content_image/382237.jpg content_image/382238.jpg content_image/382247.jpg content_image/382243.jpg content_image/382246.jpg content_image/382236.jpg content_image/382240.jpg content_image/382241.jpg content_image/382242.jpg content_image/382244.jpg content_image/382245.jpg
(containing a symmetry center) that are diffeomorphic to the product manifold 

<img src='content_image/382237.jpg'>

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 

<img src='content_image/382238.jpg'>

F 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$_{q}$ . 

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 

<img src='content_image/382247.jpg'>

## A. Quasi–local scalar variables. 

## B. Properties. 

so that 

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: 

## C. A representation of covariant quasi–local scalars. 

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 

<img src='content_image/382243.jpg'>

where, to simplify notation, we have defined 

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: 

Since the shear and electric Weyl tensors are covariant objects, then these fluctuations are also covariant. 

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) 

<img src='content_image/382246.jpg'>

where we used (27a) and (27b). Considering (30) and (31), each scalar A in (17) can be uniquely expressed in 

<img src='content_image/382236.jpg'>

<img src='content_image/382240.jpg'>

<img src='content_image/382241.jpg'>

<img src='content_image/382242.jpg'>

<img src='content_image/382244.jpg'>

<img src='content_image/382245.jpg'>

## D. Fluctuations of quasi–local duals. 

