A. Maciel, M. Le Delliou, __J. P. Mimoso__

**Abstract**

In this work we revisit the definition of matter trapping surfaces (MTS) 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 in spherical symmetry and thus isolate one preferred spatial direction that allows straightforward translation into a dual null subbasis and deduction of the geometric apparatus that follows. We 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 a generalized Tolman-Oppenheimer-Volkov functional. Further using the Misner-Sharp mass and its perfect fluid definition, we relate the spatial 2-expansion to the fluid pressure, density, and acceleration. The Raychaudhuri equations also allow 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 realize that the MTS can exist only 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 nonvanishing null expansion that 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.

**Physical Review D**

Volume 92, Issue 8, Page 083525

2015 October