  A certain identification of points in a planar Schwarzschild-anti de Sitter
(AdS) black hole generates a four-dimensional static black string. In turn, a
rotating black string can be obtained from a static one by means of an
`illegitimate coordinate transformation', a local boost in the compact
direction. On the basis of the gauge/gravity duality, these black strings are
dual to rotating thermal states of a strongly interacting conformal field
theory (CFT) that lives on a cylinder. In this work, we obtain the complete
quasinormal mode (QNM) spectrum of the gravitational perturbations of rotating
black strings. Analytic solutions for the dispersion relations are found in the
hydrodynamic limit, characterized by fluctuations with wavenumber and frequency
much smaller than the Hawking temperature of the string (or the temperature in
the CFT dual description). We obtain these dispersion relations both by
studying the gravitational perturbations of rotating black strings and by
investigating the hydrodynamic fluctuations of a moving fluid living on the
boundary of the AdS spacetime. Relativistic effects like the Doppler shift of
the frequencies, wavelength contraction, and dilation of the thermalization
time are shown explicitly in such a regime. We also investigate the behavior of
a sound wave propagating in a viscous fluid for several values of the rotation
parameter. The numerical solutions for the fundamental QNMs show a crossover (a
transition) from a hydrodynamic-like behavior to a linear relativistic scaling
for large wavenumbers. Additionally, we find a new family of QNMs which are
purely damped in the zero wavenumber limit and that does not follow as a
continuation of QNMs of the static black string, but that appears to be closely
related to the algebraically special perturbation modes.
