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

Abstract

The Einstein evolution of a dust shell universe with spatial spherical symmetry is analyzed. The implicit and parametric solutions of Tolman-Bondi equations are proposed in order to show its agreement with the rectilinear solutions of Kepler's problem. Finally, a complete systematization of Tolman-Bondi models is obtained through the classical Weierstrass approach.

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

“…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 . 5 In fact 1 r 2 represents, at any point P , the Gaussian curvature of the surface of the geodesic sphere S with its centre at the centre of symmetry O and passing through the point P [ …”

“…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 .…”

Computational Approach to Gravitational Waves Forms in Stellar Systems as Complex Structures through Keplerian Parameters

“…The main parameter is the approaching energy of the stars in the system (see also [5] and references therein). However, in a recent work [6], it is shown that the parametric form of the main evolution equation of Lemaˆıtre-Tolman-Bondi (LTB) cosmological model can be obtained considering the limiting rectilinear solutions of Keplers problem. More precisely, the analogy between the relativistic evolution of LTB shells and the classical discussion of a falling body in a Newtonian center of attraction, allows us to conclude that the relativistic orbits are clearly related to suitable astronomic coordinates.…”

“…In a generic elliptic Keplerian motion, called α the major semiaxis and E the eccentric anomaly, the orbit can be written also as (see [6], [9])…”

“…(2)-(7); in the limit the corresponding orbits are called rectilinear ellipse, hyperbola and parabola respectively. The evolution of the r-shell in LTB models is analogous to a Keplerian motion on a rectilinear ellipse, hyperbola or parabola (for more details see [6]). …”