Orbits in a Kerr geometry

Frank Wang

In Newtonian mechanics, the gravitational force is $F=GMm/r^{2}$ . Rotation of the body with mass $M$ does not affect the motion of a particle with mass $m$ orbiting around it. However, Einstein's general relativity predicts that a rotating body will drag along the spacetime surrounding it. We illustrate the frame dragging effect near an extreme Kerr black hole (a maximally spinning black hole). For a particle with energy $E=0.97$ and angular moment $l=4$ (geometrized units), if it rotates in the same direction as the black hole, it will orbit the black hole, as shown in Figure 1. If it rotates against the black hole, it will be forced to reverse its direction due to the frame dragging effect and spiral into the black hole, as shown in Figure 2.

Figure 1: Corotating orbit for extreme Kerr black hole ($a=M$ ) with $E=0.97$ and $l=4$ .

Figure 2: Counterrotating orbit for extreme Kerr black hole ($a=-M$ ) with $E=0.97$ and $l=4$ . The particle initially orbits counterclockwise, and the black hole spins clockwise. The particle will be forced to reverse its direction.