This article outlines the search for an exact general relativistic description of the exterior (vacuum) gravitational field of a rotating spheroidal black hole surrounded by a realistic axially symmetric disc of matter. The problem of multi-body stationary spacetimes is first exposed from the perspective of the relativity theory (section 1) and astrophysics (section 2), listing the basic methods employed and results obtained. Then (in section 3) basic formulas for stationary axisymmetric solutions are summarized. Sections 4 and 5 review what we have learnt with MiroslavŽáček and Tomáš Zellerin about certain static and stationary situations recently. Concluding remarks are given in section 6. Although the survey part is quite general, the list of references cannot be complete. Our main desideratum was the informative value rather than originality -novelties have been preferred, mainly reviews and those with detailed introductions.
The fields of multi-body systems involving black holesThe subject of self-gravitating sources around rotating black holes is interesting in several respects, relevant from the point of view of the relativity theory itself as well as in the astrophysical context.First, due to the non-linearity of Einstein's equations, the field of a multi-body system is a traditional challenge where one typically does not manage with a simple superposition. 1 On the first post-Newtonian level, the "celestial mechanics" of gravitationally interacting bodies can be kept linear [89,90], but in the strong-field region the interaction may bring surprising features that have only been described in a very few cases yet (for a two-body problem -today at the centre of attention because of the expected gravitational waves from colliding compact binaries, "the current state of art is the third post-Newtonian approximation" [91]).The second point in which the problem embodies the essence of general relativity is the effect of inertial frame dragging due to the rotation of the sources. Contrary to the Newtonian treatment which does not discriminate between static and stationary situation, the field is now determined not only by mass-energy configuration, but also by its motion within the bodies. The inertial space can be imagined as a viscous fluid mixed by the sources. In today's "gravitoelectromagnetic" language, gravity has not only an electric component, generated by the mass, but also a magnetic one, generated by mass currents; see [245, 103, 215, 298, 147, 79, 202, 201, 1 Even the problem of an "isolated" object is very difficult (mainly if its motion must be found as well, not mentioning the interior field) unless one makes some simple assumptions about its multipole structure; see e.g. [98,99,17] for a treatment of bodies with matter interior, [297] for that also valid for singular bodies such as black holes, and [40] for the case of point-like particles.