charge from infinity brought into black hole horizon, RN and Kerr

eherrtelle59

Let's consider the 1st Law of BH thermodynamics, which relates change of mass/energy to change of horizon area, angular momentum and charge:

[itex] dM=\frac{\kappa}{8*pi} dA + \Omega dJ + \Phi dQ [\itex]


Now

Case 1) In the Reissner Nordstrom J=0 and m>Q (but only slightly larger---it's a "near" extremal solution), we consider a small amount of charge dQ. In considering amount of work to be done to bring this charge dQ in from infinity to the BH horizon, is the extremal solution m=Q possible?

Or, would [itex]\Phi[\itex] blow up, and therefore M would explode?

Case 2) Now consider the full Kerr Newman above

Here, this seems like such a scenario is possible.

Try adding a small bit of charge from infinity, and J and A could go to zero.
Try adding a small bit of rotation dJ, and Q and A could go to zero.

Apparently there is a theorem by R. Wald which addresses this, but I'm not completely sure I understand it.

Does this make sense to anyone?