Gravitational waves in $f(Q)$ non-metric gravity via geodesic deviation

TL;DR

This paper studies gravitational waves in $f(Q)$ non-metric gravity, showing they have only two tensor modes identical to General Relativity, and their deviation behavior matches GR due to conserved energy-momentum, making them indistinguishable by polarization.

Contribution

It demonstrates that $f(Q)$ gravity produces gravitational waves with only tensor modes, identical to GR, and that their deviation equations coincide with those of GR, highlighting observational indistinguishability.

Findings

Gravitational waves in $f(Q)$ have only two tensor modes.

Deviation equations in $f(Q)$ match those in GR.

Wave polarization measurements cannot distinguish $f(Q)$ from GR.

Abstract

We investigate gravitational waves in the $f(Q)$ gravity, i.e., a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature, known as symmetric-teleparallel gravity. We show that $f(Q)$ gravity exhibits only two massless and tensor modes. Their polarizations are transverse with helicity equal to two, exactly reproducing the plus and cross tensor modes typical of General Relativity. In order to analyze these gravitational waves, we first obtain the deviation equation of two trajectories followed by nearby freely falling point-like particles and we find it to coincide with the geodesic deviation of General Relativity. This is because the energy-momentum tensor of matter and field equations are Levi-Civita covariantly conserved and, therefore, free structure-less particles follow, also in $f(Q)$ gravity, the General Relativity geodesics. Equivalently, it is possible to show that the curves are solutions of a force equation, where an extra force term of geometric origin, due to non-metricity, modifies the autoparallel curves with respect to the non-metric connection. In summary, gravitational waves produced in non-metricity-based $f(Q)$ gravity behave as those in torsion-based $f(T)$ gravity and it is not possible to distinguish them from those of General Relativity only by wave polarization measurements. This shows that the situation is different with respect to the curvature-based $f(R)$ gravity where an additional scalar mode is always present for $f(R)\neq R$.