content_image/441108.jpg content_image/441122.jpg content_image/441130.jpg content_image/441121.jpg content_image/441109.jpg content_image/441111.jpg content_image/441112.jpg content_image/441119.jpg content_image/441120.jpg content_image/441107.jpg content_image/441124.jpg
energy–momentum tensor 

<img src='content_image/441108.jpg'>

where R = R ( t, r ), R ′ = ∂R/∂r , E = E ( r ) and ρ = ρ ( t, r ) is the rest–mass density. The field equations reduce to 

where κ = 8 πG/c $^{2}$, M = M ( r ) and ˙ R = u $^{a}$∇ $_{a}$R = ∂R/∂ ( ct ). 

## A. Kinematic equivalence class and analytic solutions. 

It is common usage in the literature (see [2, 3, 24]) to classify the solutions of (3) in “kinematic equivalence classes” given by the sign of E , which determines the existence of a zero of ˙ R $^{2}$. Since E = E ( r ), the sign of this function can be, either the same in the full range of r , in which case we have LTB models of a given kine- matic class, or it can change sign in specific ranges of r , defining LTB models with regions of different kinematic class (see [24]). The solutions of the Friedman–like field equation (3) for each kinematic class take the following well known parametric form: 

Parabolic models or regions: E = 0. 

Hyperbolic models or regions: E ≥ 0. 

<img src='content_image/441122.jpg'>

Elliptic models or regions: E ≤ 0. 

<img src='content_image/441130.jpg'>

where t bb = t bb ( r ) is called “big bang time”, as it marks the coordinate locus of the central expanding curvature singularity: R ( t, r ) = 0 for r ≥ 0. It emerges as an “in- tegration constant” in the integration of (3). Notice that the locus of central curvature singularity is distinct from that of the center of symmetry R ( t, 0) = 0 (see Appendix A 1). Besides the kinematic class, LTB models that ad- mit (at least) one symmetry center can be classified as “open” or “closed”, respectively corresponding to the hy- persurfaces of constant t being topologically equivalent to R 3 or S 3 (see Appendix A 3). 

<img src='content_image/441121.jpg'>

<img src='content_image/441109.jpg'>

<img src='content_image/441111.jpg'>

<img src='content_image/441112.jpg'>

In order to deal with a given specific model by means of (5)–(7), we need to prescribe the three “conventional” free functions 

However, the metric (1) is invariant under rescalings of the radial coordinate r = r (¯ r ), hence it is always possible to eliminate one of the free functions in (8) by a suitable choice of the radial coordinate, thus leaving only two ba- sic irreducible free functions. 

Given a choice of free functions (8), the solutions (5)– (7) can be used to find the remaining relevant quantities that may be required for a specific problem, a procedure that has been used abundantly in the literature [2, 3]. 

We will assume henceforth the existence of (at least) one symmetry center marked by r = 0 (see Appendix A 1), and will denote by “regular LTB model” any LTB configuration for which the condition 

holds for all t and all r not marking a symmetry center (since M $^{′}$(0) = 0 , E (0) = 0 and R $^{′}$( t, 0) = 1 [24, 25]). Conditions (9) imply that if R $^{′}$, M $^{′}$, √$_{1 +}$$_{E}$ have a zero under regular conditions, it must be a common zero of the same order at some worldline r = r$_{tv}$ > 0. Violation of (9), i.e. R ′ = 0 for M ′ ̸ = 0 , √$_{1 +}$$_{E}$ ̸ = 0, results in a shell crossing singularity for which ρ → ∞ occurs with R > 0. Condition (9) together with (4) also implies that ρ ≥ 0 holds everywhere and that it is bounded everywhere except at the coordinate locus of a central singularity ( i.e. points for which R ( t, r ) = 0 holds that are not the worldline of a symmetry center). The restrictions on the free parameters (8) that guarantee the fulfillment of (9) are the well known Hellaby–Lake (necessary and sufficient) conditions [24, 25, 39] given by 

Parabolic and hyperbolic models or regions: 

<img src='content_image/441119.jpg'>

Elliptic models or regions: 

<img src='content_image/441120.jpg'>

where only expanding configurations are considered in (10) and the ± sign in (11) accounts for the fact that R ′ < 0 occurs in elliptic models whose that admit a second symmetry center (see Appendix A 3). The equal sign holds only at symmetry centers and at values of r where R ′ = 0. 

<img src='content_image/441107.jpg'>

## B. Regular LTB models 

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