The best one can hope is to obtain a quasi-local expression. The same applies to the angular momentum. In general, it is difficult to find physically relevant quasi-local quantities like mass and angular momentum (see the review article [@Szabados04]). However, in axial symmetry, there is a well defined notion of quasi-local angular momentum: the Komar integral of the axial Killing vector. Moreover, the angular momentum is conserved in vacuum. That is, axially symmetric gravitational waves do not carry angular momentum.

Then, for axially symmetric dynamical black holes we have two well defined quasi-local quantities: the area of the horizon $A$ and the angular momentum $J$. Note that the inequality relates only quasi-local quantities.

Using $A$ and $J$ we can define the quasi-local mass for a dynamical black hole by the Kerr formula (\[eq:3\]), that is $$\label{eq:masa}
  {m_{bh}}= \sqrt{\frac{A}{16\pi}+\frac{4\pi J^2}{A}}.$$ This is, in principle, just a definition. Since ${m_{bh}}$ is given by the Kerr formula (\[eq:3\]) it automatically satisfies the inequalities (\[eq:mj\]) and (\[eq:pen\]). However, the relevant question is: does ${m_{bh}}$ describes the quasi-local mass of a non-stationary black hole? This question is closed related to the validity of the inequality (\[eq:JA\]) in the dynamical regime. In order to answer it 