I am thinking about a way to design a process which makes a extremal Reissner-Nordström black hole out of a non extremal one, i.e. it would violate the third law of black hole thermodynamics.
- Consider the case $\vert Q \vert < M$. To achieve extremality of the black hole I either need add charges or get rid of some of the mass of the black hole. The first option is not possible because in the process the surface gravity gets weaker and weaker, hence at some point the Coulomb repulsion outweighs gravitational pull of the black hole and the charge won't cross the horizon. For the other option I'm not quite sure, isn't it theoretical possible that the black hole radiates, losing mass in the process, and achieving $\vert Q \vert = M$ eventually? What are arguments against it?
- For the other case $\vert Q \vert > M$ I'm also quite unsure. I know that this case is forbidden by the cosmic censorship conjecture but let's assume such a black hole exists. Couldn't I just drop opposite charged particles into the black hole to obtain an extremal black hole? Would that mean that the third law of black hole thermodynamics relies on the cosmic censorship conjecture? (Wouldn't this make it also "just" a conjecture?)
Edit:
For (1) came up with an idea. Since, extremal black holes don't emit any Hawking radiation, a process where the mass is decreased by radiation would lead to a configuration where less and less mass is evaporating into radiation. This means that such a process would take infinitely long in order to obtain an extremal black hole.
