187 research outputs found
Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups; Part 1: Two Killing fields spanned on u^{\alpha} and w^{\alpha }
For a rotating dust with a 3-dimensional symmetry group all possible metric
forms can be classified and, within each class, explicitly written out. This is
made possible by the formalism of Pleba\'nski based on the Darboux theorem. In
the resulting coordinates, the Killing vector fields (if any exist) assume a
special form. Each Killing vector field may be either spanned on the fields of
velocity and rotation or linearly independent of them. By considering all such
cases one arrives at the classification. With respect to the structures of the
groups, this is just the Bianchi classification, but with all possible
orientations of the orbits taken into account. In this paper, which is part 1
of a 3-part series, all solutions are considered for which two Killing fields
are spanned on velocity and rotation. The solutions of Lanczos and G\"{o}del
are identified as special cases, and their new invariant definitions are
provided. In addition, a new invariant definition is given of the Ozsvath class
III solution.Comment: 23 pages, LaTe
Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups, Part 3: All Killing fields linearly independent of u^{\alpha} and w^{\alpha}
This is the third and last part of a series of 3 papers. Using the same
method and the same coordinates as in parts 1 and 2, rotating dust solutions of
Einstein's equations are investigated that possess 3-dimensional symmetry
groups, under the assumption that each of the Killing vectors is linearly
independent of velocity and rotation at every point
of the spacetime region under consideration. The Killing fields are found and
the Killing equations are solved for the components of the metric tensor in
every case that arises. No progress was made with the Einstein equations in any
of the cases, and no previously known solutions were identified. A brief
overview of literature on solutions with rotating sources is given.Comment: One missing piece, signaled after eq. (10.7), is added after (10.21).
List of corrections: In (3.7) wrong subscript in vorticity; In (3.10) wrong
subscript in last term of g_{23}; In (4.23) wrong formulae for g_{12} and
g_{22}; In (7.17) missing factor in velocity; In (7.18) one wrong factor in
g_{22}; In (10.9) factor in vorticity; In (10.15) - (10.20) y_0 = 0; In
(10.20) wrong second term in y. The rewriting typos did not influence result
Alternative Methods of Describing Structure Formation in the Lemaitre-Tolman Model
We describe several new ways of specifying the behaviour of Lemaitre-Tolman
(LT) models, in each case presenting the method for obtaining the LT arbitrary
functions from the given data, and the conditions for existence of such
solutions. In addition to our previously considered `boundary conditions', the
new ones include: a simultaneous big bang, a homogeneous density or velocity
distribution in the asymptotic future, a simultaneous big crunch, a
simultaneous time of maximal expansion, a chosen density or velocity
distribution in the asymptotic future, only growing or only decaying
fluctuations. Since these conditions are combined in pairs to specify a
particular model, this considerably increases the possible ways of designing LT
models with desired properties.Comment: Accepted by Phys Rev D. RevTeX 4, 13 pages, no figures. Part of a
series: gr-qc/0106096, gr-qc/0303016, gr-qc/0309119. Replacement contains
very minor correction
Is the shell-focusing singularity of Szekeres space-time visible?
The visibility of the shell-focusing singularity in Szekeres space-time -
which represents quasi-spherical dust collapse - has been studied on numerous
occasions in the context of the cosmic censorship conjecture. The various
results derived have assumed that there exist radial null geodesics in the
space-time. We show that such geodesics do not exist in general, and so
previous results on the visibility of the singularity are not generally valid.
More precisely, we show that the existence of a radial geodesic in Szekeres
space-time implies that the space-time is axially symmetric, with the geodesic
along the polar direction (i.e. along the axis of symmetry). If there is a
second non-parallel radial geodesic, then the space-time is spherically
symmetric, and so is a Lema\^{\i}tre-Tolman-Bondi (LTB) space-time. For the
case of the polar geodesic in an axially symmetric Szekeres space-time, we give
conditions on the free functions (i.e. initial data) of the space-time which
lead to visibility of the singularity along this direction. Likewise, we give a
sufficient condition for censorship of the singularity. We point out the
complications involved in addressing the question of visibility of the
singularity both for non-radial null geodesics in the axially symmetric case
and in the general (non-axially symmetric) case, and suggest a possible
approach.Comment: 10 page
Geometry of the quasi-hyperbolic Szekeres models
Geometric properties of the quasi-hyperbolic Szekeres models are discussed
and related to the quasi-spherical Szekeres models. Typical examples of shapes
of various classes of 2-dimensional coordinate surfaces are shown in graphs;
for the hyperbolically symmetric subcase and for the general quasi-hyperbolic
case. An analysis of the mass function is carried out in parallel to an
analogous analysis for the quasi-spherical models. This leads to the conclusion
that determines the density of rest mass averaged over the whole space
of constant time.Comment: 19 pages, 13 figures. This version matches the published tex
Can a charged dust ball be sent through the Reissner--Nordstr\"{o}m wormhole?
In a previous paper we formulated a set of necessary conditions for the
spherically symmetric weakly charged dust to avoid Big Bang/Big Crunch, shell
crossing and permanent central singularities. However, we did not discuss the
properties of the energy density, some of which are surprising and seem not to
have been known up to now. A singularity of infinite energy density does exist
-- it is a point singularity situated on the world line of the center of
symmetry. The condition that no mass shell collapses to if it had initially thus turns out to be still insufficient for avoiding a
singularity. Moreover, at the singularity the energy density is
direction-dependent: when we approach the singular
point along a const hypersurface and when we
approach that point along the center of symmetry. The appearance of
negative-energy-density regions turns out to be inevitable. We discuss various
aspects of this property of our configuration. We also show that a permanently
pulsating configuration, with the period of pulsation independent of mass, is
possible only if there exists a permanent central singularity.Comment: 30 pages, 21 figures; several corrections after referee's comments, 4
figures modifie
Friedmann limits of rotating hypersurface-homogeneous dust models
The existence of Friedmann limits is systematically investigated for all the
hypersurface-homogeneous rotating dust models, presented in previous papers by
this author. Limiting transitions that involve a change of the Bianchi type are
included. Except for stationary models that obviously do not allow it, the
Friedmann limit expected for a given Bianchi type exists in all cases. Each of
the 3 Friedmann models has parents in the rotating class; the k = +1 model has
just one parent class, the other two each have several parent classes. The type
IX class is the one investigated in 1951 by Goedel. For each model, the
consecutive limits of zero rotation, zero tilt, zero shear and spatial isotropy
are explicitly calculated.Comment: 39 pages, LaTeX, 1 postscript figure. Subjects: General relativity,
exact solutions, cosmolog
Split structures in general relativity and the Kaluza-Klein theories
We construct a general approach to decomposition of the tangent bundle of
pseudo-Riemannian manifolds into direct sums of subbundles, and the associated
decomposition of geometric objects. An invariant structure {\cal H}^r defined
as a set of r projection operators is used to induce decomposition of the
geometric objects into those of the corresponding subbundles. We define the
main geometric objects characterizing decomposition. Invariant non-holonomic
generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All
the known types of decomposition (used in the theory of frames of reference, in
the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory
of stationary spaces, and so on) follow from the present work as special cases
when fixing a basis and dimensions of subbundles, and parameterization of a
basis of decomposition. Various methods of decomposition have been applied here
for the Unified Multidimensional Kaluza-Klein Theory and for relativistic
configurations of a perfect fluid. Discussing an invariant form of the
equations of motion we have found the invariant equilibrium conditions and
their 3+1 decomposed form. The formulation of the conservation law for the curl
has been obtained in the invariant form.Comment: 30 pages, RevTeX, aps.sty, some additions and corrections, new
references adde
First order perturbations of the Einstein-Straus and Oppenheimer-Snyder models
We derive the linearly perturbed matching conditions between a Schwarzschild
spacetime region with stationary and axially symmetric perturbations and a FLRW
spacetime with arbitrary perturbations. The matching hypersurface is also
perturbed arbitrarily and, in all cases, the perturbations are decomposed into
scalars using the Hodge operator on the sphere. This allows us to write down
the matching conditions in a compact way. In particular, we find that the
existence of a perturbed (rotating, stationary and vacuum) Schwarzschild cavity
in a perturbed FLRW universe forces the cosmological perturbations to satisfy
constraints that link rotational and gravitational wave perturbations. We also
prove that if the perturbation on the FLRW side vanishes identically, then the
vacuole must be perturbatively static and hence Schwarzschild. By the dual
nature of the problem, the first result translates into links between
rotational and gravitational wave perturbations on a perturbed
Oppenheimer-Snyder model, where the perturbed FLRW dust collapses in a
perturbed Schwarzschild environment which rotates in equilibrium. The second
result implies in particular that no region described by FLRW can be a source
of the Kerr metric.Comment: LaTeX; 29 page
Apparent horizons in the quasi-spherical Szekeres models
The notion of an apparent horizon (AH) in a collapsing object can be carried
over from the Lema\^{\i}tre -- Tolman (L--T) to the quasispherical Szekeres
models in three ways: 1. Literally by the definition -- the AH is the boundary
of the region, in which every bundle of null geodesics has negative expansion
scalar. 2. As the locus, at which null lines that are as nearly radial as
possible are turned toward decreasing areal radius . These lines are in
general nongeodesic. The name "absolute apparent horizon" (AAH) is proposed for
this locus. 3. As the boundary of a region, where null \textit{geodesics} are
turned toward decreasing . The name "light collapse region" (LCR) is
proposed for this region (which is 3-dimensional in every space of constant
); its boundary coincides with the AAH. The AH and AAH coincide in the L--T
models. In the quasispherical Szekeres models, the AH is different from (but
not disjoint with) the AAH. Properties of the AAH and LCR are investigated, and
the relations between the AAH and the AH are illustrated with diagrams using an
explicit example of a Szekeres metric. It turns out that an observer who is
already within the AH is, for some time, not yet within the AAH. Nevertheless,
no light signal can be sent through the AH from the inside. The analogue of the
AAH for massive particles is also considered.Comment: 14 pages, 9 figures, includes little extensions and style corrections
made after referee's comments, the text matches the published versio
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