99 research outputs found
Nondifferentiable Dynamic: Two Examples
Some nondifferentiable quantities (for example, the metric signature) can be
the independent physical degrees of freedom. It is supposed that in quantum
gravity these degrees of freedom can fluctuate. Two examples of such quantum
fluctuation are considered: a quantum interchange of the sign of two components
of the 5D metric and a quantum fluctuation between Euclidean and Lorentzian
metrics. The first case leads to a spin-like structure on the throat of
composite wormhole and to a possible inner structure of the string. The second
case leads to a quantum birth of the non-singular Euclidean Universe with
frozen dimension. The probability for such quantum fluctuations is
connected with an algorithmical complexity of the Einstein equations.Comment: essential changes: the initial equations in section III are changed,
as the consequence the obtained solution describes the quantum birth of the
nonsingular Universe with the matter (electromagnetic field=nondiagonal
components of the MD metric
A Renormalization Group Approach to Relativistic Cosmology
We discuss the averaging hypothesis tacitly assumed in standard cosmology.
Our approach is implemented in a "3+1" formalism and invokes the coarse
graining arguments, provided and supported by the real-space Renormalization
Group (RG) methods. Block variables are introduced and the recursion relations
written down explicitly enabling us to characterize the corresponding RG flow.
To leading order, the RG flow is provided by the Ricci-Hamilton equations
studied in connection with the geometry of three-manifolds. The properties of
the Ricci-Hamilton flow make it possible to study a critical behaviour of
cosmological models. This criticality is discussed and it is argued that it may
be related to the formation of sheet-like structures in the universe. We
provide an explicit expression for the renormalized Hubble constant and for the
scale dependence of the matter distribution. It is shown that the Hubble
constant is affected by non-trivial scale dependent shear terms, while the
spatial anisotropy of the metric influences significantly the scale-dependence
of the matter distribution.Comment: 57 pages, LaTeX, 15 pictures available on request from the Author
Note on Signature Change and Colombeau Theory
Recent work alludes to various `controversies' associated with signature
change in general relativity. As we have argued previously, these are in fact
disagreements about the (often unstated) assumptions underlying various
possible approaches. The choice between approaches remains open.Comment: REVTex, 3 pages; to appear in GR
Reply Comment: Comparison of Approaches to Classical Signature Change
We contrast the two approaches to ``classical" signature change used by
Hayward with the one used by us (Hellaby and Dray). There is (as yet) no
rigorous derivation of appropriate distributional field equations. Hayward's
distributional approach is based on a postulated modified form of the field
equations. We make an alternative postulate. We point out an important
difference between two possible philosophies of signature change --- ours is
strictly classical, while Hayward's Lagrangian approach adopts what amounts to
an imaginary proper ``time" on one side of the signature change, as is
explicitly done in quantum cosmology. We also explain why we chose to use the
Darmois-Israel type junction conditions, rather than the Lichnerowicz type
junction conditions favoured by Hayward. We show that the difference in results
is entirely explained by the difference in philosophy (imaginary versus real
Euclidean ``time"), and not by the difference in approach to junction
conditions (Lichnerowicz with specific coordinates versus Darmois with general
coordinates).Comment: 10 pages, latex, no figures. Replying to - "Comment on `Failure of
Standard Conservation Laws at a Classical Change of Signature'", S.A.
Hayward, Phys. Rev. D52, 7331-7332 (1995) (gr-qc/9606045
Ricci Flow Gravity
A theory of gravitation is proposed, modeled after the notion of a Ricci
flow. In addition to the metric an independent volume enters as a fundamental
geometric structure. Einstein gravity is included as a limiting case. Despite
being a scalar-tensor theory the coupling to matter is different from
Jordan-Brans-Dicke gravity. In particular there is no adjustable coupling
constant. For the solar system the effects of Ricci flow gravity cannot be
distinguished from Einstein gravity and therefore it passes all classical
tests. However for cosmology significant deviations from standard Einstein
cosmology will appear.Comment: 15 pages. V2: improved presentation, in particular Jordan vs.
Brans-Dicke and on viability. Added section on physical interpretation. V3:
more references. Reworked to agree with published versio
Averaging procedure in variable-G cosmologies
Previous work in the literature had built a formalism for spatially averaged
equations for the scale factor, giving rise to an averaged Raychaudhuri
equation and averaged Hamiltonian constraint, which involve a backreaction
source term. The present paper extends these equations to include models with
variable Newton parameter and variable cosmological term, motivated by the
nonperturbative renormalization program for quantum gravity based upon the
Einstein-Hilbert action. We focus on the Brans-Dicke form of the
renormalization-group improved action functional. The coupling between
backreaction and spatially averaged three-dimensional scalar curvature is found
to survive, and a variable-G cosmic quintet is found to emerge. Interestingly,
under suitable assumptions, an approximate solution can be found where the
early universe tends to a FLRW model, while keeping track of the original
inhomogeneities through three effective fluids. The resulting qualitative
picture is that of a universe consisting of baryons only, while inhomogeneities
average out to give rise to the full dark-side phenomenology.Comment: 20 pages. In the new version, all original calculations have been
improved, and the presentation has been further improved as wel
Quantum states of elementary three-geometry
We introduce a quantum volume operator in three--dimensional Quantum
Gravity by taking into account a symmetrical coupling scheme of three SU(2)
angular momenta. The spectrum of is discrete and defines a complete set of
eigenvectors which is alternative with respect to the complete sets employed
when the usual binary coupling schemes of angular momenta are considered. Each
of these states, that we call quantum bubbles, represents an interference of
extended configurations which provides a rigorous meaning to the heuristic
notion of quantum tetrahedron. We study the generalized recoupling coefficients
connecting the symmetrical and the binary basis vectors, and provide an
explicit recursive solution for such coefficients by analyzing also its
asymptotic limit.Comment: 15 pages, LaTe
How is the local-scale gravitational instability influenced by the surrounding large-scale structure formation?
We develop the formalism to investigate the relation between the evolution of
the large-scale (quasi) linear structure and that of the small-scale nonlinear
structure in Newtonian cosmology within the Lagrangian framework. In doing so,
we first derive the standard Friedmann expansion law using the averaging
procedure over the present horizon scale. Then the large-scale (quasi) linear
flow is defined by averaging the full trajectory field over a large-scale
domain, but much smaller than the horizon scale. The rest of the full
trajectory field is supposed to describe small-scale nonlinear dynamics. We
obtain the evolution equations for the large-scale and small-scale parts of the
trajectory field. These are coupled to each other in most general situations.
It is shown that if the shear deformation of fluid elements is ignored in the
averaged large-scale dynamics, the small-scale dynamics is described by
Newtonian dynamics in an effective Friedmann-Robertson-Walker (FRW) background
with a local scale factor. The local scale factor is defined by the sum of the
global scale factor and the expansion deformation of the averaged large-scale
displacement field. This means that the evolution of small-scale fluctuations
is influenced by the surrounding large-scale structure through the modification
of FRW scale factor. The effect might play an important role in the structure
formation scenario. Furthermore, it is argued that the so-called {\it
optimized} or {\it truncated} Lagrangian perturbation theory is a good
approximation in investigating the large-scale structure formation up to the
quasi nonlinear regime, even when the small-scale fluctuations are in the
non-linear regime.Comment: 15pages, Accepted for publication in Gravitation and General
Relativit
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.
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