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

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

February 2, 2024 | Salvatore Capozziello *1,2,3, Maurizio Capriolo † 4, and Shin’ichi Nojiri §5,6
This paper investigates gravitational waves in the $f(Q)$ gravity framework, a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature. The authors show that $f(Q)$ gravity exhibits only two massless and tensor modes, with transverse polarizations and helicity equal to two, identical to the plus and cross tensor modes in General Relativity (GR). They derive the deviation equation for the trajectories of nearby freely falling particles, which coincides with the geodesic deviation in GR due to the covariant conservation of the energy-momentum tensor. This conservation is a result of the Levi-Civita connection being used in the matter field equations. The curves followed by free particles in $f(Q)$ gravity are shown to be solutions of a force equation where an extra geometric force term due to non-metricity modifies the autoparallel curves. The study concludes that gravitational waves in non-metricity-based $f(Q)$ gravity behave similarly to those in torsion-based $f(T)$ gravity and cannot be distinguished from GR solely through wave polarization measurements. This contrasts with curvature-based $f(R)$ gravity, which has an additional scalar mode for $f(R) \neq R$. The paper also discusses the possibility of detecting this additional mode in future experiments like LISA and Einstein Telescope.This paper investigates gravitational waves in the $f(Q)$ gravity framework, a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature. The authors show that $f(Q)$ gravity exhibits only two massless and tensor modes, with transverse polarizations and helicity equal to two, identical to the plus and cross tensor modes in General Relativity (GR). They derive the deviation equation for the trajectories of nearby freely falling particles, which coincides with the geodesic deviation in GR due to the covariant conservation of the energy-momentum tensor. This conservation is a result of the Levi-Civita connection being used in the matter field equations. The curves followed by free particles in $f(Q)$ gravity are shown to be solutions of a force equation where an extra geometric force term due to non-metricity modifies the autoparallel curves. The study concludes that gravitational waves in non-metricity-based $f(Q)$ gravity behave similarly to those in torsion-based $f(T)$ gravity and cannot be distinguished from GR solely through wave polarization measurements. This contrasts with curvature-based $f(R)$ gravity, which has an additional scalar mode for $f(R) \neq R$. The paper also discusses the possibility of detecting this additional mode in future experiments like LISA and Einstein Telescope.
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[slides and audio] Gravitational waves in f(Q) non-metric gravity via geodesic deviation