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 $κ$ (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 $κ$ is also similar to that for rotating black holes.

Figures (7)

• Figure 1: Plot showing dependence of possible positions of circular orbits on the dimensionless acceleration $\alpha=ar_{s}$, experienced by the circular particle at a given dimensionless radius $y=r_{c}/r_{s}$ for various angular momenta. The red curve represents a value of $b=\sqrt{3}$, corresponding to ISCO in the absence of force. The plot is presented in the double logarithmic scale. Here $b=L/r_{s}$
• Figure 2: Plot showing the number of circular orbits for various accelerations and angular momenta. Here, red color means that at a given point, there are no roots, green--there is 1 root, blue--there are 2 roots, yellow--there are 3 roots. The green dot where all regions meet corresponds to ISCO.
• Figure 3: Diagram showing the acceleration $\alpha=ar_{s}$ experienced by a particle for various radial positions and angular momenta. The corresponding color represents the force acting on such a particle that can be found from the legend on the right. The dashed curve represents circular orbits that can be obtained without any force. The hatched region represents an unstable region, the blue curve--the boundary of the unstable region. Blue point, that is, the cross-section of the dashed and blue curves, represents the ISCO without a force. The panel below shows the smaller part of the same diagram.
• Figure 4: Dependence of $a^{\prime}=da/dr$ on $y=r/r_{s}$ for various angular momenta $b=L/r_{s}$.
• Figure 5: Dependence of the position of the circular orbit on the external force for various $y_{-}=r_{-}/r_{+}$, where $r_{\mp}$ are the inner and outer horizons, respectively and for various values of angular momentum $b$. Color of the curve corresponds to the same value of $b$ as on FIG. 1.