  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.
