February 2, 2024 | Salvatore Capozziello, Maurizio Capriolo, and Shin'ichi Nojiri
This paper investigates gravitational waves (GWs) in $ f(Q) $ gravity, a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature. The study shows that $ f(Q) $ gravity exhibits only two massless and tensor modes, with polarizations transverse and helicity equal to two, exactly reproducing the plus and cross tensor modes of General Relativity (GR). The deviation equation for two nearby trajectories of freely falling particles coincides with the geodesic deviation of GR, as the energy-momentum tensor is Levi-Civita covariantly conserved. This implies that free particles in $ f(Q) $ gravity follow GR geodesics. The curves are also solutions of a force equation, where an extra geometric force term modifies autoparallel curves. Thus, GWs in $ f(Q) $ gravity behave similarly to those in $ f(T) $ gravity and cannot be distinguished from GR by polarization measurements alone. In contrast, $ f(R) $ gravity has an additional scalar mode. The paper analyzes the linearized field equations in the coincident gauge, showing that $ f(Q) $ gravity produces GWs with transverse polarizations and helicity 2, identical to GR. The results indicate that $ f(Q) $ gravity is equivalent to GR in terms of GW polarization, while $ f(R) $ gravity has a distinct scalar mode. The study concludes that $ f(Q) $ gravity behaves like $ f(T) $ gravity and is indistinguishable from GR in terms of GW polarization. The paper also discusses the number of degrees of freedom in $ f(Q) $ gravity and the potential for future observations to distinguish between different gravity theories.This paper investigates gravitational waves (GWs) in $ f(Q) $ gravity, a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature. The study shows that $ f(Q) $ gravity exhibits only two massless and tensor modes, with polarizations transverse and helicity equal to two, exactly reproducing the plus and cross tensor modes of General Relativity (GR). The deviation equation for two nearby trajectories of freely falling particles coincides with the geodesic deviation of GR, as the energy-momentum tensor is Levi-Civita covariantly conserved. This implies that free particles in $ f(Q) $ gravity follow GR geodesics. The curves are also solutions of a force equation, where an extra geometric force term modifies autoparallel curves. Thus, GWs in $ f(Q) $ gravity behave similarly to those in $ f(T) $ gravity and cannot be distinguished from GR by polarization measurements alone. In contrast, $ f(R) $ gravity has an additional scalar mode. The paper analyzes the linearized field equations in the coincident gauge, showing that $ f(Q) $ gravity produces GWs with transverse polarizations and helicity 2, identical to GR. The results indicate that $ f(Q) $ gravity is equivalent to GR in terms of GW polarization, while $ f(R) $ gravity has a distinct scalar mode. The study concludes that $ f(Q) $ gravity behaves like $ f(T) $ gravity and is indistinguishable from GR in terms of GW polarization. The paper also discusses the number of degrees of freedom in $ f(Q) $ gravity and the potential for future observations to distinguish between different gravity theories.