On the mechanical analogy between the relativistic evolution of a spherical dust universe and the classical motion of falling bodies

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

“…20 Eq. (34) is formally analogous to the integral of energy for a body of unit mass, freely falling toward or from an attracting mass m e fixed in O , in classical mechanics (see [2]). …”

“…See[17, Chapter XII, Section 11, p. 408] for a precise definition of spherical symmetry around a point O 2. In accordance with[17] (but differently from[7,14, 15]) the Greek indexes will vary from 1 to 4 whereas the Latin indexes will vary from 1 to 3.Downloaded by [University of Colorado at Boulder Libraries] at 04:10 22 December 2014 ON THE EVOLUTION OF DUST SHELLS…”

Evolution of dust shells in Tolman–Bondi space–time according to the Weierstrass approach

“…To determine the influence of the initial spatial geometry on the subsequent evolution of a spherical dust universe, in this section we shall consider the curvature properties of a spherically symmetric V 3 by the method introduced by Ricci (valid for a generic V 3 only, see [23, Chapter XII §11]), taking into account 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 under the form (5). Most of the following considerations are valid for a generic spherically symmetric strictly Riemannian manifold V 3 , not necessarily limited to the case of a spatial initial manifold of a dust universe.…”

“…Its metric is given by Eq. (5), where A(t, r ) and B(t, r ), by Bondi equations (see [17,20,26]), obey the following equations:…”

“…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.…”