Gravitational wave polarization modes and the kinematical tensors in general relativity and beyond

Abstract Relations between the kinematical tensors (the expansion, the shear, and the vorticity) and the polarization modes of gravitational waves are studied within the context of metric theories of gravity by considering freely falling test particles. After analyzing exact relations, we consider slowly moving particles under the influence of a weak gravitational field. Linearized plane waves of theories representative of those determined by a general second-order Lagrangian, including General Relativity, are shown to exemplify the following interconnections: between the transverse components of the shear and the transverse tensor polarization mode; between the expansion, and both the transverse scalar and the longitudinal polarization modes; and between the longitudinal-transverse components of both the shear and the vorticity, and the vector polarization mode. These relations, from the theoretical point of view, offer a novel insight into the interpretation of the polarization modes in terms of the kinematical tensors. Additionally, the inclusion of the kinematical tensors in the study of gravitational waves might provide a new outlook on their phenomenology.