Weak field limit and gravitational waves in f(T, B) teleparallel gravity

We derive the gravitational waves for f gravity which is an extension of teleparallel gravity and demonstrate that it is equivalent to f(R) gravity by linearized the field equations in the weak field limit approximation. f(T, B) gravity shows three polarizations: the two standard of general relativity, plus and cross, which are purely transverse with two-helicity, massless tensor polarization modes, and an additional massive scalar mode with zero-helicity. The last one is a mix of longitudinal and transverse breathing scalar polarization modes. The boundary term B excites the extra scalar polarization and the mass of scalar field breaks the symmetry of the TT gauge by adding a new degree of freedom, namely a single mixed scalar polarization.