Photon Regions and Shadows of Accelerated Black Holes

In an earlier paper we have analytically determined the photon regions and the shadows of black holes of the Pleba\'nski class of metrics which are also known as the Kerr--Newman--NUT--(anti-)deSitter metrics. These metrics are characterized by six parameters: mass, spin, electric and magnetic charge, gravitomagnetic NUT charge, and the cosmological constant. Here we extend this analysis to the Pleba\'nski--Demia\'nski class of metrics which contains, in addition to these six parameters, the so-called acceleration parameter. All these metrics are axially symmetric and stationary type D solutions to the Einstein--Maxwell equations with a cosmological constant. We derive analytical formulas for the photon regions (i.e., for the regions that contain spherical lightlike geodesics) and for the boundary curve of the shadow as it is seen by an observer at Boyer--Lindquist coordinates $(r_O, \vartheta _O)$ in the domain of outer communication. Whereas all relevant formulas are derived for the whole Pleba\'nski--Demia\'nski class, we concentrate on the accelerated Kerr metric (i.e., only mass, spin and acceleration parameter are different from zero) when discussing the influence of the acceleration parameter on the photon region and on the shadow in terms of pictures. The accelerated Kerr metric is also known as the rotating $C$-metric. We discuss how our analytical formulas can be used for calculating the horizontal and vertical angular diameters of the shadow and we estimate these values for the black holes at the center of our Galaxy and at the center of M87.