Abstract
At first the thermodynamic and evolutionary properties of Kerr black holes are clarified using the M–J plane, where M is the hole’s mass and J is its angular momentum. In this plane Schwarzschild black holes with h = 0 are distributed along the M-axis and extreme Kerr holes with h = 1 lie on the line J = M2, where $h \equiv J/4S$ is a non-dimensional parameter and S is the entropy. Taking into account possible accretion processes, we then derive the condition under which the third law of black-hole thermodynamics for Kerr holes is not violated. The condition is given in the form of as $\alpha \ge 1$, where the rate of change of a hole’s state, dh/dM, is proportional to $(1-h)^\alpha$ in the neighbourhood of $h \simeq 1$. If the rate is proportional to the vanishing surface gravity, gH, with which the hole has to accrete matter and angular momentum, α is given by $\alpha= 1+2/C$, where $dh/dM=Cg_\text H=C(1-h^2)/4M$, and C is a proportionality constant. In this case M, J and S diverge to infinity as a power law for $h \to 1$, and therefore no Kerr holes can reach the extreme Kerr state with the absolute zero temperature by accreting finite amounts of mass and angular momentum.
Gedanken experiments on nearly extremal black holes and the third law
