Co-accelerated particles in the C-metric

With appropriately chosen parameters, the C-metric represents two uniformly accelerated black holes moving in the opposite directions on the axis of t he axial symmetry (the z-axis). The acceleration is caused by nodal singula rities located on the z-axis. In the present paper, geodesics in the C-metric are examined. In general, t here exist three types of timelike or null geodesics in the C-metric: geode sics describing particles (a) falling under the black hole horizon; (b) cro ssing the acceleration horizon; and (c) orbiting around the z-axis and co-a ccelerating with the black holes. Using an effective potential, it can be shown that there exist stable timel ike geodesics of the third type if the product of the parameters of the C-m etric, mA, is smaller than a certain critical value. Null geodesics of the third type are always unstable. Special timelike and null geodesics of the third type are also found in an analytical form.