allows us to illustrate several local and global dynamical systems techniques
involving, e.g., blow ups of nilpotent fixed points, center manifold analysis,
averaging, and use of monotone functions. As a result of applying dynamical
systems methods to globally state space adapted dynamical systems formulations,
we obtain pictures of the entire solution spaces in both the Jordan and the
Einstein frames. This shows, e.g., that due to the domain of the conformal
transformation between the Jordan and Einstein frames, not all the solutions in
the Jordan frame are completely contained in the Einstein frame. We also make
comparisons with previous dynamical systems approaches to cosmology and
discuss their advantages and disadvantages.Comment: 36 pages, 7 figures. v2: references added, matches published versio
Static self-gravitating Newtonian elastic balls
The existence of static self-gravitating Newtonian elastic balls is proved
under general assumptions on the constitutive equations of the elastic
material. The proof uses methods from the theory of finite-dimensional
dynamical systems and the Euler formulation of elasticity theory for
spherically symmetric bodies introduced recently by the authors. Examples of
elastic materials covered by the results of this paper are Saint
Venant-Kirchhoff, John and Hadamard materials.Comment: 30 pages, 2 figures. The order of presentation of the results and the
notation have been changed considerably to improve the reading flow of the
article. Some assumptions and theorems have been reformulated in a more clear
way and several new remarks have been adde
Quintessential -attractor inflation: A dynamical systems analysis
The equations for quintessential -attractor inflation with a single
scalar field, radiation and matter in a spatially flat FLRW spacetime are
recast into a regular dynamical system on a compact state space. This enables a
complete description of the solution space of these models. The inflationary
attractor solution is shown to correspond to the unstable center manifold of a
de Sitter fixed point, and we describe connections between slow-roll and
dynamical systems approximations for this solution, including Pad\'e
approximants. We also introduce a new method for systematically obtaining
initial data for quintessence evolution by using dynamical systems properties;
in particular, this method exploits that there exists a radiation dominated
line of fixed points with an unstable quintessence attractor submanifold, which
plays a role that is reminiscent of that of the inflationary attractor solution
for inflation.Comment: 34 pages, 23 figure
Spherical linear waves in de Sitter spacetime
We apply Christodoulou's framework, developed to study the Einstein-scalar
field equations in spherical symmetry, to the linear wave equation in de Sitter
spacetime, as a first step towards the Einstein-scalar field equations with
positive cosmological constant. We obtain an integro-differential evolution
equation which we solve by taking initial data on a null cone. As a corollary
we obtain elementary derivations of expected properties of linear waves in de
Sitter spacetime: boundedness in terms of (characteristic) initial data, and a
Price law establishing uniform exponential decay, in Bondi time, to a constant.Comment: 9 pages, 1 figure; v2: minor changes, references added, matches final
published versio
The Einstein-Friedrich-nonlinear scalar field system and the stability of scalar field Cosmologies
A frame representation is used to derive a first order quasi-linear symmetric
hyperbolic system for a scalar field minimally coupled to gravity. This
procedure is inspired by similar evolution equations introduced by Friedrich to
study the Einstein-Euler system. The resulting evolution system is used to show
that small nonlinear perturbations of expanding
Friedman-Lema\^itre-Robertson-Walker backgrounds, with scalar field potentials
satisfying certain future asymptotic conditions, decay exponentially to zero,
in synchronous time.Comment: Version 4: Matches final published versio