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

Abstract

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

THE WEIERSTRASS CRITERION AND THE LEMAÎTRE–TOLMAN–BONDI MODELS WITH COSMOLOGICAL CONSTANT Λ

“…Let us give now a brief summary of the main features of LTB-models according to [14,7,15,16]. We will consider a dust system C which, during its evolution, generates a Riemannian manifold, with locally spatial spherical symmetry around a physical point O; c the metric can then be given the Levi Civita's form [17]:…”

“…It has been shown [18] that it is possible to break the corresponding Cauchy problem into two separate intrinsically formulated invariant problems: the problem of initial conditions and the restricted problem of evolution. By taking into account [14,7,18,15,16], we can immediately translate the restricted evolution problem into the following equations d…”

“…the dust initially distributed on the surface of the geodesic sphere with center at O and radius r (O-sphere) S(r)). According to [14,15,16], we assign, at each particle of a r-shell, the initial intrinsic radius r = R(r, 0) as the radial co-moving coordinate. Hence the metric of the initial spatial manifold V 3 takes the form…”

Lemaître-Tolman-Bondi cosmological wormhole

“…ρ 0 (r) is the energy density on an initial hypersurface, a prime denotes differentiation with respect to r, and dΩ 2 ≡ dϑ 2 + sin 2 ϑ dϕ 2 . The line element (1) describes a spherical inhomogeneity in a dust-dominated universe ( [20][21][22]; for a recent review see [24]). Consider now a wormhole shell Σ at r = r Σ (t) which joins two identical copies of an LTB spacetime (this shell describes a wormhole created with the universe and not formed as the result of a dynamical process after the Big Bang).…”

Cosmological expansion and local systems: a Lemaître–Tolman–Bondi model

“…The Lemaître-Tolman-Bondi (LTB) class of solutions (e.g., [9]) describes a spherical inhomogeneity embedded in a dust-dominated (pressureless) FLRW universe. Here we propose, in the LTB class of solutions, a particularly clear example of a local system composed of two spherical shells with density merging smoothly with the cosmic substratum, which is embedded in a FLRW universe.…”