Abstract. As the velocity of a rocket in a circular orbit near a black hole increases, the outwardly directed rocket thrust must increase to keep the rocket in its orbit. This feature might appear paradoxical from a Newtonian viewpoint, but we show that it follows naturally from the equivalence principle together with special relativity and a few general features of black holes. We also derive a general relativistic formalism of inertial forces for reference frames with acceleration and rotation. The resulting equation relates the real experienced forces to the time derivative of the speed and the spatial curvature of the particle trajectory relative to the reference frame. We show that an observer who follows the path taken by a free (geodesic) photon will experience a force perpendicular to the direction of motion that is independent of the observers velocity. We apply our approach to resolve the submarine paradox, which regards whether a submerged submarine in a balanced state of rest will sink or float when given a horizontal velocity if we take relativistic effects into account. We extend earlier treatments of this topic to include spherical oceans and show that for the case of the Earth the submarine floats upward if we take the curvature of the ocean into account. I Introduction Consider a rocket in a circular orbit outside the event horizon of a black hole. 1 If the orbit lies within the photon radius, the radius where free photons can move on circular orbits, 2 a greater outward rocket thrust is required to keep the rocket in orbit the faster the rocket moves. However, outside of the photon radius the outward thrust decreases as the orbital speed increases just as it would for a similar scenario in Newtonian mechanics (the thrust will be inward directed for sufficiently high speeds, see Analogous to the situation in Newtonian mechanics we can introduce in general relativity a gravitational force that is velocity independent. This force is fictitious (unlike in Newtonian mechanics). We can also introduce a velocity dependent (fictitious) centrifugal force that together with the gravitational force balances the real force from the jet engine of the rocket. By this definition, the centrifugal force is directed inward inside of the photon radius and directed outward outside of the photon radius. This reversal of the direction of the fictitious centrifugal force is described by the formalism of optical geometry (see Appendix A) in which the phenomena has been discussed. We start by illustrating how the fact that the rocket thrust increases with increasing orbital speed (sufficiently close to the black hole) follows naturally from the equivalence principle (reviewed in Appendix B). We do so by first considering an idealized special relativistic scenario of a train moving relative to an (upward) accelerating platform. We then consider a more general but still effectively two-dimensional discussion of forces perpendicular to the direction of motion for motion relative to an accelerated reference frame in special relativity. In a static spacetime, the reference frame connected to the static observers behaves locally like an accelerating reference frame in special relativity and the formalism can therefore be applied also to this case. We then illustrate how to apply the formalism of this paper to the submarine paradox. Next we generalize the formalism of forces and curvature of spatial paths to include three-dimensional cases, forces parallel to the direction of motion, and rotating reference frames. The acceleration and rotation of the reference frame will be shown (as in Newtonian mechanics) to introduce terms that can be interpreted as iner-
