Gravitational Waves in Higher Order Teleparallel Gravity

Abstract

The teleparallel equivalent of higher order Lagrangians like $L_{\square R}=-R+a_0{R}^2+a_{1}R\square R$ 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 $\square$ 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 $k_1^2=0$; 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 $k_3^2=M_3^2$. 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 ${\hat{B}}_3\left(\mathbf{k}\right)$.

27 pages, accepted for publication in Classical and Quantum Gravity