50 research outputs found
The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models
The purpose of this paper is to further investigate the solution space of
self-similar spherically symmetric perfect-fluid models and gain deeper
understanding of the physical aspects of these solutions. We achieve this by
combining the state space description of the homothetic approach with the use
of the physically interesting quantities arising in the comoving approach. We
focus on three types of models. First, we consider models that are natural
inhomogeneous generalizations of the Friedmann Universe; such models are
asymptotically Friedmann in their past and evolve fluctuations in the energy
density at later times. Second, we consider so-called quasi-static models. This
class includes models that undergo self-similar gravitational collapse and is
important for studying the formation of naked singularities. If naked
singularities do form, they have profound implications for the predictability
of general relativity as a theory. Third, we consider a new class of
asymptotically Minkowski self-similar spacetimes, emphasizing that some of them
are associated with the self-similar solutions associated with the critical
behaviour observed in recent gravitational collapse calculations.Comment: 24 pages, 12 figure
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
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
Convergence to a self-similar solution in general relativistic gravitational collapse
We study the spherical collapse of a perfect fluid with an equation of state
by full general relativistic numerical simulations. For 0, it has been known that there exists a general relativistic counterpart
of the Larson-Penston self-similar Newtonian solution. The numerical
simulations strongly suggest that, in the neighborhood of the center, generic
collapse converges to this solution in an approach to a singularity and that
self-similar solutions other than this solution, including a ``critical
solution'' in the black hole critical behavior, are relevant only when the
parameters which parametrize initial data are fine-tuned. This result is
supported by a mode analysis on the pertinent self-similar solutions. Since a
naked singularity forms in the general relativistic Larson-Penston solution for
0, this will be the most serious known counterexample against
cosmic censorship. It also provides strong evidence for the self-similarity
hypothesis in general relativistic gravitational collapse. The direct
consequence is that critical phenomena will be observed in the collapse of
isothermal gas in Newton gravity, and the critical exponent will be
given by , though the order parameter cannot be the black
hole mass.Comment: 22 pages, 15 figures, accepted for publication in Physical Review D,
reference added, typos correcte
Stability criterion for self-similar solutions with a scalar field and those with a stiff fluid in general relativity
A stability criterion is derived in general relativity for self-similar
solutions with a scalar field and those with a stiff fluid, which is a perfect
fluid with the equation of state . A wide class of self-similar
solutions turn out to be unstable against kink mode perturbation. According to
the criterion, the Evans-Coleman stiff-fluid solution is unstable and cannot be
a critical solution for the spherical collapse of a stiff fluid if we allow
sufficiently small discontinuity in the density gradient field in the initial
data sets. The self-similar scalar-field solution, which was recently found
numerically by Brady {\it et al.} (2002 {\it Class. Quantum. Grav.} {\bf 19}
6359), is also unstable. Both the flat Friedmann universe with a scalar field
and that with a stiff fluid suffer from kink instability at the particle
horizon scale.Comment: 15 pages, accepted for publication in Classical and Quantum Gravity,
typos correcte
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
Lax pair tensors and integrable spacetimes
The use of Lax pair tensors as a unifying framework for Killing tensors of
arbitrary rank is discussed. Some properties of the tensorial Lax pair
formulation are stated. A mechanical system with a well-known Lax
representation -- the three-particle open Toda lattice -- is geometrized by a
suitable canonical transformation. In this way the Toda lattice is realized as
the geodesic system of a certain Riemannian geometry. By using different
canonical transformations we obtain two inequivalent geometries which both
represent the original system. Adding a timelike dimension gives
four-dimensional spacetimes which admit two Killing vector fields and are
completely integrable.Comment: 10 pages, LaTe
Invariant construction of solutions to Einstein's field equations - LRS perfect fluids II
The properties of LRS class II perfect fluid space-times are analyzed using
the description of geometries in terms of the Riemann tensor and a finite
number of its covariant derivatives. In this manner it is straightforward to
obtain the plane and hyperbolic analogues to the spherical symmetric case. For
spherically symmetric static models the set of equations is reduced to the
Tolman-Oppenheimer-Volkoff equation only. Some new non-stationary and
inhomogeneous solutions with shear, expansion, and acceleration of the fluid
are presented. Among these are a class of temporally self-similar solutions
with equation of state given by , and a class of
solutions characterized by . We give an example of geometry
where the Riemann tensor and the Ricci rotation coefficients are not sufficient
to give a complete description of the geometry. Using an extension of the
method, we find the full metric in terms of curvature quantities.Comment: 24 pages, 1 figur
A complete classification of spherically symmetric perfect fluid similarity solutions
We classify all spherically symmetric perfect fluid solutions of Einstein's
equations with equation of state p/mu=a which are self-similar in the sense
that all dimensionless variables depend only upon z=r/t. For a given value of
a, such solutions are described by two parameters and they can be classified in
terms of their behaviour at large and small distances from the origin; this
usually corresponds to large and small values of z but (due to a coordinate
anomaly) it may also correspond to finite z. We base our analysis on the
demonstration that all similarity solutions must be asymptotic to solutions
which depend on either powers of z or powers of lnz. We show that there are
only three similarity solutions which have an exact power-law dependence on z:
the flat Friedmann solution, a static solution and a Kantowski-Sachs solution
(although the latter is probably only physical for a1/5, there are
also two families of solutions which are asymptotically (but not exactly)
Minkowski: the first is asymptotically Minkowski as z tends to infinity and is
described by one parameter; the second is asymptotically Minkowski at a finite
value of z and is described by two parameters. A complete analysis of the dust
solutions is given, since these can be written down explicitly and elucidate
the link between the z>0 and z<0 solutions. Solutions with pressure are then
discussed in detail; these share many of the characteristics of the dust
solutions but they also exhibit new features.Comment: 63 pages. To appear in Physical Review