  At second-order, scalar perturbations can source traceless and transverse
perturbations to the metric, called induced gravitational waves (IGW). The
apparent gauge-dependence of the IGW obscures the interpretation of the
stochastic gravitational-wave signal. To elucidate the gauge dependence, we
study the IGW from manifestly gauge-invariant scalar perturbations, namely, the
relative velocity between the baryon and cold dark matter. From this relative
velocity perturbation, we compute the dimensionless gravitational wave power
spectrum and the corresponding expected angular power spectrum of the B-mode
polarization of the cosmic microwave background. Although the effect turns out
to be unobservably small, the calculation demonstrates both the importance of
using observable quantities to remove the gauge ambiguity and the observable
consequences of tensor perturbations which are not propagating gravitational
waves.
