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

TL;DR

This work develops a general framework linking the kinematical tensors of a cloud of freely falling test particles to the polarization content of gravitational waves, applicable to GR and a broad class of metric theories. It first derives exact relations between the kinematical tensors $\theta$, $\sigma_{ab}$, and $\omega_{ab}$ and the tidal tensor $K_{ab}$, then specializes to the weak-field, linearized regime to express these quantities in terms of metric perturbations. By analyzing linearized plane waves in GR, $f(R)$ gravity, and Einstein-Bach gravity, the paper shows how GR excites only the transverse-traceless tensor mode, while $f(R)$ gravity also activates a massive scalar (affecting expansion and the longitudinal/transverse-scalar modes), and Einstein-Bach gravity introduces a massive spin-2 sector that contributes to all kinematical components, including vorticity and vector modes. The results provide a practical bridge between observable kinematical effects and GW polarization content, offering a pathway to test gravity theories with GW data and to explore gauge-invariant formulations in future work.

Abstract

Interrelationships between the expansion, the shear, and the vorticity -- the kinematical tensors -- on the one hand, and the polarization modes of gravitational waves on the other hand, are studied by considering freely falling test particles. After studying exact relations, we consider slowly moving particles under the influence of a weak gravitational field. Linearized plane waves of metric theories of gravity 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 the shear and the vorticity, and the vector polarization mode.