Ultra-fast outflows (UFOs) have become an established feature in analyses of the X-ray spectra of active galactic nuclei (AGN). According to the standard picture, they are launched at accretion disc scales with relativistic velocities, up to 0.3−0.4 times the speed of light. Their high kinetic power is enough to induce an efficient feedback on a galactic scale, possibly contributing to the co-evolution between the central supermassive black hole (SMBH) and the host galaxy. It is, therefore, of paramount importance to gain a full understanding of UFO physics and, in particular, of the forces driving their acceleration and the relation to the accretion flow from which they originate.

In this paper, we investigate the impact of special relativity effects on the radiative pressure exerted onto the outflow. The radiation received by the wind decreases for increasing outflow velocity, v, implying that the standard Eddington limit argument has to be corrected according to v. Due to the limited ability of the radiation to counteract the black hole gravitational attraction, we expect to find lower typical velocities with respect to the non-relativistic scenario.

We integrated the relativistic-corrected outflow equation of motion for a realistic set of starting conditions. We concentrated on a range of ionisations, column densities, and launching radii consistent with those typically estimated for UFOs. We explore a one-dimensional, spherical geometry and a three-dimensional setting with a rotating, thin accretion disc.

We find that the inclusion of special relativity effects leads to sizeable differences in the wind dynamics and that v is reduced up to 50% with respect to the non-relativistic treatment. We compare our results with a sample of UFOs from the literature and we find that the relativistic-corrected velocities are systematically lower than the reported ones, indicating the need for an additional mechanism, such as magnetic driving, to explain the highest velocity components. Finally, we note that these conclusions, derived for AGN winds, are generally applicable.