Circular orbits in spherically symmetric spacetimes and BSW effect with nonzero force

Abstract

We consider circular particle motion under the action of an unspecified force in a static spherically symmetric spacetime. We derive the machinery that allows one to find the force acting on a circular particle and deduce whether its position is stable or not. This also allows one to extend the definition of ISCO to the case of a non-zero external force. By conducting the near-horizon expansion, we obtain that for any non-extremal black holes, the acceleration for extremal ones is finite, and for ultraextremal (multiple) horizons it tends to zero. Applying the derived machinery to the case of the Schwarzschild metric assuming that a force is constant, we scrutiny how the number of orbits for a given force depends on its value. In particular, if a force is big enough, an additional branch of solutions appears that was absent in the case of geodesic motion. Then, for various circular orbits, we numerically investigate their stability. A similar problem is solved for the Reissner-Nordstrom (RN) metric and uncharged particles. It appears that for the near-extremal and extremal RN black holes, there exist near-horizon circle trajectories (in contrast to the nonextremal case). For the ISCO, the dependence of the orbit radius on $\kappa$ (the surface gravity) is similar to that in the case of neutral particles moving in the background of rotating black holes. In addition, two scenarios of high-energy particle collisions near such orbits are considered, and it is found that dependence on $\kappa$ is also similar to that for rotating black holes.