45 research outputs found
The Averaging Problem in Cosmology and Macroscopic Gravity
The averaging problem in cosmology and the approach of macroscopic gravity to
resolve the problem is discussed. The averaged Einstein equations of
macroscopic gravity are modified on cosmological scales by the macroscopic
gravitational correlation tensor terms as compared with the Einstein equations
of general relativity. This correlation tensor satisfies a system of structure
and field equations. An exact cosmological solution to the macroscopic gravity
equations for a constant macroscopic gravitational connection correlation
tensor for a flat spatially homogeneous, isotropic macroscopic space-time is
presented. The correlation tensor term in the macroscopic Einstein equations
has been found to take the form of either a negative or positive spatial
curvature term. Thus, macroscopic gravity provides a cosmological model for a
flat spatially homogeneous, isotropic Universe which obeys the dynamical law
for either an open or closed Universe.Comment: 8 pages, LaTeX, ws-ijmpa.cls, few style and typo corrections. Based
on the plenary talk given at the Second Stueckelberg Workshop, ICRANet
Coordinating Center, Pescara, Italy, September 3-7, 2007. To appear in
International Journal of Modern Physics A (2008
Cosmological Solutions in Macroscopic Gravity
In the macroscopic gravity approach to the averaging problem in cosmology,
the Einstein field equations on cosmological scales are modified by appropriate
gravitational correlation terms. We present exact cosmological solutions to the
equations of macroscopic gravity for a spatially homogeneous and isotropic
macroscopic space-time and find that the correlation tensor is of the form of a
spatial curvature term. We briefly discuss the physical consequences of these
results.Comment: 5 page
The Spatial Averaging Limit of Covariant Macroscopic Gravity - Scalar Corrections to the Cosmological Equations
It is known that any explicit averaging scheme of the type essential for
describing the large scale behaviour of the Universe, must necessarily yield
corrections to the Einstein equations applied in the Cosmological setting. The
question of whether or not the resulting corrections to the Einstein equations
are significant, is still a subject of debate, partly due to possible
ambiguities in the averaging schemes available. In particular, it has been
argued in the literature that the effects of averaging could be gauge
artifacts. We apply the formalism of Zalaletdinov's Macroscopic Gravity (MG)
which is a fully covariant and nonperturbative averaging scheme, in an attempt
to construct gauge independent corrections to the standard
Friedmann-Lemaitre-Robertson-Walker (FLRW) equations. We find that whereas one
cannot escape the problem of dependence on \emph{one} gauge choice -- which is
inherent in the assumption of large scale homogeneity and isotropy -- it is
however possible to construct \emph{spacetime scalar} corrections to the
standard FLRW equations. This partially addresses the criticism concerning the
corrections being gauge artifacts. For a particular initial choice of gauge
which simplifies the formalism, we explicitly construct these scalars in terms
of the underlying inhomogeneous geometry, and incidentally demonstrate that the
formal structure of the corrections with this gauge choice is identical to that
of analogous corrections derived by Buchert in the context of spatial averaging
of scalars.Comment: 18 pages, no figures, revtex4; v2 - minor clarifications added; v3 -
minor changes in presentation to improve clarity, reference added, to appear
in Phys. Rev.
An exact quantification of backreaction in relativistic cosmology
An important open question in cosmology is the degree to which the
Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions of Einstein's equations
are able to model the large-scale behaviour of the locally inhomogeneous
observable universe. We investigate this problem by considering a range of
exact n-body solutions of Einstein's constraint equations. These solutions
contain discrete masses, and so allow arbitrarily large density contrasts to be
modelled. We restrict our study to regularly arranged distributions of masses
in topological 3-spheres. This has the benefit of allowing straightforward
comparisons to be made with FLRW solutions, as both spacetimes admit a discrete
group of symmetries. It also provides a time-symmetric hypersurface at the
moment of maximum expansion that allows the constraint equations to be solved
exactly. We find that when all the mass in the universe is condensed into a
small number of objects (<10) then the amount of backreaction in dust models
can be large, with O(1) deviations from the predictions of the corresponding
FLRW solutions. When the number of masses is large (>100), however, then our
measures of backreaction become small (<1%). This result does not rely on any
averaging procedures, which are notoriously hard to define uniquely in general
relativity, and so provides (to the best of our knowledge) the first exact and
unambiguous demonstration of backreaction in general relativistic cosmological
modelling. Discrete models such as these can therefore be used as laboratories
to test ideas about backreaction that could be applied in more complicated and
realistic settings.Comment: 13 pages, 9 figures. Corrections made to Tables IV and
Relative entropy as a measure of inhomogeneity in general relativity
We introduce the notion of relative volume entropy for two spacetimes with
preferred compact spacelike foliations. This is accomplished by applying the
notion of Kullback-Leibler divergence to the volume elements induced on
spacelike slices. The resulting quantity gives a lower bound on the number of
bits which are necessary to describe one metric given the other. For
illustration, we study some examples, in particular gravitational waves, and
conclude that the relative volume entropy is a suitable device for quantitative
comparison of the inhomogeneity of two spacetimes.Comment: 15 pages, 7 figure
Averaging in Cosmology
In this paper we discuss the effect of local inhomogeneities on the global
expansion of nearly FLRW universes, in a perturbative setting. We derive a
generic linearized averaging operation for metric perturbations from basic
assumptions, and we explicify the issue of gauge invariance. We derive a gauge
invariant expression for the back-reaction of density inhomogeneities on the
global expansion of perturbed FLRW spacetimes, in terms of observable
quantities, and we calculate the effect quantitatively. Since we do not adopt a
comoving gauge, our result incorporates the back-reaction on the metric due to
scalar velocity and vorticity perturbations. The results are compared with the
results by other authors in this field.Comment: 24 pages, Latex, accepted for publication in Phys. Rev.
Covariant coarse-graining of inhomogeneous dust flow in General Relativity
A new definition of coarse-grained quantities describing the dust flow in
General Relativity is proposed. It assigns the coarse--grained expansion, shear
and vorticity to finite-size comoving domains of fluid in a covariant,
coordinate-independent manner. The coarse--grained quantities are all
quasi-local functionals, depending only on the geometry of the boundary of the
considered domain. They can be thought of as relativistic generalizations of
simple volume averages of local quantities in a flat space. The procedure is
based on the isometric embedding theorem for S^2 surfaces and thus requires the
boundary of the domain in question to have spherical topology and positive
scalar curvature. We prove that in the limit of infinitesimally small volume
the proposed quantities reproduce the local expansion, shear and vorticity. In
case of irrotational flow we derive the time evolution for the coarse-grained
quantities and show that its structure is very similar to the evolution
equation for their local counterparts. Additional terms appearing in it may
serve as a measure of the backreacton of small-scale inhomogeneities of the
flow on the large-scale motion of the fluid inside the domain and therefore the
result may be interesting in the context of the cosmological backreaction
problem. We also consider the application of the proposed coarse-graining
procedure to a number of known exact solutions of Einstein equations with dust
and show that it yields reasonable results.Comment: 17 pages, 5 figures. Version accepted in Classical and Quantum
Gravity
BACK-REACTION IN RELATIVISTIC COSMOLOGY
We introduce the concept of back-reaction in relativistic cosmological modeling. Roughly speaking, this can be thought of as the difference between the large-scale behavior of an inhomogeneous cosmological solution of Einstein’s equations, and a homogeneous and isotropic solution that is a best-fit to either the average of observables or dynamics in the inhomogeneous solution. This is sometimes paraphrased as “the effect that structure has of the large-scale evolution of the universe.” Various different approaches have been taken in the literature in order to try and understand back-reaction in cosmology. We provide a brief and critical summary of some of them, highlighting recent progress that has been made in each case
Spherically Symmetric Solutions in Macroscopic Gravity
Schwarzschild's solution to the Einstein Field Equations was one of the first
and most important solutions that lead to the understanding and important
experimental tests of Einstein's theory of General Relativity. However,
Schwarzschild's solution is essentially based on an ideal theory of
gravitation, where all inhomogeneities are ignored. Therefore, any
generalization of the Schwarzschild solution should take into account the
effects of small perturbations that may be present in the gravitational field.
The theory of Macroscopic Gravity characterizes the effects of the
inhomogeneities through a non-perturbative and covariant averaging procedure.
With similar assumptions on the geometry and matter content, a solution to the
averaged field equations as dictated by Macroscopic Gravity are derived. The
resulting solution provides a possible explanation for the flattening of
galactic rotation curves, illustrating that Dark Matter is not real but may
only be the result of averaging inhomogeneities in a spherically symmetric
background.Comment: 14 pages, added and updated references, some paragraphs rewritten for
clarity, typographical errors fixed, results have not change