Hyperbolic Motion in Curved Space Time

The differential equations of motion for a test particle moving with uniform acceleration in a curved space time are proposed. They are obtained by generalizing the differential-geometric characteristics of a rectangular hyperbola in Minkowski space time. The problem is proposed, though not solved, of deriving these equations of motion from the field equations of general relativity. However, it is suggested that they also hold independently of general relativity in cosmological space times based on the Robertson-Walker metric. The equations are solved in detail for the particular case of de Sitter space time, which is relevant to the steady-state theory. It is found, inter alia, that in this space time a particle moving radially with uniform acceleration ultimately moves with constant relative velocity through the substratum; that there is a critical first fundamental particle (galaxy) on its line of motion which it will never overtake; that, in turn, a light signal emitted at or after a certain critical time will not catch up with the accelerating particle; and that, if a particle with a given available acceleration $\alpha$ passes beyond a certain proper distance (the $\alpha$ horizon) it can no longer return to its place of origin. Possible applications to intergalactic rocketry are examined.