2,739 research outputs found
Internal Time Formalism for Spacetimes with Two Killing Vectors
The Hamiltonian structure of spacetimes with two commuting Killing vector
fields is analyzed for the purpose of addressing the various problems of time
that arise in canonical gravity. Two specific models are considered: (i)
cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes,
which respectively involve open and closed universe boundary conditions. For
each model canonical variables which can be used to identify points of space
and instants of time, {\it i.e.}, internally defined spacetime coordinates, are
identified. To do this it is necessary to extend the usual ADM phase space by a
finite number of degrees of freedom. Canonical transformations are exhibited
that identify each of these models with harmonic maps in the parametrized field
theory formalism. The identifications made between the gravitational models and
harmonic map field theories are completely gauge invariant, that is, no
coordinate conditions are needed. The degree to which the problems of time are
resolved in these models is discussed.Comment: 36 pages, Te
The Gowdy T3 Cosmologies revisited
We have examined, repeated and extended earlier numerical calculations of
Berger and Moncrief for the evolution of unpolarized Gowdy T3 cosmological
models. Our results are consistent with theirs and we support their claim that
the models exhibit AVTD behaviour, even though spatial derivatives cannot be
neglected. The behaviour of the curvature invariants and the formation of
structure through evolution both backwards and forwards in time is discussed.Comment: 11 pages, LaTeX, 6 figures, results and conclusions revised and
(considerably) expande
On the area of the symmetry orbits in symmetric spacetimes
We obtain a global existence result for the Einstein equations. We show that
in the maximal Cauchy development of vacuum symmetric initial data with
nonvanishing twist constant, except for the special case of flat Kasner initial
data, the area of the group orbits takes on all positive values. This
result shows that the areal time coordinate which covers these spacetimes
runs from zero to infinity, with the singularity occurring at R=0.Comment: The appendix which appears in version 1 has a technical problem (the
inequality appearing as the first stage of (52) is not necessarily true), and
since the appendix is unnecessary for the proof of our results, we leave it
out. version 2 -- clarifications added, version 3 -- reference correcte
Numerical Investigation of Cosmological Singularities
Although cosmological solutions to Einstein's equations are known to be
generically singular, little is known about the nature of singularities in
typical spacetimes. It is shown here how the operator splitting used in a
particular symplectic numerical integration scheme fits naturally into the
Einstein equations for a large class of cosmological models and thus allows
study of their approach to the singularity. The numerical method also naturally
singles out the asymptotically velocity term dominated (AVTD) behavior known to
be characteristic of some of these models, conjectured to describe others, and
probably characteristic of a subclass of the rest. The method is first applied
to the unpolarized Gowdy T cosmology. Exact pseudo-unpolarized solutions
are used as a code test and demonstrate that a 4th order accurate
implementation of the numerical method yields acceptable agreement. For generic
initial data, support for the conjecture that the singularity is AVTD with
geodesic velocity (in the harmonic map target space) < 1 is found. A new
phenomenon of the development of small scale spatial structure is also
observed. Finally, it is shown that the numerical method straightforwardly
generalizes to an arbitrary cosmological spacetime on with one
spacelike U(1) symmetry.Comment: 37 pp +14 figures (not included, available on request), plain Te
Mixmaster Behavior in Inhomogeneous Cosmological Spacetimes
Numerical investigation of a class of inhomogeneous cosmological spacetimes
shows evidence that at a generic point in space the evolution toward the
initial singularity is asymptotically that of a spatially homogeneous spacetime
with Mixmaster behavior. This supports a long-standing conjecture due to
Belinskii et al. on the nature of the generic singularity in Einstein's
equations.Comment: 4 pages plus 4 figures. A sentence has been deleted. Accepted for
publication in PR
Evidence for an oscillatory singularity in generic U(1) symmetric cosmologies on
A longstanding conjecture by Belinskii, Lifshitz, and Khalatnikov that the
singularity in generic gravitational collapse is locally oscillatory is tested
numerically in vacuum, U(1) symmetric cosmological spacetimes on . If the velocity term dominated (VTD) solution to Einstein's equations is
substituted into the Hamiltonian for the full Einstein evolution equations, one
term is found to grow exponentially. This generates a prediction that
oscillatory behavior involving this term and another (which the VTD solution
causes to decay exponentially) should be observed in the approach to the
singularity. Numerical simulations strongly support this prediction.Comment: 15 pages, Revtex, includes 12 figures, psfig. High resolution
versions of figures 7, 8, 9, and 11 may be obtained from anonymous ftp to
ftp://vela.acs.oakland.edu/pub/berger/u1genfig
The Geroch group in the Ashtekar formulation
We study the Geroch group in the framework of the Ashtekar formulation. In
the case of the one-Killing-vector reduction, it turns out that the third
column of the Ashtekar connection is essentially the gradient of the Ernst
potential, which implies that the both quantities are based on the ``same''
complexification. In the two-Killing-vector reduction, we demonstrate Ehlers'
and Matzner-Misner's SL(2,R) symmetries, respectively, by constructing two sets
of canonical variables that realize either of the symmetries canonically, in
terms of the Ashtekar variables. The conserved charges associated with these
symmetries are explicitly obtained. We show that the gl(2,R) loop algebra
constructed previously in the loop representation is not the Lie algebra of the
Geroch group itself. We also point out that the recent argument on the
equivalence to a chiral model is based on a gauge-choice which cannot be
achieved generically.Comment: 40 pages, revte
G1 Cosmologies with Gravitational and Scalar Waves
I present here a new algorithm to generate families of inhomogeneous massless
scalar field cosmologies. New spacetimes, having a single isometry, are
generated by breaking the homogeneity of massless scalar field models
along one direction. As an illustration of the technique I construct
cosmological models which in their late time limit represent perturbations in
the form of gravitational and scalar waves propagating on a non-static
inhomogeneous background. Several features of the obtained metrics are
discussed, such as their early and late time limits, structure of singularities
and physical interpretation.Comment: 24 pages, 2 figure
Coordinate Singularities in Harmonically-sliced Cosmologies
Harmonic slicing has in recent years become a standard way of prescribing the
lapse function in numerical simulations of general relativity. However, as was
first noticed by Alcubierre (1997), numerical solutions generated using this
slicing condition can show pathological behaviour. In this paper, analytic and
numerical methods are used to examine harmonic slicings of Kasner and Gowdy
cosmological spacetimes. It is shown that in general the slicings are prevented
from covering the whole of the spacetimes by the appearance of coordinate
singularities. As well as limiting the maximum running times of numerical
simulations, the coordinate singularities can lead to features being produced
in numerically evolved solutions which must be distinguished from genuine
physical effects.Comment: 21 pages, REVTeX, 5 figure
Hunting Local Mixmaster Dynamics in Spatially Inhomogeneous Cosmologies
Heuristic arguments and numerical simulations support the Belinskii et al
(BKL) claim that the approach to the singularity in generic gravitational
collapse is characterized by local Mixmaster dynamics (LMD). Here, one way to
identify LMD in collapsing spatially inhomogeneous cosmologies is explored. By
writing the metric of one spacetime in the standard variables of another,
signatures for LMD may be found. Such signatures for the dynamics of spatially
homogeneous Mixmaster models in the variables of U(1)-symmetric cosmologies are
reviewed. Similar constructions for U(1)-symmetric spacetimes in terms of the
dynamics of generic -symmetric spacetime are presented.Comment: 17 pages, 5 figures. Contribution to CQG Special Issue "A Spacetime
Safari: Essays in Honour of Vincent Moncrief
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