One can imagine other expressions for quasi-local mass, may be more involved, in axial symmetry. On the contrary, reversing the argument, a proof of will certainly suggest that the mass has physical meaning for non-stationary black holes as a natural quasi-local mass (at least in axial symmetry). Also, the inequality provide a non trivial control of the size of a black hole valid at any time.

In a seminal article Penrose [@Penrose73] proposed a remarkably physical argument that connects global properties of the gravitational collapse with geometric inequalities on the initial conditions. That argument lead to the well known Penrose inequality (\[eq:pen\]) for dynamical black holes (without any symmetry assumption). In the following we review this argument imposing axial symmetry, where angular momentum is conserved. And, more important, we include a relevant new ingredient: we assume that the inequality (\[eq:JA\]) holds.

We will assume that the following statements hold in a gravitational collapse:

-   Gravitational collapse results in a black hole (weak cosmic censorship).

-   The spacetime settles down to a stationary final state. We will further assume that at some finite time all the matter have fallen into the black hole and hence the exterior region is vacuum.

Conjectures 