Unchanged direction motion in general relativity: The problems of prescribing acceleration and the extensibility of trajectories

Abstract

The notion of unchanged direction (UD) motion in general relativity is introduced, extending widely the concept of uniformly accelerated motion. An observer which obeys an UD motion is characterized as a pointing future unit timelike curve with all its curvatures identically zero up to the first one. The initial value problem when the acceleration of the motion is prescribed is analysed. It is also studied that the completeness of inextensible UD motions, that can be physically interpreted saying that the observers which obey an UD motion lives forever. For certain spacetimes with relevant symmetries that includes the generalized Robertson-Walker spacetimes, a geometric approach leads to the completeness. On the other hand, a more analytical approach permits to prove completeness of inextensible UD motions in a plane wave spacetime.