  It was recently argued that certain relativistic theories at finite density
can exhibit an unconventional spectrum of Goldstone excitations, with gapped
Goldstones whose gap is exactly calculable in terms of the symmetry algebra. We
confirm this result as well as previous ones concerning gapless Goldstones for
non-relativistic systems via a coset construction of the low-energy effective
field theory. Moreover, our analysis unveils additional gapped Goldstones,
naturally as light as the others, but this time with a model-dependent gap.
Their exact number cannot be inferred solely from the symmetry breaking pattern
either, but rather depends on the details of the symmetry breaking mechanism--a
statement that we explicitly verify with a number of examples. Along the way we
provide what we believe to be a particularly transparent interpretation of the
so-called inverse-Higgs constraints for spontaneously broken spacetime
symmetries.
