Hawking radiation with angular momentum and the entropy variation in a Kerr black hole

Considering Hawking radiation with angular momentum, we propose a modified Stefan–Boltzmann law about the particles with both energy and angular momentum radiated from a Kerr black hole. Based on the modified Stefan–Boltzmann law, we calculate the entropy in the interior volume of the Kerr black hole and investigate the proportional relation between the variation of this entropy and the variation of Bekenstein–Hawking entropy under Hawking radiation. Comparing to Hawking radiation without angular momentum, we find that whether the particles radiated from the black hole take angular momentum cannot affect the proportional relation between the two types of entropy. Furthermore, the proportionality coefficient of the two types of entropy is investigated. It is found that this coefficient is approximately a constant except the late stage of Hawking radiation. Moreover, the proportional relation can degenerate to the Schwarzschild case when the angular momentum of the black hole completely disappears. For a Kerr black hole, the extremal black hole can technically not prevent and stop the evaporation anymore under Hawking radiation with angular momentum.