  We formulate and solve by semi-analytic means the axisymmetric equilibria of
relativistic self-similar disks of infinitesimal vertical thickness. These
disks are supported in the horizontal directions against their self-gravity by
a combination of isothermal (two-dimensional) pressure and a flat rotation
curve. The dragging of inertial frames restricts possible solutions to rotation
speeds that are always less than 0.438 times the speed of light, a result first
obtained by Lynden-Bell and Pineault in 1978 for a cold disk. We show that
prograde circular orbits of massive test particles exist and are stable for all
of our model disks, but retrograde circular orbits cannot be maintained with
particle velocities less than the speed of light once the disk develops an
ergoregion. We also compute photon trajectories, planar and non-planar, in the
resulting spacetime, for disks with and without ergoregions. We find that all
photon orbits, except for a set of measure zero, tend to be focused by the
gravity of the flattened mass-energy distribution toward the plane of the disk.
This result suggests that strongly relativistic, rapidly rotating, compact
objects may have difficulty ejecting collimated beams of matter or light along
the rotation axes until the flows get well beyond the flattened parts of the
relativistic mass distribution (which cannot happen in the self-similar models
considered in this paper).
