Rigidity - There is a body of opinion that would hold that rigidity is not compatible with (special) relativity. There are two ways of getting to this conclusion. One is based on the Ehrenfest paradox, whereas the other, is that a rigid body just cannot co-exist with special relativity. I will treat the Ehrenfest paradox first.

The Ehrenfest paradox claims that if a stationary disk is rigid, it cannot be spun because in special relativity, the circumference would become shorter and hence the material would need to be compressed. Consequently, there can be no completely rigid body. At no point do I disagree with this: what I claim is that because as a black hole is rigid, it is impossible to change its' spin. To do so would change its' circumference in accordance with special relativity; this is impossible for a truly rigid body.

The second argument is more tricky. It is argued that a rigid body will move as one; hit it with a hammer and the far side will move instantaneously. This implies a speed of sound inside the material that is infinite, clearly in contravention of the speed of light is the highest attainable speed, in special relativity.

The first line of enquiry is to think about what would happen if we hit a black hole with a hammer. If you believe in a central singularity, the hammer will traverse the event horizon without impediment. You will, of course, lose the hammer in the process(, so there are no means of making a measurement of the rigidity of the black hole. In the case of a solid core (as proposed here) the hammer would never actually hit the event horizon. It would not suffer any resistance, nor impart any momentum to the black hole until an infinite time has elapsed.

The claim made here for the rigidity of black holes is based on the impossibility of moving any two points inside the event horizon closer together. Does special relativity stop this happening? No, because in a region of space where time is unchanging, we cannot apply special relativity. Just look at the equations of special relativity and then set the relative velocity to zero. There is then no effect on the length of anything, or on clocks which are all stationary. No effects and hence no restrictions.