Abstracts
- On the Propagation of Jump Discontinuities in
Relativistic Cosmology (work with George F R Ellis
and Bernd G Schmidt) (Uuencoded /
Compressed Postscript File, 19 Pages)
A recent dynamical formulation at derivative level $\ptl^{3}g$ for
fluid spacetime geometries $({\cal M},{\bf g},{\bf u})$, that
employs the concept of evolution systems in
first-order symmetric hyperbolic format, implies the existence in
the Weyl curvature branch of a set of timelike characteristic
3-surfaces associated with propagation speed $|v| =
\sfrac{1}{2}$ relative to fluid-comoving observers. We show it is
the physical role of the constraint equations to prevent
realisation of jump discontinuities in the derivatives of the
related initial data so that Weyl curvature modes propagating along
these 3-surfaces cannot be activated. In addition we introduce a
new, illustrative first-order symmetric hyperbolic evolution system
at derivative level $\ptl^{2}g$ for baryotropic perfect fluid
cosmological models that are invariant under the transformations of
an Abelian $G_{2}$ isometry group.
- Partially Locally Rotationally Symmetric
Perfect Fluid Cosmologies (work with Nazeem Mustapha,
George F R Ellis and Mattias Marklund)
(Uuencoded / Compressed Postscript File,
20 Pages)
We show that there are no new consistent cosmological perfect fluid
solutions when in an open neighbourhood ${\cal U}$ of an event the
fluid kinematical variables and the electric and magnetic Weyl
curvature are all assumed rotationally symmetric about a common
spatial axis, specialising the Weyl curvature tensor to algebraic
Petrov type D. The consistent solutions of this kind are either
locally rotationally symmetric, or are subcases of the Szekeres
dust models. Parts of our results require the assumption of a
barotropic equation of state. Additionally we demonstrate that
local rotational symmetry of perfect fluid cosmologies follows from
rotational symmetry of the Riemann curvature tensor and of its
covariant derivatives only up to second order, thus strengthening a
previous result.
[ Class. Quantum Grav. 17 (2000), 3135. ]
- Longitudinal Weyl Curvature Modes: Abelian G2
Perfect Fluid Examples (work with George F R Ellis)
(Uuencoded / Compressed Postscript File,
8 Pages)
We study the initial value problem for spacetime geometries in
Wainwright's general class A(i) of Abelian $G_{2}$ perfect fluid
cosmologies with barotropic equation of state in terms of a
first-order symmetric hyperbolic evolution system, which is based
on the orthonormal frame {\em connection\/} formulation of
gravitational fields. We find that in this case the only {\em
non-zero\/} characteristic propagation speeds for physically
relevant discontinuities in the initial data are $|v| = \cs$
and $|v| = 1$, despite the fact that the {\em longitudinal\/}
Weyl curvature eigenfields (which generically are associated with
characteristic 3-surfaces defining propagation speeds $|v|=
\sfrac{1}{2}$) are typically non-vanishing.
- Transcript of George Ellis' Cargèse
Lectures 1998: Cosmological Models
(Uuencoded / Compressed Postscript File,
90 Pages)
The aim of this set of lectures is a systematic presentation of a
$1+3$ covariant approach to studying the geometry, dynamics, and
observational properties of relativistic cosmological models. In
giving (i) the basic $1+3$ covariant relations for a cosmological
fluid, the present lectures cover some of the same ground as a
previous set of Carg\`{e}se lectures \cite{ell73}, but they then go
on to give (ii) the full set of corresponding tetrad equations,
(iii) a classification of cosmological models with exact
symmetries, (iv) a brief discussion of some of the most useful
exact models and their observational properties, and (v) an
introduction to the gauge-invariant and $1+3$ covariant
perturbation theory of
almost-Friedmann--Lema\^{\i}tre--Robertson--Walker universes, with
a fluid description for the matter and a kinetic theory description
of the radiation.
[ gr-qc/9812046 ]
- Causal Propagation of Geometrical Fields
in Relativistic Cosmology (work with George F R Ellis)
(Uuencoded / Compressed Postscript File,
25 Pages)
We employ the extended $1+3$ orthonormal frame formalism for fluid
spacetime geometries $({\cal M},{\bf g},{\bf u})$, which contains
the Bianchi field equations for the
Weyl curvature, to derive a 44-D evolution system of first-order
symmetric hyperbolic form for a set of geometrically defined
dynamical field variables. Describing the matter source fields
phenomenologically in terms of a barotropic perfect fluid, the
propagation velocities $v$ (with respect to matter-comoving
observers that Fermi-propagate their spatial reference frames) of
disturbances in the matter and the gravitational field, represented
as wavefronts by the characteristic 3-surfaces of the system, are
obtained. In particular, the Weyl curvature is found to account for
two (non-Lorentz-invariant) Coulomb-like characteristic eigenfields
propagating with $v = 0$ and four transverse characteristic
eigenfields propagating with $|v| = 1$, which are well known, and
four (non-Lorentz-invariant) longitudinal characteristic
eigenfields propagating with $|v| = \sfrac{1}{2}$. The implications
of this result are discussed in some detail and a parallel is drawn
to the propagation of irregularities in the matter distribution. In
a worked example, we specialise the equations to cosmological
models in locally rotationally symmetric class II and include the
constraints into the set of causally propagating dynamical
variables.
[ Phys. Rev. D 59 (1999), 024013. ]
- Quasi-Newtonian Dust Cosmologies
(work with George F R Ellis) (Uuencoded /
Compressed Postscript File, 25 Pages)
Exact dynamical equations for a generic dust matter source field in
a cosmological context are formulated with respect to a
non-comoving Newtonian-like timelike reference congruence and
investigated for internal consistency. On the basis of a lapse
function $N$ (the relativistic acceleration scalar potential) which
evolves along the reference congruence according to $\dot{N} =
\alpha\,\Theta\,N$ ($\alpha = \mbox{const}$), we find that
consistency of the quasi-Newtonian dynamical equations is not
attained at the first derivative level. We then proceed to show
that a self-consistent set can be obtained by linearising the
dynamical equations about a (non-comoving) FLRW background. In this
case, on properly accounting for the first-order momentum density
relating to the non-relativistic peculiar motion of the matter,
additional source terms arise in the evolution and constraint
equations describing small-amplitude energy density fluctuations
that do not appear in similar gravitational instability scenarios
in the standard literature.
[ Class. Quantum Grav. 15 (1998), 3545. ]
- Deviation of Geodesics in FLRW Spacetime
Geometries (work with George F R Ellis)
(Uuencoded / Compressed Postscript File,
17 Pages)
The geodesic deviation equation (`GDE') provides an elegant tool to
investigate the timelike, null and spacelike structure of spacetime
geometries. Here we employ the GDE to review these structures
within the Friedmann--Lema\^{\i}tre--Robertson--Walker (`FLRW')
models, where we assume the sources to be given by a
non-interacting mixture of incoherent matter and radiation, and we
also take a non-zero cosmological constant into account. For each
causal case we present examples of solutions to the GDE and we
discuss the interpretation of the related first integrals. The de
Sitter spacetime geometry is treated separately.
[ Contribution to the Engelbert Schücking Festschrift.
Also: gr-qc/9709060. ]
- Extensions and Applications of 1+3
Decomposition Methods in General Relativistic Cosmological
Modelling (A Ph.D. Thesis Accepted by the University of
London, November 1996) (Uuencoded / Compressed
Postscript File, 181 Pages)
$1+3$ ``threading'' decomposition methods of the
pseudo-Riemannian spacetime manifold $\left(\,{\cal M}, \,{\bf g}
\,\right)$ and all its geometrical objects and dynamical relations
with respect to an invariantly defined preferred timelike reference
congruence ${\bf u}/c$ have been useful tools in general
relativistic cosmological modelling for more than three decades. In
this thesis extensions of the $1+3$ decomposition formalism are
developed, partially in fully covariant form, and partially on the
basis of choice of an arbitrary Minkowskian orthonormal reference
frame, the timelike direction of which is aligned with ${\bf
u}/c$. After introductory remarks, in Chapter \r{ch2} first an
exposition is given of the general $1+3$ covariant dynamical
equations for the fluid matter and Weyl curvature variables, which
arise from the Ricci and second Bianchi identities for the Riemann
curvature tensor of $\left(\, {\cal M}, \,{\bf g}, \,{\bf u}/c\,
\right)$. New evolution equations are then derived for all
spatial derivative terms of geometrical quantities orthogonal to
${\bf u}/c$. The latter are used to demonstrate in
$1+3$ covariant terms that the spatial constraints restricting
relativistic barotropic perfect fluid spacetime geometries are
preserved along the integral curves of ${\bf u}/c$. The
integrability of a number of different special subcases of interest
can easily be derived from this general result.
In Chapter \r{ch3}, $1+3$ covariant representations of two classes
of well-known cosmological models with a barotropic perfect fluid
matter source are obtained. These are the families of the locally
rotationally symmetric (LRS) and the orthogonally spatially
homogeneous (OSH) spacetime geometries, respectively. Subcases
arising from either dynamical restrictions or the existence of
higher symmetries are systematically discussed. For example, models
of purely ``magnetic'' Weyl curvature and, in the LRS case, a
transparent treatment of tilted spatial homogeneity can be
obtained. The $1+3$ covariant discussion of the OSH models requires
completion.
Chapter \r{ch4} reviews the complementary $1+3$ orthonormal frame
(ONF) approach and extends it to include the second Bianchi
identities, which provide dynamical relations for the physically
interesting Weyl curvature variables. Then, possible choices of
local coordinates within the $1+3$ ONF framework are introduced,
taking both the $1+3$ threading and the ADM $3+1$ slicing
perspectives.
The $1+3$ ONF method is employed in Chapter \r{ch5} to investigate
the integrability of the dynamical equations describing ``silent''
irrotational dust spacetime geometries, for which the ``magnetic''
part of the Weyl curvature is required to vanish. Evidence is
obtained that these equations may not be consistent in the generic
case, but that only either algebraically special or spatially
homogeneous classes of solutions may be covered. Furthermore, this
chapter uses the extended $1+3$ ONF dynamical equations to describe
LRS models with an imperfect fluid matter source and contrasts
the perfect fluid subcase with the results obtained in Chapter
\r{ch3}.
In Chapter \r{ch6}, a brief detour is taken into considering those
classical theories of gravitation in which the Lagrangean density
of the gravitational field is assumed to be proportional to a
general differentiable function $f(R)$ in the Ricci curvature
scalar. The generalisations of the relativistic $1+3$ covariant
dynamical equations to the $f(R)$ case are derived and a few
examples of applications are commented on.
Finally, Chapter \r{ch7} investigates in detail features of the
dynamical evolution of the cosmological density parameter $\Omega$
in anisotropic inflationary models of Bianchi Type--I and Type--V
and points out important qualitative changes as compared to the
idealised standard FLRW situation. A related analysis employing the
same spacetime geometries addresses the occurrence of restrictions
on the permissible functional form of the inflationary expansion
length scale parameter $S$ as a consequence of the so-called
reality condition for Einstein--Scalar-Field configurations. Again,
the effect of the (exact) anisotropic perturbations on the FLRW
case is thoroughly studied and found to have significant
effects. Both cases can be treated as examples of structural
instability.
This thesis ends with concluding remarks and an appendix section
containing the conventions employed and mathematical relations
relevant to derivations given in various chapters.
PACS number(s): 04.20.-q, 98.80.Hw, 98.80.Dr, 04.20.Jb
- Integrability of Irrotational Silent Cosmological
Models (work with Claes Uggla, William M Lesame, George F R Ellis
and Roy Maartens) (Uuencoded / Compressed
Postscript File, 11 Pages)
We revisit the issue of integrability conditions for the
irrotational silent cosmological models. We formulate the problem
both in 1+3 covariant and 1+3 orthonormal frame notation, and
show there exists a series of constraint equations that need to be
satisfied. These conditions hold identically for FLRW-linearised
silent models, but not in the general exact non-linear case.
Thus there is a linearisation instability, and it is highly unlikely
that there is a large class of silent models.
We conjecture that there are no spatially inhomogeneous
solutions with Weyl curvature of Petrov
type I, and indicate further issues that await clarification.
[ Class. Quantum Grav. 14 (1997), 1151. ]
- 1+3 Orthonormal Frame Approach to Perfect
Fluid Spacetime Geometries: The Equations
(Uuencoded / Compressed Postscript File,
11 Pages)
These notes provide in explicit detail the differential equations
of the $1+3$ orthonormal frame approach to relativistic
gravitation, assuming the matter source of the gravitational field
can be modelled as a single barotropic perfect fluid. The set of
equations consists of the commutation relation for the basis
vectors $\{\,\p_{a}\,\}$, the Jacobi and Ricci identities for the
frame commutation functions $\{\,\gamma^{a}{}_{bc}\,\}$, and the
second Bianchi identity for the Riemann curvature components
$\{\,R^{a}{}_{bcd}\,\}$. The latter three sets of equations as
listed in these notes were generated by the computer algebra
package {\tt CLASSI}.
[ Unpublished private notes (1997/1999). ]
- General Relativistic 1+3 Orthonormal Frame
Approach (work with Claes Uggla)
(Uuencoded / Compressed Postscript File,
21 Pages)
The dynamical equations of an extended 1+3 orthonormal frame
approach to the relativistic description of spacetime geometries
are explicitly presented and discussed in detail. In particular,
the Bianchi identities for the Weyl curvature tensor occur in fully
expanded form, as they are given a central role in the extended
formalism. It is shown how one can naturally introduce local
coordinates, both in the 1+3 threading and the ADM 3+1 slicing
context. By specialising the general 1+3 dynamical equations it
is demonstrated how a number of problems of interest can be
obtained. In particular, the simplest choices of spatial frames for
spatially homogeneous cosmological models, locally rotationally
symmetric spacetime geometries, cosmological models with an Abelian
isometry group G_{2} and ``silent'' dust cosmological models are
discussed.
[ Class. Quantum Grav. 14 (1997), 2673. ]
- The Covariant Approach to LRS Perfect Fluid
Spacetime Geometries (work with George Ellis)
(Uuencoded / Compressed Postscript File,
25 Pages)
The dynamics of perfect fluid spacetime geometries which exhibit
Local Rotational Symmetry (LRS) are reformulated in the
language of a 1+3 "threading" decomposition of the spacetime
manifold, where covariant fluid and curvature variables are
used. This approach presents a neat alternative to the orthonormal
frame formalism. The dynamical equations reduce to a set of
differential relations between purely scalar quantities. The
consistency conditions are worked out in a transparent way. We
discuss their various subcases in detail and focus in particular
on models with higher symmetries within the class of expanding
spatially inhomogeneous LRS models, via a consideration of
functional dependencies between the dynamical variables.
[ Class. Quantum Grav. 13 (1996), 1099. ]
- Kinematics and Dynamics of f(R) Theories of
Gravity (work with Steve Rippl, Reza Tavakol and David Taylor)
(Uuencoded / Compressed Postscript File,
9 Pages)
We generalise the equations governing relativistic fluid dynamics
given by Ehlers and Ellis for general relativity, and by Maartens
and Taylor for quadratic theories, to generalised f(R) theories
of gravity. In view of the usefulness of this alternative framework
to general relativity, its generalisation can be of potential
importance for deriving analogous results to those obtained in
general relativity. We generalise, as an example, the results of
Maartens and Taylor to show that within the framework of general
f(R) theories, a perfect fluid spacetime with vanishing
vorticity, shear and acceleration is
Friedmann--Lema\^{\i}tre--Robertson--Walker only if the fluid has
in addition a barotropic equation of state. It then follows that
the Ehlers--Geren--Sachs theorem and its ``almost'' extension also
hold for f(R) theories of gravity.
[ Gen. Rel. Grav. 28 (1996), 193. ]
- Quantum Cosmology and Higher-Order Lagrangean
Theories of Gravity (work with Jim Lidsey and Reza Tavakol)
(Uuencoded / Compressed Postscript File,
13 Pages)
In this paper the quantum cosmological consequences
of introducing a term cubic in the Ricci curvature scalar $R$ into
the Einstein--Hilbert action are investigated. It is argued that
this term represents a more generic perturbation to the action
than the quadratic correction usually considered. A qualitative
argument suggests that there exists a region of parameter space in
which neither the tunneling nor the no-boundary boundary
conditions predict an epoch of inflation that can solve the
horizon and flatness problems of the standard big bang model. This
is in contrast to the $R^{2}$--theory.
[ Class. Quantum Grav. 11 (1994), 2483. ]
- Evolution of the Density Parameter in
Inflationary Cosmology in the Presence of Shear and Bulk
Viscosity (work with Reza Tavakol)
(Uuencoded / Compressed Postscript File,
10 Pages)
We study the dynamical evolution of the cosmological density
parameter $\Omega$ in presence of shear and bulk viscosity
(in first order as well as in extended non-equilibrium
thermodynamical settings) perturbations, both analytically
and numerically. Our results show that overall the main
conclusions of Madsen and Ellis within the FLRW framework
regarding the likelihood of $\Omega$ taking values close to
unity remain robust with respect to such perturbations,
particularly at late times. There are, however, important
detailed changes to the dynamics at early times.
[ Phys. Rev. D 49 (1994), 6460. ]
- Jacobi Metric for the Bartnik/McKinnon
SU(2)-EYM Field (Uuencoded / Compressed
Postscript File)
Motivated by the possibility of applying the Killing vector fields
of the Jacobi metric for a particular spacetime--matter
configuration to generate new solutions by continuous 1-parameter
point transformations, we examine the associated Jacobi metric for
the SU(2)--EYM field as given by Bartnik and McKinnon to see if
internal symmetries of this kind exist. However, the only solution
of the Killing equations we find is $\xi^{a}=0$. In addition we
mention more general symmetry properties of the Jacobi metric.
[ Gen. Rel. Grav. 25 (1993), 1295. ]
- Relativistic Kinematics in 2-D Minkowski
Spacetime
(Uuencoded / Compressed Postscript File)
In these notes the difference between the geometric (co-ordinate
invariant) concept of tensors and the co-ordinate dependent values
of their components is briefly described in the case of 2--D
Special Relativity for kinematical quantities such as velocity and
acceleration by discussing their behaviour under the {\em linear}
co-ordinate transformation introduced by Lorentz. In reducing the
``absolute space'' of Special Relativity --- flat, 4--D Minkowski
spacetime --- to 2--D, the problem can be simplified significantly
without losing its essential features. In this way, relative motion
between co-ordinate frames is automatically put into so-called
standard configuration, that is frames are in rectilinear motion
relativ to each other along a mutual x--axis, with all other
co-ordinate axes also oriented parallel, and their origins
coinciding initially. From this an extension to the full 4--D case
is straightforward. The main part of the discussion presented
concentrates on a kinematical setting, in which one observer is in
{\em non-uniform} (that is accelerated) motion with respect to a
second, inertial one. Concluding these notes it is demonstrated in
an example, that the individual clocks, both these observers carry
along their world lines, will read a {\em different} elapsed rate
of time, after the observer in non-uniform motion returns from a
round trip through 2--D Minkowski spacetime and meets the inertial
observer again.
henk@gmunu.mth.uct.ac.za
Work
Last revision: Thu, 3-8-2000 (This page is under construction)