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provided a theoretical context for an Omega parameter for LTB models, which has been defined in the literature as an ansatz [15, 16]. Given the widespread use of LTB dust solutions for dealing with a wide variety of prob- lems and models in General Relativity and Cosmology, the new approach to these solutions presented here has a significant potential for applications. 

In this article we have mostly introduced, discussed and justified the use of a new scalar representation for studying LTB models. However, the formalism that we have presented is potentially useful in future work using these models, hence we are looking at its direct applica- tion in separate articles, on the make, dealing with new results on important theoretical and practical issues, such as: (i) the asymptotic properties and boundary condi- tions of scalars in the radial direction, (ii) the conditions for constructing LTB models with scalars having radial profiles of “clumps” or “voids”, (iii) the conditions for the existence of a positive “back–reaction” in LTB mod- els, which would allow us to mimmic the effect of an accelerated cosmic expansion in the context of Buchert’s scalar averaging [21, 22]. Together with these articles under elaboration, we are also conducting a comprehen- sive numeric study of the models, based on the evolu- tion equations derived here. Finally, as shown by [45], the extension of the formalism of quasi–local scalars can be suitably modified in order to apply it to the quasi– spherical Szekeres spacetimes. The main justification of the present article is then to motivate and induce researchers to consider the introduced variables in the study and application of LTB models, as well as to serve as the theoretical reference for the use of these variables. 

## Appendix A: Regularity issues of the new variables. 

The regularity of LTB models in the conventional variables has been extensively discussed in the litera- ture [2, 3, 24, 25, 39]. We examine this issue in terms of the quasi–local variables and their fluctuations. 

We have only considered in this article LTB models having (at least) one symmetry center, which is a regular timelike worldline corresponding to a fixed point of the rotation group SO(3). This is a sufficient condition for integrals in (24) to be finite in a domain ϑ ( r ) defined by (23). The symmetry center can be marked as r = 0 (and r = r$_{c}$ if there is a second one). The following conditions hold: R ( t, 0) = ˙ R ( t, 0) = 0. Considering (36) and (40) and R ′ → 1 as r → 0, we can mark the symmetry center by a zero of R$_{i}$ ( r ) (distinct from the locus of a curvature singularity: L = 0) [25]. Hence 

## 1. Symmetry centers. 

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Since E (0) = M (0) = M $^{′}$(0) = E $^{′}$(0) = 0, then m$_{qi}$ (0) = m$_{i}$ (0) and k$_{qi}$ (0) = k$_{i}$ (0) must hold, and thus H $_{qi}$(0) = H $_{i}$(0). For any scalar A and its dual function A$_{q}$ we have 

<img src='content_image/387377.jpg'>

The same conditions hold at a second symmetry cen- ter r = r$_{c}$ in closed elliptic models (see Appendix A 3). Notice that a central singularity is now associated with L ( ct, r ) = 0. 

Regardless of which parameters or variables we may use, the regularity at each spacetime point or surface in LTB models can be characterized by the continu- ity and finiteness of the Riemann tensor R (e)(f)(g)(h) = R $_{abcd}$e a $_{(e)}$e b $_{(f)}$e c $_{(g)}$e d $_{(h)}$, in an orthonormal tetrad basis e a (e) [3, 24]. Considering the natural tetrad associated with the LTB metric in its form (39) 

the nonzero Riemann tensor basis components are given readily in terms of the new variables m$_{q}$ and δ ( m ) by 

<img src='content_image/387379.jpg'>

These basis components also involve other scalars like H , H $_{q}$, k, k$_{q}$ and δ ( H $^{)}$, δ( k $^{)}$, which are related to m, m$_{q}$ and δ ( m ) by the constraints (37), (38), (33) and the Hamiltonian constraint (20) for the local covariant scalars 

<img src='content_image/387382.jpg'>

Clearly, it is sufficient for the regularity of LTB models based on (A4) that all the involved scalars (not only m and m$_{q}$ ) are bounded and continuous, but it is not a nec- essary condition, since H and H q could diverge because k and k$_{q}$ diverge, with m and m$_{q}$ remaining bounded. Also, we note that the relative fluctuations δ ( A ) might diverge in some cases when A and A$_{q}$ are bounded. 

Given the existence of (at least) one symmetry center, the admissible topologies (homeomorphic classes) of the $^{3}$T [ t ] are 

## 2. The Riemann tensor. 

<img src='content_image/387378.jpg'>

## 3. Topology of the space slices. 

