  We study the elastic response of planar black hole (BH) solutions in a simple
class of holographic models with broken translational invariance. We compute
the transverse quasi-normal mode spectrum and the propagation speed of the
lowest energy mode. We find that the speed of the lowest mode relates to the BH
rigidity modulus as dictated by elasticity theory. This allows to identify
these modes as transverse phonons---the pseudo Goldstone bosons of
spontaneously broken translational invariance. In addition, we show that these
modes have a mass gap controlled by an explicit source of the translational
symmetry breaking. These results provide a new confirmation that the BHs in
these models do exhibit solid properties that become more manifest at low
temperatures. Also, by the AdS/CFT correspondence, this allows to extend the
standard results from the effective field theory for solids to quantum-critical
materials.
