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We introduce a quasi-local integral functional and scalar quasi-localvariables to examine a wide class of spherically symmetric inhomogeneousspacetimes that generalize the Lemaitre-Tolman-Bondi (LTB) dust solutions("LTB" spacetimes). By using these variables, we can transform the fluid flowevolution equations into evolution equations for non-linear, covariant,gauge--invariant perturbations of Friedman-Lemaitre-Robertson-Walker (FLRW)cosmologies. In the linear limit, we obtain spherical perturbations in thesynchronous gauge under the long wavelength approximation. The formalism has asignificant potential for cosmological applications, as it allows one toexamine a wide variety of sources with different "equations of state",generalizing known FLRW solutions to idealized but non-trivial and non-linearinhomogeneous conditions. The quasi-local functional can be reformulated as aweighed proper volume average distribution, with the weight factor given by ascalar invariant related to the quasi-local mass-energy function. Theback-reaction terms, emerging in Buchert's proper averaging formalism, can beexpressed as differences between fluctuations of averaged and quasi-localenergy densities. By comparing this average with the weighed quasi-local one,we can define a binding energy functional related to spatial gradients of theaveraged and quasi-local variables that appear in the back-reaction terms.

Quasi-local variables, non-linear perturbations and back-reaction in spherically symmetric spacetimes