### 5.4 Rings around black holes

The problem of uniformly rotating rings surrounding a black hole can be viewed as an intermediate step
between one-body axisymmetric configurations and the two body problem. Indeed, even if one has to deal
with two components, the problem is still axisymmetric. In [13], configurations of a black hole surrounded
by a uniformly rotating ring of matter are computed in general relativity. The matter is assumed to be a
perfect fluid. To solve the equations, space is divided into five computational domains. One of them
describes the ring itself, another one the region around the black hole and another extends up to infinity.
The two other domains are used to connect these regions. One of the difficulties is that the
surface of the ring is not know a priori and so the domains must be dynamically adapted to its
surface. Cylindrical-type coordinates are used and, in each domain, are mapped onto squares of
numerical coordinates. The actual mappings depend on the domain and can be found in Section IV
of [13].
Numerical coordinates are expanded in terms of Chebyshev polynomials. The system to be solved is
obtained by writing Einstein’s equations in collocation space including regularity conditions
on the axis and appropriate boundary conditions on both the horizon of the black hole and
at spatial infinity. As in [9, 10], the system is solved iteratively, using the Newton–Raphson
method.

Both the Newtonian and relativistic configurations are computed. The ratio between the
mass of the black hole and the mass of the ring is varied from zero (no black hole) to 144. The
inner mass shedding of the ring can be obtained. One of the most interesting results is the
existence of configurations for which the ratio of the black hole angular momentum
and the square of its mass exceeds one, contrary to what can be achieved for an isolated black
hole.