  The spacetime in the interior of a black hole can be described by an
homogeneous line element, for which the Einstein--Hilbert action reduces to a
one-dimensional mechanical model. We have shown in [SciPost Phys. 10, 022
(2021), [2010.07059]] that this model exhibits a symmetry under the
$(2+1)$-dimensional Poincar\'e group. Here we explain how this can be
understood as a broken infinite-dimensional BMS$_3$ symmetry. This is done by
reinterpreting the action for the model as a geometric action for BMS$_3$,
where the configuration space variables are elements of the algebra
$\mathfrak{bms}_3$ and the equations of motion transform as coadjoint vectors.
The Poincar\'e subgroup then arises as the stabilizer of the vacuum orbit. This
symmetry breaking is analogous to what happens with the Schwarzian action in
AdS$_2$ JT gravity, although in the present case there is no direct
interpretation in terms of boundary symmetries. This observation, together with
the fact that other lower-dimensional gravitational models (such as the BTZ
black hole) possess the same broken BMS$_3$ symmetries, provides yet another
illustration of the ubiquitous role played by this group.
