Dust universes with spatial spherical symmetry and euclidean initial hypersurfaces

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

“…Moreover this result allows the generalization of some of the results found in the previous papers [4], [8] in the spatially euclidean case (at least in an a suitable interval of time), also to the not euclidean case.…”

“…Since in the following we are going to consider dust universes with spatial spherical symmetry, we want to briefly summarize the previous main results in a form inspired by [3,4]. n/k where m and k are integers such that k ≥ 1 is called a Puiseux series or a fractional power series (see e. g. [5,6,7]).…”

“…3 In accordance with [1] (but differently from [3,4]) the latin indices will vary from 1 to 3, whereas the greek indices will vary from 1 to 4. 4 At any point 1 B 2 represents the gaussian curvature of the geodesic sphere with its centre at the centre of symmetry O and passing through the point [1, Chap.…”

“…the dust initially distributed on the surface of the geodesic sphere with center at O and radius r (O-sphere) S(r)); in accordance with [3,4] we assign each particle of an r-shell the initial intrinsic radius B(0, r) as radial co-moving coordinate r…”

“…where μ(t, r) is the mass density, μ 0 = μ(0, r) is the initial mass density, and m(r) is the so-called "Euclidean mass" [3,4] m(r) = 4π…”