Is a Kerr black hole related to the Hopf fibration? The Hopf fibration is a way of divided space into circles. There is a planar circle at the "centre" with other circle wrapped round it in a stack of toroidal surfaces.
The Kerr rotating black hole it has a central ring singularity.
It seems like the metric has some similarities to the Hopf fibration. I wonder if the tidal forces converted to a vector field would form a Hopf fibration? Or some way to convert the torsion field into a vector field.
As far as I can tell the two options for a smooth vector field going round a ring singularity would either take the form ofa Hopf fibration or alternatively all the flow lines would have to spiral around or into the singularity.
Do you know of any connection?
(I don't think the orbital paths would form a Hopf fibration since these would be elliptical and also have precession.)
 A: 
Is a Kerr black hole related to the Hopf fibration?

Yes. Kerr metric is related to the Hopf fibration via certain twisting shear–free null geodesic congruence.
Hopf fibration is a realization of sphere $S_3$ as a principal fiber bundle over $S_2$ base space with $U(1)$ structure group. By applying stereographic projection to the sphere $S_3$ we would obtain description of space $\mathbb{R}^3$ in terms of disjointed and singly linked circles organized in nested tori. 
To give this structure interpretation in relativistic context we could consider the Hopf fibration a projection on a time-slice of a congruence of null geodesics, the Robinson congruence. There is Kerr's theorem that relates this geodesic shear-free null congruence in Minkowski spacetime to the zero set of an arbitrary analytic function, and amazingly this theorem could also be extended to the curved Kerr–Schild spaces.
Kerr–Schild spacetime is the spacetime with the following metric:
$$
 g_{\mu\nu}=\eta_{\mu\nu}+2 h(x^\mu) k_\mu k_\nu,
$$
where $\eta_{\mu\nu}$ is Minkowski spacetime metric, $h$ is a scalar function, and $k_\mu$ is a null vector field. (Note, that $k$ is null relative to both the full spacetime metric $g$ and auxiliary Minkowski metric $\eta$). A result by Debney, Kerr & Schild provides a class of solution of Einstein or Einstein–Maxwell field equations in Kerr–Schild form (with $k$ being the tangent vector to null and shear-free geodesic congruence) parametrized by two arbitrary functions of complex variable (one mainly associated with the metric, the other with EM field). Among the solutions thus generated are Kerr and Kerr–Newman spacetimes.
In order to interpret this geodesic congruence as related to Hopf fibration one needs to do the projection on a time–slice of auxiliary Minkowski spacetime.
More detailed account could be found in the thesis of J.W. Dalhuisen:


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*Dalhuisen, J.W., The Robinson congruence in electrodynamics and general relativity, (2014), Doctoral Thesis, Leiden University, OA pdf.

