  The magnetorotational instability originates from the elastic coupling of
fluid elements in orbit around a gravitational well. Since inertial
accelerations play a fundamental dynamical role in the process, one may expect
substantial modifications by strong gravity in the case of accretion on to a
black hole. In this paper, we develop a fully covariant, Lagrangian
displacement vector field formalism with the aim of addressing these issues for
a disk embedded in a stationary geometry with negligible radial flow. This
construction enables a transparent connection between particle dynamics and the
ensuing dispersion relation for MHD wave modes. The MRI--in its incompressible
variant-- is found to operate virtually unabated down to the marginally stable
orbit; the putative inner boundary of standard accretion disk theory. To get a
qualitative feel for the dynamical evolution of the flow below $r_{\rm ms}$, we
assume a mildly advective accretion flow such that the angular velocity profile
departs slowly from circular geodesic flow. This exercise suggests that the
turbulent eddies will occur at spatial scales approaching the radial distance
while tracking the surfaces of null angular velocity gradients. The implied
field topology, namely large-scale horizontal field domains, should yield
strong mass segregation at the displacement nodes of the non-linear modes when
radiation stress dominates the local disk structure (an expectation supported
by quasi-linear arguments and by the non-linear behavior of the MRI in a
non-relativistic setting). Under this circumstance, baryon-poor flux in
horizontal field domains will be subject to radial buoyancy and to the Parker
instability, thereby promoting the growth of poloidal field.
