Alice, Bob and Charge falling into Black Hole

[Dmitry67]

Alice and Bob are hovering at say 10Rs at opposite sides of the non-rotarting black hole. Bob drops charged body and it is free falling into black hole (again, at opposite side from Alice). They both measure the intensity of electromagnetc field. Initially, of course, intensity at Bobs location is much higher.

Question: do Alice and Bobs measurements become equal in finite time (as Black Hole does not have hair and it can't have any non-symemtric charge distribution), or, as for Bob and Alce object 'freezes' at the event horizon, Bob measures slightly more intensive EM field forever?

My bet is they never become equal: even body reaches the singularity and situation becomes symmetric, the change in EM field propagates from EH very slowly.

Now slightly different situation: Bob and Alice observe neutron star with non-symmetric charge distribution, where the unbalanced charge is already below future horizon. Now star suddenly collapses. I conclude that Bob and Alice wll still measure different values. Am I right?

 

[bcrowell]

There's a good discussion of the kind of issue you're talking about in ch. 6 and 7 of Black Holes and Time Warps, by Kip Thorne.

I don't think the nonzero charge on the object really changes the answer to your question. With an uncharged object, we get all the same issues, but there is only gravitational radiation, not gravitational+EM.

The various no-hair theorems are all theorems about static solutions. They don't have anything to say about the process of collapse.

The Penrose-Hawking singularity theorems prove geodesic incompleteness, i.e., they only talk about the experience of an infalling observer, not about how the process appears from the outside.

In your example, I believe what happens is that gravitational and electromagnetic radiation very quickly remove the nonuniformity produced by the charged object that Bob drops in. Bob has only added a finite amount of mass-energy to the system, so radiation can only carry away a finite amount of energy. As the object approaches the horizon, its radiation (both electromagnetic and gravitational) becomes infinitely weak and redshifted. To a distant observer, the externally observable fields converge asymptotically to those of a static black hole.