Evolution of a Spherical Universe in a Short Range Collapse/Generation Interval

Bochicchio, Ivana; Laserra, Ettore

doi:10.1007/978-3-540-72586-2_141

Cited by 4 publications

(5 citation statements)

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“…As such, we can interpret B(t, r) as the intrinsic radius of the rsphere S(r) at time t [17, Chapter XII, Section 11, p. 411], so that, in the metric (1), each evolving r -sphere S(r) will have B(t, r) as the intrinsic radius at time t and r as the initial intrinsic radius 3 Here we depart somewhat from Levi-Civita's notation in that we use ρ, r, a(r) in place of r, R, A(R) respectively. 4 As it is known [17, Chapter XII, Section 11, p. 411] formula (3) gives the most general expression for the metric of a V 3 , which is symmetric around a point O .…”

Section: Levi-civita's Spherical Coordinates As Spatial Coordinatesmentioning

confidence: 99%

“…In this section we want to consider the curvature properties of a spherically symmetric V 3 12 by the method introduced by Ricci (valid for a generic V 3 only, see [17, Chapter XII, Section 11]), remembering that the metric d σ 2 = γ i j d x i d x j of a spherically symmetric manifold V 3 can be given the form (3).…”

Section: Three-dimensional Spatial Initial Metric With Spherical Symmmentioning

confidence: 99%

“…In particular the plan of the paper is as follows: in Section 2 we recall the model of Dust Universe in accordance with [14,15,1,3,4]; in Section 3 we study the curvature properties of the initial spatial hypersurface V 3 and their relationship with the principal curvature ω 1 ; in Section 4 we recall the different exact solutions of the evolution equations [11,12,13,10] and we analyze their behavior in correspondence with the different signs of the first principal curvature ω 1 of V 3 .…”

Section: Introductionmentioning

confidence: 99%

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On the evolution of dust shells in Lemaître-Tolman-Bondi models

Bochicchio¹,

Laserra²

2009

Journal of Interdisciplinary Mathematics

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Section: Levi-civita's Spherical Coordinates As Spatial Coordinatesmentioning

confidence: 99%

Section: Three-dimensional Spatial Initial Metric With Spherical Symmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the evolution of dust shells in Lemaître-Tolman-Bondi models

Bochicchio¹,

Laserra²

2009

Journal of Interdisciplinary Mathematics

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“…It is not possible to expand the second member of (5.18) in a power series with respect to ρ, but following [21] we can expand it in fractional Puiseux series t 1 cr we obtain the temporal law of evolution in the following parametric form…”

Section: Exact Solutions In Implicit Formmentioning

confidence: 99%

A Review on the Geometric Formulation of Tolman–bondi Equations in General Relativity

Bochicchio

Francaviglia

Laserra

2009

Int. J. Geom. Methods Mod. Phys.

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This work is focused on spherically symmetric space-times. More precisely, geometric and structural properties of spatially spherical shells of a dust universe are analyzed in detail considering recent results of our research. Moreover, exact solutions, obtained for constant Ricci principal curvatures, are inferred and qualitatively analyzed through suitable classic analogies.Any physical theory is based partly on a space-time structure, which is needed to locate events (space-time points) and to provide a domain of definition for variables describing particles and field.Hence, we begin by recalling the Riemannian space-time of General Relativity in order to introduce Einstein's equations under the hypothesis of a spherically symmetric space-time. Space-time structure and Einstein equationsIn Einstein's general theory of relativity, space-time is represented mathematically as a pair (M, g) where M is a connected four-dimensional, Hausdorff C ∞ manifold and g is a Lorentzian metric (i.e. a metric of signature +2) on M. The manifold Int. J. Geom. Methods Mod. Phys. 2009.06:595-617. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 06/10/15. For personal use only. A Review on the Geometric Formulation of T-B Equations 597M is assumed to be connected since we would have no classical knowledge of any disconnected components. It is taken to be Hausdorff since this seems to agree with normal experience. Together with the existence of a Lorentz metric the Hausdorff condition implies that M is paracompact [8,9]. The Lorentzian signature of the metric tensor field g implies that at each point of M the set {X | g(X, X) = 0} of null vectors is a cone in the tangent space. This null cone separates the tangent space into three open parts: the set of futuredirected timelike vectors, that of the past-directed timelike vectors and that of the spacelike vectors.With any kind of matter or radiation there is associated a second rank tensor field T, called energy momentum tensor, which describes the distribution of energy, momentum and stress. It is also assumed that the curvature associated with the metric g is related to matter by Einstein's field equation

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“…This model has been analyzed within an asymptotically flat background in [21]. However, in widely accepted cosmological models describing the present scenario of the expanding universe, space-time curvatures are nonvanishing through the universe [3][4][5]; so, extending and generalizing all the previous ideas in several directions, we can analyze the intrinsic geometrical properties of the initial spatial manifold V 3 , pointing out its Ricci principal curvatures ω 1 , ω 2 , ω 3 . Our considerations here show that the geometry of V 3 and its curvature properties are completely determined by the principal curvature ω 1 only.…”

mentioning

confidence: 99%