Description of magnetic field in black holes

[Srr]

Given a Schwarzschild BH. A neutron fall into the BH. The neutron having non zero magnetic moment will carry a magnetic field B with it.
How do I describe the new system, on which parameters will the metric depend?
In term of classical GR, Kerr Newman solution provides a B in term of
the charge Q and angular momentum J (and the no hair theorem is satisfied).
I see a paradox might emerge.
1) If the B from the added neutron can be described as a Kerr Newman solution, then I don't know how to explain charge conservation. For both the initial systems schwarzschild BH and neutron, Q is zero.
2) If I have instead a solution for the metric with zero Q and J, the metric will have to depend on the intrinsic magnetic moment. This would violate the no hair theorem

Another way to state the problem is the following. Consider a macroscopic neutral magnet. Somehow it shirks to form a BH. Again, which will be the parameter in the metric?
If Q and J, then I cannot explain Q conservation.
If the magnetic moment, then no hair theorem is violated.

I have been thinking about this for a while, maybe classical GR is not enough.

 

[lbrits]

The no-hair theorem is actually a family of no-hair theorems, one for each set of fields that you are considering. So the Kerr case has to do with gravity coupled to the electromagnetic field.

I don't think there's a classical field that gives rise to the neutron's spin (whereas the electron's Dirac field has spin, the neutron is a composite object). You could probably force one out and maybe the no-hair theorem would apply to that. But I think your final conclusion is probably correct.

 

[Entropia]

It is possible that additional theories such as quantum gravity or string theory may be needed to fully understand the behavior of magnetic fields in black holes. However, there are some proposed solutions within classical GR that address this paradox.

One possible solution is that the magnetic field of the neutron is not actually carried into the black hole, but instead is "frozen" in the event horizon. This means that the magnetic field remains on the surface of the black hole and does not affect the interior of the black hole. In this case, the metric would still depend on Q and J, and charge conservation would not be violated.

Another solution is that the magnetic field of the neutron is actually converted into gravitational energy within the black hole, as suggested by some theories. This would mean that the magnetic field is no longer a separate parameter in the metric, but is instead incorporated into the overall mass and energy of the black hole. In this case, the no hair theorem would still hold true.

It's also important to note that the Kerr Newman solution is a classical solution, and does not take into account the effects of quantum mechanics. As we know, at the quantum level, particles can have both spin and magnetic moment, which may play a role in the behavior of magnetic fields in black holes.

In conclusion, the behavior of magnetic fields in black holes is still an area of active research and there may not be a definitive answer within classical GR. However, it is possible that a combination of classical and quantum theories may provide a more complete understanding of this phenomenon.