Infinitesimal relative position vector fields for observers in a reference frame and applications to conformally stationary spacetimes

We introduce and analyze the concept of infinitesimal relative position vector field between “infinitesimally nearby” observers, showing the equivalence between different definitions. Through the Fermi–Walker derivative of infinitesimal relative position vector fields along an observer in a reference frame, we characterize spacetimes admitting an umbilic foliation. Sufficient and necessary conditions for those spacetimes to be a conformally stationary spacetime are given. Finally, the important class of cosmological models known as generalized Robertson–Walker spacetimes is characterized.