  So far, Brillouin scattering and cavity optomechanics were mostly
disconnected branches of research -- although both deal with photon-phonon
coupling. This begs for the development of a broader theory that contains both
fields. Here, we derive the dynamics of optomechanical cavities from that of
Brillouin-active waveguides. This explicit transition elucidates the link
between phenomena such as Brillouin amplification and electromagnetically
induced transparency. It proves that effects familiar from cavity optomechanics
all have traveling-wave partners, but not vice versa. We reveal a close
connection between two parameters of central importance in these fields: the
Brillouin gain coefficient and the zero-point optomechanical coupling rate.
This enables comparisons between systems as diverse as ultracold atom clouds,
plasmonic Raman cavities and nanoscale silicon waveguides. In addition,
back-of-the-envelope calculations show that unobserved effects, such as
photon-assisted amplification of traveling phonons, are now accessible in
existing systems. Finally, we formulate both circuit- and cavity-oriented
optomechanics in terms of vacuum coupling rates, cooperativities and gain
coefficients, thus reflecting the similarities in the underlying physics.
