let us analyze the evolution of ${m_{bh}}$.

For a dynamical black hole, by the area theorem, we know that the horizon area $A$ increase with time, see figure \[fig:4\].

![The area theorem. The horizon area of a dynamical black hole increase with time.[]{data-label="fig:4"}](area-th-p.pdf){width="6cm"}

In general, the quasi-local mass of the black hole is not expected to be a monotonically increasing quantity. Energy can be extracted from a rotating black hole by the Penrose process. However, if we assume axial symmetry then the angular momentum will be conserved at the quasi-local level. On physical grounds, one would expect that in this situation the quasi-local mass of the black hole should increase with the area, since there is no mechanism at the classical level to extract mass from the black hole. In effect, the Penrose process involves an interchange of angular momentum between the black hole and the exterior. But the angular momentum transfer is forbidden in axial symmetry. Then, both the area $A$ and the quasi- local mass ${m_{bh}}$ should monotonically increase with time in axial symmetry.

Let us take a time derivative of ${m_{bh}}$. To analyze this, it is illustrative to write down the complete differential, namely the first law 