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A dual null formalism for the collapse of fluids in a cosmological background

In this work we revisit the definition of Matter Trapping Surfaces (MTS) introduced in previous investigations and show how it can be expressed in the so-called dual null formalism developed for Trapping Horizons (TH). With the aim of unifying both approaches, we construct a 2+2 threading from the 1+3 flow, and thus isolate one prefered spatial direction, that allows straightforward translation into a dual nul subbasis, and to deduce the geometric apparatus that follows. We remain as general as possible, reverting to spherical symmetry only when needed, and express the MTS conditions in terms of 2-expansion of the flow, then in purely geometric form of the dual null expansions. The Raychadhuri equations that describe both MTS and TH are written and interpreted using the previously defined gTOV (generalized Tolman-Oppenheimer-Volkov) functional introduced in previous work. Further using the Misner-Sharp mass and its previous perfect fluid definition, we relate the spatial 2-expansion to the fluid pressure, density and acceleration. The Raychaudhuri equations also allows us to define the MTS dynamic condition with first order differentials so the MTS conditions are now shown to be all first order differentials. This unified formalism allows one to realise that the MTS can only exist in normal regions, and so it can exist only between black hole horizons and cosmological horizons. Finally we obtain a relation yielding the sign, on a TH, of the non-vanishing null expansion which determines the nature of the TH from fluid content, and flow characteristics. The 2+2 unified formalism here investigated thus proves a powerful tool to reveal, in the future extensions, more of the very rich and subtle relations between MTS and TH.