inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) extend (under appropriate assumptions) to the fully dynamical regime. Moreover, these inequalities are deeply connected with properties of the global evolution of Einstein equations, in particular with the cosmic censorship conjecture.

To discuss the physical arguments that support these inequalities in the dynamical regime it is convenient to start with the inequality . For a dynamical black hole, the physical quantities that are well defined are the total ADM mass $m$ of the spacetime and the area $A$ of the black hole horizon. The total mass $m$ of the spacetime measures the sum of the black hole mass and the mass of the gravitational waves surrounding it. In the stationary case, the mass of the black hole is equal to the total mass of the spacetime, but this is no longer true for a dynamical black hole. The mass $m$ is a global quantity, it carries information on the whole spacetime. In contrast, the area of the horizon $A$ is a quasi-local quantity, it carries information on a bounded region of the spacetime.

It is well known that the energy of the gravitational field cannot be represented by a local quantity (i.e. a scalar field). 