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

We introduce a quasi-local integral functional and scalar quasi-local variables to examine a wide class of spherically symmetric inhomogeneous spacetimes that generalize the Lemaitre-Tolman-Bondi (LTB) dust solutions ("LTB" spacetimes). By using these variables, we can transform the fluid flow evolution 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 the synchronous gauge under the long wavelength approximation. The formalism has a significant potential for cosmological applications, as it allows one to examine a wide variety of sources with different "equations of state", generalizing known FLRW solutions to idealized but non-trivial and non-linear inhomogeneous conditions. The quasi-local functional can be reformulated as a weighed proper volume average distribution, with the weight factor given by a scalar invariant related to the quasi-local mass-energy function. The back-reaction terms, emerging in Buchert's proper averaging formalism, can be expressed as differences between fluctuations of averaged and quasi-local energy densities. By comparing this average with the weighed quasi-local one, we can define a binding energy functional related to spatial gradients of the averaged and quasi-local variables that appear in the back-reaction terms.