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 $f(R)neq
R$.
The Future Roadmap for Gravitational Waves in $f(Q)$ Gravity
Introduction
In this article, we explore the behavior of gravitational waves in $f(Q)$ gravity, a geometric theory of gravity described by a non-metric compatible connection known as symmetric-teleparallel gravity. We analyze the properties of these waves and compare them to gravitational waves in General Relativity.
Two Massless and Tensor Modes
Our findings reveal that $f(Q)$ gravity exhibits only two massless and tensor modes. These modes have transverse polarizations with helicity equal to two, which is consistent with the plus and cross tensor modes observed in General Relativity.
Geodesic Deviation and Trajectory Analysis
To further study these gravitational waves, we examine the deviation equation of two nearby freely falling point-like particles. Surprisingly, we discover that this deviation equation coincides with the geodesic deviation observed in General Relativity. This suggests that free particles without any structure follow the geodesics of General Relativity even in $f(Q)$ gravity.
Force Equation and Geometric Origin
Alternatively, we can interpret the particle trajectories as solutions of a force equation. In this equation, an extra force term of geometric origin arises due to non-metricity. This modification to the autoparallel curves introduced by the non-metric connection showcases how non-metricity affects the behavior of gravitational waves in $f(Q)$ gravity.
Comparison with Torsion and Curvature-Based Gravity Theories
We compare the behavior of gravitational waves in $f(Q)$ gravity to torsion-based $f(T)$ gravity and curvature-based $f(R)$ gravity. Our analysis reveals that gravitational waves in $f(Q)$ gravity behave similarly to those in $f(T)$ gravity, where wave polarization measurements alone cannot distinguish them from waves in General Relativity. However, this differs from gravitational waves in $f(R)$ gravity, where an additional scalar mode is always present for $f(R)neq R$.
Conclusion and Future Challenges
This research demonstrates the similarity between gravitational waves in $f(Q)$ gravity and General Relativity. The absence of additional modes and the reproduction of the plus and cross tensor modes suggest that $f(Q)$ gravity may provide a consistent framework for describing gravitational waves. However, further investigation is needed to fully understand the implications and potential differences of gravitational wave behavior in $f(Q)$ gravity compared to General Relativity. Continued research in this area may uncover new challenges and opportunities, ultimately shaping the future of gravitational wave study.