First of all, I will use a conclusion given by gravitomagnetism, but the example will concern Kerr black holes, and therefore lies in the realm of strong gravitational fields.

Well, gravitomagnetism gives us two Maxwell's equations analogs $[1]$, $$\vec{\nabla} \cdot \vec{B}_{g} = 0 \tag{1}$$ $$\vec{\nabla} \times \vec{B}_{g} = \frac{16\pi G}{c^2}\vec{J} \tag{2}$$

This fact motivated a friend of mine to pose a interesting question:

Why there are no gravitational magnets? $\tag{3}$

This question, on the other hand, motivated me to think about a situation:

Suppose a large region of intergactic space (i.e. a region with almost no curvature, therefore a region with almost no gravity). Then fill this region with identical Kerr black holes rotating with the same angular velocity and in the same axis of rotation. By analogy, this intuitive picture occurs with spins of atoms of a magnetic material: the alignment of spins. $\tag{4}$

Well, concerning the situation (4), can we say that this situation generates an "apparent gravitational magnet"?

$[1]$ HOBSON.M.P; et al. General Relativity: An Introduction for Physicists. Cambridge. pages 490-492

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    $\begingroup$ Gravitoelectromagnetism is a weak-field approximation, and in particular isn't valid near Kerr black holes. $\endgroup$ – probably_someone May 19 at 22:01

As I mentioned here, you can certainly have gravitational analogues of magnets. Since the source of gravitomagnetic field is momentum, a rotating cylinder looks like a gravitational magnet from far away for precisely the same reason that solenoid looks like an ordinary magnet. So if you have two parallel rotating cylinders, for instance, there should be a gravitomagnetic repulsion force.

The reason that you don't hear about gravitational magnets often is that in ordinary electromagnetism, magnetic fields are generally smaller than electric fields by a factor of $v/c$. So in general, you would expect magnetic effects to be negligible.

Now, in electromagnetism you can have objects whose magnetic effects dominate because their total charge can be zero, and we call those objects magnets. But in gravity, there is no such thing as negative charge, so the gravitoelectric field (i.e. the ordinary gravitational field $\mathbf{g})$ is always much larger than the gravitomagnetic field.

That's no longer true when the source matter is moving quickly, $v \sim c$, but in that case gravitoelectromagnetism is not a good approximation. Other things could also happen when the effects of gravity become strong, but gravitoelectromagnetism doesn't apply in those cases, either. So there is no case where something perfectly analogous to a magnet exists.

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The paper [1] claims that two spinning black holes can repel each other:

the remarkable feature of the newly found binary black-hole configurations is that the black holes in them can repel each other due to prevailing of the spin-spin repulsion over gravitational attraction, which prevents the two black holes from merging into a single black hole.

I don't know if this claim has been independently checked yet (I haven't checked it in any detail), but if it is correct, then the fact that spin is essential suggests some kind of analogy with magnetism. Can't be a perfect analogy, because gravity is a spin-2 interaction and electromagnetism is a spin-1 interaction, but maybe it still qualifies as some kind of analogy.

[1] Manko, Ruiz, "'Black hole-naked singularity' dualism and the repulsion of two Kerr black holes due to spin-spin interaction," https://arxiv.org/abs/1803.03301

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