22 5. Restrictions of the radial range due to a curvature singularity. 4. Possible blowing up of the δ ( A $^{)}$. Appendix B: Local vs. quasi–local spatial curvature. The kinematic class of LTB models is determined by the sign of the initial quasi–local spatial curvature k$_{qi}$ (or E = − k$_{qi}$R 2 $_{i}$). From (37), the sign of k$_{qi}$ determines the sign of k$_{q}$ . It is evident that parabolic models (contain- ing a symmetry center) are spatially flat, as k$_{qi}$ = k$_{i}$ = 0 trivially implies k$_{q}$ = k = 0. However, it is not obvi- ous if a given sign of k$_{q}$ in regular hyperbolic or elliptic LTB models also determines the sign of the local spatial curvature k = $^{3}$R / 6. In order to examine this point, we rewrite (41b) as We examine the relation between k$_{q}$ and k and their initial values for hyperbolic and elliptic models. k = k$_{q}$ [ 1 + δ ( k ) ] = k$_{qi}$ 3Γ L 2 [ Γ + 3 ( δ ( k ) i + 2 3 )] . (B1) where r$_{bb}$ is the intersection of t bb ( r ) and the constant t value associated with the $^{3}$T [ t ]. As a consequence, the integration range in the definition (24) for these $^{3}$T [ t ] must be ¯ ϑ ( r ), not ϑ ( r ). For a collapsing singularity in elliptic models we have exactly the same situation, but the involved hypersurfaces are those with t ≥ t coll ( r ), with the lower radial bound given by r = r$_{coll}$ marking the intersection of t coll ( r ) and the $^{3}$T [ t ]. However, this range restriction has no consequence in the definition and usage of the quasi–local scalars A$_{q}$ , as the involved integrals can be treated simply as stan- dard improper integrals. We define at each $^{3}$T [ t ] with t ≤ t bb ( r ) the incumbent integrals with their lower in- tegration limit as y = r$_{bb}$ + ϵ , for an arbitrarily small ϵ > 0, and then obtain the limit as ϵ → 0. Off course, since scalars like m, k, H diverge in this limit, m$_{q}$, k$_{q}$ and H q might diverge as well, but the functions are well de- fined and behaved in the range ¯ ϑ ( r ). This restriction only prevents the A$_{q}$ from taking values r < r$_{bb}$ for t ≤ t bb ( r ), and so all results that involve these scalars can be trivially extended to include hypersurfaces $^{3}$T [ t ] for t ≤ t bb ( r ) and t ≥ t coll ( r ). specified on an arbitrary fiducial or “initial” hypersurface $^{3}$T [ t$_{i}$ ]. must hold for every r and for any arbitrary pair of distinct hypersurfaces $^{3}$T [ t$_{i}$ ] , $^{3}$T [ t$_{j}$ ]. As consequence, (9) can be given as an initial condition Since (9) is valid in each $^{3}$T [ t ] and F = √$_{1 +}$ E does not depend on t , a zero of F (characteristic of S 3 topology) or the fulfillment of F > 0 (characteristic of R 3 topology) will be common to all $^{3}$T ( t ). Hence, all $^{3}$T [ t ] belong to the same topological class and thus When we introduced the quasi–local scalars in (24), we assumed that the integration range ϑ ( r ) defined by (23) was fully regular. However, in general, the coordinate locus (58a) of the central singularity (expanding or col- lapsing) is not simultaneous ( i.e. not marked by t = t bb = const., so that t ′ bb = 0). Consider the case of an expand- ing non–simultaneous singularity (big bang), marked by the curve [ t bb ( r ) , r ], with t ′ bb ≤ 0, in the ( ct, r ) coordinate plane, so that L ( t bb ( r ) , r ) = 0 (the collapsing singularity is analogous). The hypersurfaces $^{3}$T [ t ] for t ≤ t bb are only regular for the semi open subset ¯ ϑ ( r ) ≡ { x | r$_{bb}$ < x ≤ r } ⊂ ϑ ( r ) , (A8) sign[ R $^{′}$( t$_{i}$, r )] = sign[ R $^{′}$( t$_{j}$, r )] , (A6) sign ( R ′ $_{i}$) = sign M ′ = sign F , (A7) From their definition (31), it is evident that relative fluctuations δ ( A ) will diverge under certain regular con- ditions ( i.e continuous and finite A and A$_{q}$ ) if in the integration domain ϑ ( r ) of (24) A$_{q}$ has a zero that is not a common same order zero of A . This situation occurs in various situations that do not violate regularity, such as the zero of H q at t = t max as elliptic models or regions pass from expansion ( H q > 0) to collapse ( H q < 0), or in mixed elliptic/hyperbolic configurations where k$_{q}$ passes from positive to negative at some comoving radius r = r$_{b}$ (see section XIV D). However, the quantities δ ( m $^{)}$, δ( k ) and δ ( H ) in all basis components in (A4) only appear in terms such as m$_{q}$δ ( m $^{)}$, k$_{q}$ δ ( k ) and H $_{q}$δ ( H $^{)}$, which do not diverge at a zero of m$_{q}$ , k$_{q}$ or H $_{q}$. Hence, the blowing up of δ ( H ) and δ ( k ) because of zeroes of k$_{q}$ or H q does not affect the regularity of the solutions, as conveyed by the continuity and finiteness of (A4). Nevertheless, in these cases it will be preferable to use A$_{q}$δ ( A $^{)}$, instead of δ ( A $^{)}$, for studying the time evolution of scalars (which justifies the use of (103) over (102) for numeric work involving expanding/collapsing regions or models). • “Open models” : $^{3}$T [ t ] homeomorphic to R $^{3}$. There is only one symmetry center, at r = 0. As a consequence of the regularity condition (9), R ′ > 0 and E > − 1 must hold for all r , and so this topology is compatible with regions or models of all kinematic classes (hyperbolic, parabolic or elliptic with − 1 < E < 0). We discuss the asymptotic radial regime associated with this topology in a separate article. • “Closed models” : $^{3}$T [ t ] homeomorphic to S $^{3}$. There are two symmetry centers, at r = 0 and r = r$_{c}$ . Since R ( t, 0) = R ( t, r$_{c}$ ) = 0 for all t , then there must exist a turning value of R so that R $^{′}$( r$_{tv}$ ) = 0 where 0 < r$_{tv}$ < r$_{c}$ .