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A new spinorial strategy for the construction of geometric inequalitiesinvolving the Arnowitt-Deser-Misner (ADM) mass of black hole systems in generalrelativity is presented. This approach is based on a second order ellipticequation (the approximate twistor equation) for a valence 1 Weyl spinor. Thishas the advantage over other spinorial approaches to the construction ofgeometric inequalities based on the Sen-Witten-Dirac equation that it allows tospecify boundary conditions for the two components of the spinor. This greatercontrol on the boundary data has the potential of giving rise to new geometricinequalities involving the mass. In particular, it is shown that the mass isbounded from below by an integral functional over a marginally outer trappedsurface (MOTS) which depends on a freely specifiable valence 1 spinor. Fromthis main inequality, by choosing the free data in an appropriate way, oneobtains a new nontrivial bounds of the mass in terms of the inner expansion ofthe MOTS. The analysis makes use of a new formalism for the $1+1+2$decomposition of spinorial equations.

New spinorial approach to mass inequalities for black holes in general relativity