5 (containing a symmetry center) that are diffeomorphic to the product manifold so that where x = 0 marks a symmetry center. Because of spher- ical symmetry every scalar function on a domain D ( r ) is equivalent to a real valued function on the real interval ϑ ( r ). Consider now the following Definition 1: Quasi–local scalar functions A$_{q}$ . Let ϑ ( r ) be a radial domain in a regular $^{3}$T [ t ]. For every scalar function A [ t$_{0}$ ] : ϑ ( r ) → R , there is a “quasi–local dual function” A$_{q}$ [ t$_{0}$ ] : ϑ ( r ) R such that Comment 1: Definition (24) is invariant under arbitrary rescalings of the radial coordinate r = r (¯ r ) that do not violate (9) and (A7). In particular, the proper length ℓ = ∫ √ g$_{rr}$ d r can be used as the integration variable at each $^{3}$T [ t ] (though it is not useful as a global integral variable because ℓ = ℓ ( t, r )). Comment 2: In general, not all hypersurfaces $^{3}$T [ t ] in LTB models are fully regular, since the central singular- ities (expanding or collapsing) are not simultaneous and will intersect some of the $^{3}$T [ t ]. For these $^{3}$T [ t ] the do- main (23) must be suitably restricted and the integrals in (24) must be treated as improper integrals. We discuss this issue in A 5. Comment 3. The quasi–local functions A$_{q}$ in (24) are not “averages”, as they do not comply with the prop- erties of average distributions of a continuous random variable. They can be recast as proper volume averages with “weight factor” F if we define them as functionals. See [22] for discussion and clarification of this issue. The definition (24) leads to the definition of a “quasi– local volume” given by where we used (27a) and (27b). Considering (30) and (31), each scalar A in (17) can be uniquely expressed in A convenient way to relate the local covariant scalars A in (17) and their quasi–local duals A$_{q}$ follows by intro- ducing the relative deviations (fluctuations) Since the shear and electric Weyl tensors are covariant objects, then these fluctuations are also covariant. Notice that M, R, ˙ R = u $^{a}$∇ $_{a}$R and F = √$_{1 +}$$_{E}$ are invariants in spherical symmetry [26–28], hence m$_{q}$, k$_{q}$ and H q are covariant quantities. As a consequence of (28), the scalars Σ and E in (15) and (16), associated with the shear and electric Weyl tensors, become expressible as deviations or fluctuations of the local scalars H , m (or Θ , ρ ) with respect to their quasi–local duals: where, to simplify notation, we have defined The introduced quasi–local variables provide a com- plete and very useful representation of covariant scalars that is alternative to (17). Applying the definition (24) to (3), (13) and (21) we obtain the quasi–local duals of ρ, $^{3}$R and Θ where we expressed (14) as Θ = [ln( R $^{2}$R $^{′}$)]˙ and the com- mutation of ∂/∂t with the integrals in (24). The quasi– local functions comply with the following properties: ˙ V q q = 3 ˙ R R = Θ$_{q}$ , V ′ q q = 3 R ′ R , (26) D ( r ) = ϑ ( r ) × S $^{2}$( θ, φ ) , ϑ ( r ) ≡ { x | 0 ≤ x ≤ r } , (23) A$_{q}$ [ t$_{0}$ ]( r ) = ∫ r 0 A F d V p r 0 0 d p = ∫ r 0 A ( t$_{0}$, x ) R $^{2}$( t$_{0}$, x ) R $^{′}$( t$_{0}$, x ) d x ∫ r 0 R $^{2}$( t$_{0}$, x ) R $^{′}$( t$_{0}$, x )d x , (24) where F is given by (22). Since it is clear that t is an arbitrary but fixed parameter in these integrals, we will omit (unless it is necessary) to express explicitly the time dependence of scalars A and A$_{q}$ . A ′ q = ( A$_{q}$ ) ′ = V ′ q q [ A − A$_{q}$ ] , (27a) A ( r ) − A$_{q}$ ( r ) = 1 V $_{q}$( r ) ∫ r 0 A ′ V q d x, (27b) ˙ A$_{q}$ = ( A$_{q}$ )˙ = ( ˙ A )$_{q}$ + (Θ A )$_{q}$ − Θ$_{q}$ A$_{q}$, (27c) 2 m$_{q}$ = 2 M R 3 , k$_{q}$ = − E R $_{2}$, H 2 q = ˙ R 2 R 2 = 2 m$_{q}$ − k$_{q}$. (28) 2 m ≡ κ 3 ρ, k ≡ $^{3}$R 6 , H ≡ Θ 3 , (29) Σ = − ( H − H $_{q}$) , E = − ( m − m$_{q}$ ) . (30) δ ( A ) ≡ A − A$_{q}$ A$_{q}$ = A ′ $_{q}$/A$_{q}$ 3 R $^{′}$/R = 1 A$_{q}$ ( r ) ∫ r 0 A $^{′}$V$_{q}$ d x, (31) V $_{q}$( r ) = ∫ D ( r ) F d V p = 4 π ∫ r 0 R $^{2}$R $^{′}$d x = 4 π 3 R $^{3}$( r ) , (25) B. Properties. D. Fluctuations of quasi–local duals. C. A representation of covariant quasi–local scalars. A. Quasi–local scalar variables.