# Description of magnetic field in black holes

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.

## Answers and Replies

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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.