Gravitational waves in higher order teleparallel gravity

Abstract

Abstract The teleparallel equivalent of higher order Lagrangians like L □R = −R + a 0 R 2 + a 1 R□R can be obtained by means of the boundary term B = 2∇ μ T μ . 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 □ 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 × 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 (GWs) exhibit four oscillation modes related to four degrees of freedom: the two classical + and × tensor modes of frequency ω 1, related to the standard Einstein waves with k 1 2 = 0 ; two mixed longitudinal-transverse scalar modes for each frequencies ω 2 and ω 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. ϵ ̂ ( + ) ω 1 , ϵ ̂ ( × ) ω 1 , B ̂ 2 k , and B ̂ 3 k . To describe a general teleparallel gravity model of order (2p + 2), we used the teleparallel Lagrangian L □ k T neq = e T + a 0 T 2 + ∑ k = 1 p a k T □ k T which demonstrates to be not equivalent to L □ k R = − g R + ∑ k = 0 p a k R □ k R . By varying its action, the related field equations in the presence of matter are derived. Hence we obtain the GWs for these teleparallel gravity models, i.e. L □ k T neq , that are exactly the Einstein GWs as in f(T) teleparallel gravity. In conclusion, the boundary term B generates extra polarizations and additional modes beyond the standard ones. In particular B 2 and B□B terms generate both the additional scalar polarization and the extra scalar modes.