  Spectral distortions of the cosmic microwave background (CMB) may become a
powerful probe of primordial perturbations at small scales. Existing studies of
spectral distortions focus almost exclusively on primordial scalar metric
perturbations. Similarly, vector and tensor perturbations should source CMB
spectral distortions. In this paper, we give general expressions for the
effective heating rate caused by these types of perturbations, including
previously neglected contributions from polarization states and higher
multipoles. We then focus our discussion on the dissipation of tensors, showing
that for nearly scale invariant tensor power spectra, the overall distortion is
some six orders of magnitudes smaller than from the damping of adiabatic scalar
modes. We find simple analytic expressions describing the effective heating
rate from tensors using a quasi-tight coupling approximation. In contrast to
adiabatic modes, tensors cause heating without additional photon diffusion and
thus over a wider range of scales, as recently pointed out by Ota et. al 2014.
Our results are in broad agreement with their conclusions, but we find that
small-scale modes beyond k< 2x10^4 Mpc^{-1} cannot be neglected, leading to a
larger distortion, especially for very blue tensor power spectra. At small
scales, also the effect of neutrino damping on the tensor amplitude needs to be
included.
