Centrifugal Force Paradox

The general theory of relativity (GTR) predicts that in certain circumstances the centrifugal force may be directed toward, not away from, the center of circular motion. In fact, if an astronaut could steer a spacecraft sufficiently near to a black hole, the astronaut would feel a centrifugal force pushing inward, not outward, contrary to everyday experience! What is the conceptual explanation of the unusual result?

Answer

The conceptual resolution of this paradox starts with the consideration of light paths near the black hole. The general theory of relativity (GTR) predicts that there should be light paths around the black hole that are circular at a radial distance of 1.5 times the gravitational radius r_g = 2GM/c². Around one of these circular light paths imagine a circular tube centered exactly on the path of the circular ray

so that the axis of the tube and the path of the ray coincide. Measurements with a straight ruler verify that the axis of the tube is circular, yet because of the bending of the light rays, the tube is seen as absolutely straight by an observer on the axis. A lamp held at the axis by a colleague will appear dimmer to you as he or she walks away along the axis, but the lamp is never obscured, so you must conclude that the tube is straight. Therefore, along this circular path one would expect no centrifugal force effects.

Instead of the tube being around the circular light path, consider the tube to be around a smaller circular path centered on the black hole. With rulers one can again verify that the tube curves to the left, with the black hole on the left as one walks forward. The outward direction is to the right. Everyday experience tells us that the centrifugal force pushes outward. Again your colleague walks away with the lamp held along the axis of the tube. If somehow the light rays were not bent by the gravitational field of the black hole, you would see the lamp disappear behind the left side of the tube, and you would conclude that the tube bends to the left. If the path is the one discussed above, the lamp is always in view. But the tube is so close to the black hole that the light rays bend even more than circular rays. So you actually see the lamp disappear to the right. Therefore the outward direction is to the left, and you must predict that the centrifugal force would push to the left!

Several related paradoxes for paths near a massive body are discussed in the references, including the fact that for a rocket to maintain a constant speed the boosters would need to fire perpendicularly to the path, but their force is not dependent on how fast the rocket is moving.