  Local conformal symmetry introduces the conformal curvature (Weyl tensor)
that gets split into its (gravito-) electric and magnetic (tensor) parts.
Newtonian tidal forces are expected from the gravitoelectric field, whereas
general-relativistic frame-dragging effects emerge from the gravitomagnetic
field. The symmetric, traceless gravitoelectric and gravitomagnetic tensor
fields can be visualized by their eigenvectors and eigenvalues. In this essay,
we depict the gravitoelectric and gravitomagnetic fields around a slowly
rotating black hole. This suggests that the phenomenon of ultra-fast outflows
observed at the centers of active galaxies may give evidence for the
gravitomagnetic fields of spinning supermassive black holes. We also question
whether the current issues in our contemporary observations might be resolved
by the inclusion of gravitomagnetism on large scales in a perturbed FLRW model.
