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## III. COVARIANT OBJECTS AND THEIR “1+3” EVOLUTION EQUATIONS. 

The normal geodesic 4–velocity in (1)–(2) defines a natural time slicing in which the space slices are the hypersurfaces $^{3}$T [ t ], orthogonal to u a and marked by arbitrary constant values of t [46]. The metric of the $^{3}$T [ t ] is simply h$_{ab}$ = g$_{ab}$ + u$_{a}$u$_{b}$ = g$_{ij}$δ i $_{a}$δ j b (or (1) with d t = 0). The proper volume element and 3–dimensional Ricci scalar associated with these hypersurfaces are 

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<img src='content_image/382957.jpg'>

Given this time splitting, the covariant objects in dust LTM models are ρ and $^{3}$R in (4) and (13), together with the shear and electric Weyl tensors, σ$_{ab}$ = ˜ ∇ ( $_{a}$u$_{b}$$_{)}$ − (Θ / 3) h$_{ab}$ and E$_{ab}$u$_{c}$u$_{d}$C $^{abcd}$, where ˜ ∇ a = h b a $_{b}$, and C abcd is the Weyl tensor, and the expansion scalar, Θ, is given by 

Since LTB models (as all spherically symmetric space- times) are locally rotationally symmetric (LRS), all their covariant objects are either scalars or entirely charac- terized by covariant scalars [36]. Therefore, the spatial trace–less shear and electric Weyl tensors can be given in terms of single scalar functions, Σ , E , in a covariant manner: 

<img src='content_image/382952.jpg'>

<img src='content_image/382953.jpg'>

where Ξ ab ≡ h ab − 3 χ $^{a}$χb and χ a = √$_{h}$ rr δ a r is the unit vector orthogonal to u a and to the 2–spheres orbits of SO(3). 

A description of the dynamics of LTB models, which is an equivalent alternative to the field equations and their analytic solutions (5)–(7), follows by considering the following set of covariant scalars 

which completely characterize these models. Hence, their dynamics is completely determined by suitable evolution equations and constraints for these scalars. Following the “fluid flow” or covariant “1+3” framework of Ehlers, Ellis, Bruni, Dunsbury and van Ellst [32–34, 36], and considering (15) and (16), the 1+3 evolution equations 

<img src='content_image/382964.jpg'>

<img src='content_image/382950.jpg'>

for the scalars (17) are 

while the spacelike and Hamiltonian constraints are 

<img src='content_image/382958.jpg'>

Solving this system of partial differential equations de- termines the metric functions and remaining quantities though the definitions (3), (4) and (12)–(16) that relate these functions to the scalars in (17). However, it can be a difficult system to use in numerical work because time and radial derivatives cannot be (in general) de- coupled. In the following section we will introduce an alternative set of covariant scalars whose corresponding evolution equations (see section XI) are wholly equivalent to (18)–(20) but are easier to handle. 

## IV. QUASI–LOCAL SCALAR FUNCTIONS AND FLUCTUATIONS. 

The function M that appears in (3)–(4) is for LTB models the Misner–Sharp quasi–local mass–energy func- tion, which is an important invariant in spherically sym- metric spacetimes [26–28]. It is basically the integral along a spherical comoving domain of the field equation (4), which will be well defined if we assume the existence of a symmetry center (at r = 0) and can be given as 

<img src='content_image/382954.jpg'>

where the spherical comoving domain D ( r ) is defined fur- ther below and we have used the notation ∫ r 0 ... d x = ∫ x = r x =0 ... d x . It is evident from (21) and (22) that M is a proper volume integral “weighed” by the invariant scalar F which generalizes the “ γ ” factor of Special Relativity [27, 28]. The integral definition (21) motivates the in- troduction of similar “weighed” proper volume integral functions which turn out to be very useful in the study of LTB models. In order to define such integrals, we re- mark that every regular $^{3}$T [ t ] contains comoving regions 

<img src='content_image/382960.jpg'>

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<img src='content_image/382961.jpg'>

<img src='content_image/382963.jpg'>

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