Gravitational Waves in Higher Order Teleparallel Gravity

The teleparallel equivalent of higher order Lagrangians like \(L_{\Box R}=-R+a_{0}R^{2}+a_{1}R\Box R\) can be obtained by means of the boundary term \(B=2\nabla_{\mu}T^{\mu}\). In this perspective, we derive the field equations in presence of matter for higher-order teleparallel gravity considering, in particular, sixth-order theories where the \(\Box\) operator is linearly included. In the weak field approximation, gravitational wave solutions for these theories are derived. Three states of polarization are found: the two standard \(+\) and \(\times\) polarizations, namely 2-helicity massless transverse tensor polarizations, and a 0-helicity massive, with partly transverse and partly longitudinal scalar polarization. Moreover, these gravitational waves exhibit four oscillation modes related to four degrees of freedom: the two classical \(+\) and \(\times\) tensor modes of frequency \(\omega_{1}\), related to the standard Einstein waves with \(k^{2}_{1}=0\); two mixed longitudinal-transverse scalar modes for each frequencies \(\omega_{2}\) and \(\omega_{3}\), related to two different 4-wave vectors, \(k^{2}_{2}=M_{2}^{2}\) and \(k^{2}_{3}=M^{2}_{3}\). The four degrees of freedom are the amplitudes of each individual mode, i.e. \(\hat{\epsilon}^{(+)}\left(\omega_{1}\right)\), \(\hat{\epsilon}^{(\times)}\left(\omega_{1}\right)\), \(\hat{B}_{2}\left(\mathbf{k}\right)\), and \(\hat{B}_{3}\left(\mathbf{k}\right)\).