Lemaître–Tolman–Bondi solutions in the Newtonian gauge: from strong to weak fields

Lematre-Tolman-Bondi (LTB) solutions are used frequently to describe the collapse or expansion of spherically symmetric inhomogeneous mass distributions in the Universe. These exact solutions are obtained in the synchronous gauge where non-linear dynamics (with respect to the Friedmann-Lematre-Roberston-Walker (FLRW) background) induce large deviations from the FLRW metric. In this paper we show explicitly that this is a gauge artefact (for realistic sub-horizon inhomogeneities). We write down the non-linear gauge transformation from synchronous to Newtonian gauge for a general LTB solution using the fact that the peculiar velocities are small. In the latter gauge we recover the solution in the form of a weakly perturbed FLRW metric that is assumed in standard cosmology. Furthermore we show how to obtain the LTB solutions directly in Newtonian gauge and illustrate how the Newtonian approximation remains valid in the non-linear regime where cosmological perturbation theory breaks down. Finally we discuss the implications of our results for the backreaction scenario.