Black holes are very small objects by astronomical standards, so in many circumstances they interact with their surroundings like a Newtonian mass point. However, if black holes are present in X-ray binary systems, the X-rays emitted in the inner part of the accretion disk probe the highly curved spacetime geometry near the horizon, particularly if the black hole is rapidly rotating. Some of the properties of circular orbits near the black hole are quite sensitive to the amount of angular momentum. The relativistic corrections remove a Newtonian degeneracy between several of the characteristic frequencies associated with perturbations of the circular orbits.
Hot spots in the inner part of the disk can produce dramatic fluctuations in intensity, since the frequency shifts of photons emitted by a given point on the disk are strongly time-dependent. The bending of the photon trajectories by the strong gravitational field can drastically affect the energy balance of the disk; much of the radiation emitted by the inner part of the disk is reabsorbed. The dragging of inertial frames by the angular momentum of the black hole can have striking consequences for the structure of the disk at quite large radii if the angular momentum of the accreting matter is not in the same direction as the angular momentum of the black hole.
Dynamic perturbations of black holes are now being intensively studied to see if there are any surprising physical effects associated with the rotation of the black hole. Unfortunately, though quite interesting methods of extracting energy from the black hole exist in principle most of them are unlikely to be realized to an important extent in the real astrophysical world.