Will [1] solved the perturbation of a Schwarzschild black hole due to a slowly rotating light concentric thin ring, using Green's functions expressed as infinite-sum expansions in multipoles and in the small mass and rotational parameters. In a previous paper [2], we expressed the Green functions in closed form containing elliptic integrals, leaving just summation over the mass expansion. Such a form is more practical for numerical evaluation, but mainly for generalizing the problem to extended sources where the Green functions have to be integrated over the source. We exemplified the method by computing explicitly the first-order perturbation due to a slowly rotating thin disc lying between two finite radii. After finding basic parameters of the system -mass and angular momentum of the black hole and of the disc -we now add further properties, namely those which reveal how the disc gravity influences geometry of the black-hole horizon and those of circular equatorial geodesics (specifically, radii of the photon, marginally bound and marginally stable orbits). We also realize that, in the linear order, no ergosphere occurs and the central singularity remains point-like, and check the implications of natural physical requirements (energy conditions and subluminal restriction on orbital speed) for the single-stream as well as counter-rotating double-stream interpretations of the disc.

The task of finding the potential of a thin circular disk with power-law radial density profile is revisited. The result, given in terms of infinite Legendre-type series in the above reference, has now been obtained in closed form thanks to the method of Conway employing Bessel functions. Starting from a closed-form expression for the potential generated by the elementary density term ρ 2l , we cover more generic—finite solid or infinite annular—thin disks using superposition and/or inversion with respect to the rim. We check several specific cases against the series-expansion form by numerical evaluation at particular locations. Finally, we add a method to obtain a closed-form solution for finite annular disks whose density is of “bump” radial shape, as modeled by a suitable combination of several powers of radius. Density and azimuthal pressure of the disks are illustrated on several plots, together with radial profiles of free circular velocity.

We continue to study black holes subjected to strong sources of gravity, again paying special attention to the behaviour of geometry in the black-hole interior. After examining, in previous two papers, the deformation arising in the Majumdar-Papapetrou binary of extremally charged black holes and that of a Schwarzschild black hole due to a surrounding (Bach-Weyl) ring, we consider here the system of two Schwarzschild-type black holes held apart by the Appell ring. After verifying that such a configuration can be in a strut-free equilibrium along certain lines in a parameter space, we compute several basic geometric characteristics of the equilibrium configurations. Then, like in previous papers, we calculate and visualize simple invariants determined by the metric (lapse or, equivalently, potential), by its first derivatives (gravitational acceleration) and by its second derivatives (Kretschmann scalar). Extension into the black-hole interior is achieved along particular null geodesics starting tangentially to the horizon. In contrast to the case involving the Bach-Weyl ring, here each single black hole is placed asymmetrically with respect to the equatorial plane (given by the Appell ring) and the interior geometry is really deformed in a non-symmetrical way. Inside the black holes, we again found regions of negative Kretschmann scalar in some cases.

We give a full metric describing the gravitational field of a static and axisymmetric thin disk without radial pressure encircling a Schwarzschild black hole. The disk density profiles are astrophysically realistic, stretching from the horizon to radial infinity, yet falling off quickly at both these locations. The metric functions are expressed as finite series of Legendre polynomials. Main advantages of the solution are that (i) the disks have no edges, so their fields are everywhere regular (outside the horizon), and that (ii) all non-trivial metric functions are provided analytically and in closed forms. We examine and illustrate basic properties of the black-hole-disk space-times.

We provide an explicit, closed and compact expression for the Debye superpotential of a circular source. This superpotential is obtained by integrating the Green function of Teukolsky Master Equation (TME). The Debye potential itself is then, for a particular configuration, calculated in the same manner as the ϕ0 field component is calculated from the Green function of the TME -by convolution of the Green function with sources. This way we provide an exact field of charged ring and circular current on the Kerr background, finalizing thus the work of Linet.