Properties of a congruence of time-like curves are investigated, by means of a projection technique due to Cattaneo. Born's requirement of vanishing spatial rate of strain ensures that the congruence defines a rigid frame. Einstein's field equations impose further conditions, whose physical consequences are explored. Sufficient conditions are given for the extension of the Herglotz-Noether classification theorem to general relativity. The case is investigated in which the curves of the congruence are the orbits of a group of motions. Validity of the Herglotz-Noether theorem is shown to be associated with the existence of a certain type of motion. Finally, some properties of two important special cases of rigid frames-isometric and irrotational motions-are examined.