theorem we have that the area of the black hole increase with time and hence $$\label{eq:15}
  A_0\geq A.$$ Since gravitational waves carry positive energy, the total mass of the spacetime should be bigger than the final mass of the remainder Kerr black hole $$\label{eq:4}
  m\geq m_0.$$ The difference $m-m_0$ is the total amount of gravitational radiation emitted by the system.

To related the initial angular momentum $J$ with the final angular momentum $J_0$ is much more complicated. Angular momentum is in general non-conserved. There exists no simple relation between the total angular momentum $J$ of the initial conditions and the angular momentum $J_0$ of the final black hole. For example, a system can have $J=0$ initially, but collapse to a black hole with final angular momentum $J_0\neq 0$. We can imagine that on the initial conditions there are two parts with opposite angular momentum, one of them falls in to the black hole and the other escape to infinity. Axially symmetric vacuum spacetimes constitute a remarkable exception because the angular momentum is conserved. In that case we have $$\label{eq:59}
  J=J_0.$$ For a discussion of this conservation law in detail see [@dain12] and reference therein.

We have assumed that the inequality 