6 A. A FLRW–like metric and scaling laws for covariant scalars. Consider the dimensionless scale factor where R$_{i}$ ≡ R ( t$_{i}$, r ) and t = t$_{i}$ marks a fiducial initial hypersurface $^{3}$T i = $^{3}$T [ t$_{i}$ ]. We shall denote henceforth all quantities evaluated at t = t$_{i}$ by the subindex $_{i}$. Bearing in mind (36), we can re–write (28) as By applying the definition (31) to the Friedman–like equation for H 2 q in (28) we obtain the following important constraint terms of its associated A$_{q}$ and δ ( A $^{)}$: L ≡ R R$_{i}$ , (36) m = m$_{q}$ [ 1 + δ ( m ) ] , H = H q [ 1 + δ ( H ) ] , k = k$_{q}$ 1 + δ ( k ) ] , Σ = −H q δ ( H $^{)}$, E = − m$_{q}$ δ ( m $^{)}$. (32) m$_{q}$ = m$_{qi}$ L 3 , k$_{q}$ = k$_{qi}$ L $_{2}$, (37) H 2 q = ˙ L 2 L 2 = 2 m$_{q}$ − k$_{q}$ = 2 m$_{qi}$ − k$_{qi}$L L 3 , (38) 2 H 2 $_{q}$δ ( H ) = 2 m$_{q}$ δ ( m ) − k$_{q}$ δ ( k $^{)}$, (33) M ′ M = 3 R ′ R [ 1 + δ ( m ) ] , E ′ E = 3 R ′ R [ 2 3 + δ ( k ) ] , (34) which, if we identify L with a position dependent FLRW scale factor, are formally identical to the scaling laws for density, spatial curvature and Hubble factor in FRLR dust universes. Applying this parametrization to the LTB metric (1) yields the FLRW–like line element while the relation between δ ( m ) and δ ( k ) and radial gra- dients of the conventional variables M and E follows readily from (28) and (31). Since m, k, H and m$_{q}$, k$_{q}$, H q are all covariant scalars, their δ ( A ) are also covariant. In fact, these fluctuations can be understood in the context of a covariant gauge in- variant and non–linear perturbation formalism (see sec- tion XII and references [30, 31]). Hence, the set where the new dimensionless metric function Γ is d s 2 = − c $^{2}$d t $^{2}$+ L 2 [ Γ $^{2}$R ′ i $^{2}$d r 2 1 − k$_{qi}$ R 2 i + R 2 i ( d θ 2 + sin $^{2}$θ d φ $^{2}$) ] , (39) { A$_{q}$, δ ( A $^{)}$} = { m$_{q}$, k$_{q}$, H $_{q}$, δ ( m $^{)}$, δ( k $^{)}$, δ( H $^{)}$} , (35) Γ ≡ R $^{′}$/R R ′ $_{i}$/R$_{i}$ = 1 + L $^{′}$/L R ′ $_{i}$/R$_{i}$ , Γ$_{i}$ = 1 . (40) The local density ( m = κρ/ 3) and spatial curvature ( k = $^{3}$R / 6) satisfy the following scaling laws in terms of L, Γ and initial value functions m = m$_{qi}$ L 3 [1 + δ ( m $^{)}$] = m$_{i}$ L $^{3}$Γ , (41a) k + k$_{qi}$ L 2 [1 + δ ( k $^{)}$] = k$_{i}$ L $^{2}$Γ [ 1 + Γ − 1 3 (1 + δ ( k ) i ) ] . (41b) where we used (4), (13) and (32). Comparing (41) with (37), and bearing in mind (29) and (31), we obtain the following scaling laws for δ ( m ) and δ ( k ) constitutes a complete covariant scalar representation that is equivalent and alternative to (17). There are important regularity issues associated with variables (35), which we discuss in detail in Appendix A. For example, any one of the δ ( A ) might diverge if A$_{q}$ → 0 under regular conditions (no violation of (9) and finite curvature scalars). However, as we show in Appendix A 4 this does not lead to a curvature singularity, as the Riemann tensor frame components do not diverge (see Appendix A 2). Also, the regular radial range of some of the hypersurfaces $^{3}$T [ t ] is necessarily restricted by a central singularity, which does not invalidate the regu- larity of these scalars away from the singular coordinate surface (see Appendix A 5). While these issues must be taken into consideration, they do not affect the usage of the variables (35). V. SOME THEORETICAL AND PRACTICAL IMPLICATIONS OF QUASI–LOCAL SCALARS. 1 + δ ( m ) = 1 + δ ( m ) i Γ , (42a) 2 3 + δ ( k ) = 2 / 3 + δ ( k ) i Γ , (42b) δ ( m ) − 3 2 δ ( k ) = δ ( m ) i − (3 / 2) δ ( k ) i Γ . (42c) Applying (31) to the Hamiltonian constraint (38) yields the scaling law for δ ( H ) 2 δ ( H ) = 2 m$_{q}$ δ ( m ) − k$_{q}$ δ ( k ) 2 m$_{q}$ k$_{q}$ = 2 m$_{qi}$ δ ( m ) − k$_{qi}$ L δ ( k ) 2 m$_{qi}$ k$_{qi}$ L . (43) The quasi–local variables and their relative fluctua- tions introduced in the previous section lead in a natural way to an initial value parametrization of LTB models, which can be very helpful for numeric and qualitative work. Various elements of this parametrization has been already considered in the literature [14–17], but only as a useful set of coordinate ansatzes whose justification is merely their resemblance with FLRW parameters. How- ever, these ansatzes can acquire a clear covariant meaning by its relation with quasi–local scalars.