content_image/376272.jpg content_image/376273.jpg content_image/376282.jpg content_image/376278.jpg content_image/376271.jpg
which (up to leading order) is the form of Γ in (59). Hence, Γ passes smoothly from its elliptic to its hyper- bolic form, approaching the form for parabolic models wherever k$_{qi}$ ≈ 0 for r ≈ r$_{b}$ . Since the limit k$_{qi}$ 0 implies x$_{i}$ 0 in either the 

→ → elliptic ( r < r$_{b}$ ) or hyperbolic side ( r < r$_{b}$ ), we should also obtain for L the approximated parabolic form (46) for r ≈ r$_{b}$ from the analytic solutions (50) and (55) in the limit x$_{i}$ → 0. Expanding Z$_{h}$ ( u ) and Z$_{e}$ ( u ) around u = 0 yields 

Hence, taking only the leading term in Z$_{h}$ and Z$_{e}$ and substituting into (50) and the expanding phase of (55) yields for both elliptic and hyperbolic sides 

which up to the leading term coincides with (46). We did not consider the collapsing phase in (55) because, from (57), we have ct max → ∞ as k$_{qi}$ → 0, hence the worldline marking the interface r = r$_{b}$ is contained in the expanding phase for all t . 

The conditions for the fulfillment of (61) can be ob- tained jointly for the elliptic and hyperbolic regions, bearing in mind that for r ≈ r$_{b}$ we have δ ( k ) i → −∞ but also Γ and L take the forms (143) and (145). For both the elliptic and hyperbolic regions we have as r → r$_{b}$ 

hence δ ( m ) i ≤ 0 (with δ ( m ) i = 0 only at r = 0) is a suf- ficient condition for ct ′ bb < 0 in (68) and (72) to hold in the full radial range (note, from (31), that this condition is equivalent to m ′ qi ≤ 0). Since ct coll ( r ) → ∞ as r → r$_{b}$ , the collapsing singularity is contained entirely in the el- liptic region, hence the existence of the hyperbolic region for r > r$_{b}$ has no effect on the condition ct ′ coll 0 in (72). 

≥ The only remaining condition is δ ( k ) i ≥ − 2 / 3 in the hyperbolic region. Evidently, this condition does not hold in the limit r → r$_{b}$ because δ ( k ) i → −∞ , but in this limit Γ has the form (143), and so (61) is not violated. As long as δ ( k ) i remains negative we have k ′ qi > 0 (from (31) with k$_{qi}$ < 0), and thus E ′ > 0 holds, hence condition (10) (equivalent to (61)) also holds. In the elliptic side, we have k$_{qi}$ → 0 with k ′ qi < 0, which implies from (B3) that E ′ > 0 and k$_{i}$ = $^{3}$R $_{i}$/ 6 < 0 near r$_{b}$ . In fact local spatial curvature k = / 6 is negative for all times near r = r$_{b}$ , even if k$_{q}$ > 0 all the way up to r = r$_{b}$ . This follows directly by applying the integral property (27b) with r = r$_{b}$ as integration limit and considering that 

<img src='content_image/376272.jpg'>

<img src='content_image/376273.jpg'>

<img src='content_image/376282.jpg'>

<img src='content_image/376278.jpg'>

<img src='content_image/376271.jpg'>

This provides an example of how elliptic dynamics does not (necessarily) imply positive local curvature (see Ap- pendix B). 

Notice that if instead of a hyperbolic region in (141), we have a parabolic region ( k$_{q}$ = 0) for r > r$_{b}$ , then the results above still apply to the elliptic region in 0 ≤ r < r$_{b}$ . Hence, we would still have k < 0 near r = r$_{b}$ in this region, and thus k < 0 would also hold in the parabolic region. This is an example of how local spatial curvature can be negative in a parabolic region not containing a symmetry. 

## XV. CONCLUSION AND FURTHER WORK. 

We have provided a comprehensive examination of LTB dust models in terms of quasi–local scalars and their fluctuations, which are covariant objects that can be re- lated to covariant scalars in the “fluid flow” or “1+3” formalism. The motivation, and contents of the article, together with a summary of previous work, concepts and ideas, have been given in detail in the introduction. As we have shown throughout the article, the initial value parametrization that emerges from these scalars is use- ful in analytic, qualitative and numerical studies of the models. The “fluid flow” evolution equations for these scalars are an appealing and practical alternative to the analytic solutions conventionally used in the literature, as they are fully general and at the same time techni- cally simple: they can be effectively handled as ordinary differential equations, and thus can be very handy for a numeric treatment of the models. These evolution equa- tions can also be understood in terms of a gauge invariant and covariant (GIC) perturbation formalism, consistent with the covariant fluid flow approach of Ellis et al [32– 34], as well as the traditional gauge invariant approach [37]. Under this formalism, the dynamics of LTB mod- els can be cast in terms of the dynamics of spherical, non–linear GIC perturbations on a FLRW background. 

Regarding analytic and qualitative work, the quasi– local scalar representation and its associated initial value description provide an appealing theoretical context to understand the conventional parameters of the models, leading to an understanding and appreciation of previ- ous work under a new perspective. This is specially evi- dent in the fact that all scalars can be expressed by sim- ple scaling laws that can be analyzed qualitatively, as well as in the formulation of regularity conditions (the Hellaby–Lake conditions [24, 25, 39]) as restrictions on initial value functions. We have also looked at the prob- lem, proposed by Krasi´ nski and Hellaby [7, 9], of finding the existence of a unique LTB model that is consistent with the “mapping” of arbitrary radial profiles of den- sity or velocity at different cosmic times. Also, we have 

