Completeness of unchanged direction trajectories in Galilean spacetimes

The notion of Unchanged Direction (UD) motion in an arbitrary Galilean spacetime is introduced providing a mathematical framework for the intuitive notion that “proper acceleration does not change its direction”. Given a prescribed scalar acceleration, the corresponding initial value problem is analyzed and used to characterize the UD trajectories as the projected curves on the spacetime of the integral curves of a smooth vector field defined on a certain fiber bundle. This key fact allows us to prove a result on the completeness of inextensible UD motions, that can be physically interpreted saying that observers which obey a UD motion live forever. Concretely, it is shown that a UD observer in a globally synchronizable Galilean spacetime can be extended if there exists a complete spatially conformally Leibnizian field of observers with conformal factor and gravitational field bounded along finite times. Finally, another completeness result is presented in the case of the existence of a compact absolute spacelike leaf.