content_image/401842.jpg content_image/401847.jpg content_image/401841.jpg content_image/401844.jpg content_image/401834.jpg
The system (117) clearly suggests a dynamical systems approach, as discussed in [44] for FLRW and Bianchi models. This approach is based on the fluid flow equa- tions [32–34] and an Omega parameter constructed as the ratio of density to the the squared expansion scalar H 2 = Θ $^{2}$/ 9. As shown in [44], if the evolution parameter is defined by ∂/∂ξ = (1 / H ) ∂/∂τ the Raychaudhuri equa- tion decouples from the remaining evolution equations, leading to a reduced system that can be analyzed qualita- tively: critical points, invariant subspaces, etc. For LTB models associated with evolution equations like (117), it is more convenient (see [29]) to define ξ in terms of H in (118) by means of the coordinate transformation [29] 

so that for every scalar A ( τ, r ) = A ( τ ( ξ, r ) , r ) = A ( ξ, r ) and derivatives associated with the 4–velocity flow ∂/∂τ (with r constant) become 

Hence, the three equations (117b)–(117d) become inde- pendent of H and effectively decouple from (117a), lead- ing to the reduced system 

<img src='content_image/401842.jpg'>

which is very similar to that analyzed in [29] (where δ ( m ) and a dimensionless Σ were used instead of Δ and δ ( H $^{)}$). Notice that this dynamical system is character- ized by a 3–dimensional phase space parametrized by { ˆ Ω , Δ , δ ( H $^{)}$} , which contains the FLRW dust case as an invariant set given by the line [ ˆ Ω = ˆ Ω( ξ ) , Δ = 0 , δ ( H ) = 0]. Therefore, this dynamical systems approach also pro- vides a nice and appealing theoretical context to justify the role of ˆ Ω as an inhomogeneous generalization of the FLRW dust Omega parameter. 

## XIV. REGULARITY OF SPECIAL LTB CONFIGURATIONS. 

## A. Closed elliptic models and regular zeroes of R ′ = 0 . 

In closed elliptic models the $^{3}$T [ t ] are homeomorphic to S 3 (see Appendix A 3). There are two symmetry centers, 

## B. A dynamical system 

<img src='content_image/401847.jpg'>

<img src='content_image/401841.jpg'>

at r = 0 and r = r$_{c}$ . Since R ( t, 0) = R ( t, r$_{c}$ ) = 0 for all t , then R $^{′}$( r$_{tv}$ ) = 0 where 0 < r$_{tv}$ < r$_{c}$ must hold for all t . The regularity conditions (A6) and (A7) imply that for all r the sign of R ′ is the same for all $^{3}$T [ t ]. Hence, (9) requires the function R$_{i}$ ( r ) to be selected so that R$_{i}$ (0) = R$_{i}$ ( r$_{c}$ ) = 0 and R ′ $_{i}$( r$_{tv}$ ) = 0, with 

<img src='content_image/401844.jpg'>

where F = ± √$_{1 +}$ E , with F (0) = 1 , F ( r$_{c}$ ) = − 1 and F $^{′}$(0) = F $^{′}$( r$_{c}$ ) = F ( r$_{tv}$ ) = 0. Since R, R ′ and F are bounded, if (9) holds then the maximal coordinate range is 0 ≤ r ≤ r$_{c}$ in all regular $^{3}$T [ t ]. Equations (122) provide the coordinate ranges where 

the “+” or “ − ” signs hold in specifying the Hellaby–Lake conditions (11) in the conventional variables. However, the ± signs are no longer needed with the initial value parametrization, as it is clear that fulfillment of (72) (the equivalent of (11)) implies by its construction that t ′ bb and t ′ coll must have, respectively, the opposite and same sign (and a common zero) as R ′ $_{i}$/R$_{i}$ , and from (A6), this sign will be the same for R $^{′}$/R at all $^{3}$T [ t ]. If only the first sign equality in (9) holds, then there 

would exist a range of r for which 

so that M ′ and R ′ i have a common zero, but either F has a zero that is not common to that zero, or F has no zeroes. In these situations, ρ in (4) remains bounded, and so there is no curvature singularity, but we have a surface layer discontinuity at r = r$_{tv}$ [24, 25, 40]. This is the reason why when R ′ vanishes regularly at some r = r$_{tv}$ , this worldline must lie, either in an elliptic model or in an elliptic region. Besides the surface layer, (123) also implies an ill–defined metric coefficient g$_{rr}$ in (1) and (39) (which leads in turn to an ill–defined proper radial length between different comoving dust layers becomes). 

Once these issues are taken under consideration, initial conditions complying with the Hellaby–Lake conditions (72) can be provided for this class of models, whether for analytic/qualitive or numeric work. 

In general, the initial curvature singularity given by (58a) is not simultaneous, since it is marked by the curve ct = ct bb ( r ) in the ( ct, r ) coordinate plane. For parabolic models, the condition ct ′ bb = 0 in (63a) implies δ ( m ) i = 0, and thus m ′ qi = 0, which corresponds to the FLRW limit. However, non–trivial hyperbolic and elliptic LTB models follow by setting the initial value functions so that ct bb is a constant (see [10]). In order to examine these cases, we use (52), (57) and (63b) to rewrite Γ in (60) in terms of ct bb and ct ′ $_{bb}$, which after setting t ′ bb = 0 and t bb = t (0) bb = 

<img src='content_image/401834.jpg'>

## B. Simultaneous big–bang. 

