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 diverges, while for extremal ones it is finite. 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 and the position of the ISCO particles depending on the external force. In particular, we show that the force allows ISCO particles to become closer to the horizon, but they cannot reach it. 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), however, they are unstable. Analysis of the ISCO particles in the RN case gives qualitatively the same results as for Schwarzschild. In addition, high-energy particle collisions of circular particles are considered, and it is found that they may give an increase in the collisional energy only for near-horizon circular orbits that exist only in near-extremal and extremal cases.