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.