Black hole encircled by a thin disk: fully relativistic solution

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.