Blandford Znajek Mechanism

The Blandford Znajek mechanism is a process by which energy can be extracted by a rotating black hole in a magnetic field $B$. The details are quite complicated, but useful results can be achieved with merely dimensional analysis.

Suppose there's a slowly rotating black hole. The rotational energy is of the order of $U_r = a^2 M c^2$, where $a$ is the spin parameter, $M$ is the mass of the black hole, and $c$ is the speed of light. The time scale over which this energy is radiated can be evaluated in the following way. The energy is radiated in the form of electromagnetic radiation, so the power would be of the form

$P \propto B^2 R^2 c$

where $R$ is the Schwarzschild radius. To change the properties of a black hole considerably, the radiated energy must be comparable to the black hole's energy, hence the relevant time scale is

$\tau \propto \frac{M c^2}{B^2 R^2 c}$

We assume that the radiated energy mostly comes at the expanse of the rotational energy, so the power is proportional to the ratio of the rotational energy and the time scale calculated earlier

$P \propto \frac{U_r}{\tau} \propto a^2 B^2 R^2 c \propto a^2 B^2 \frac{G^2 M^2}{c^3}$