  Black hole perturbation theory is useful for studying the stability of black
holes and calculating ringdown gravitational waves after the collision of two
black holes. Most previous calculations were carried out at the level of the
field equations instead of the action. In this work, we compute the
Einstein-Hilbert action to quadratic order in linear metric perturbations about
a spherically symmetric vacuum background in Regge-Wheeler gauge. Using a 2+2
splitting of spacetime, we expand the metric perturbations into a sum over
scalar, vector, and tensor spherical harmonics, and dimensionally reduce the
action to two dimensions by integrating over the two sphere. We find that the
axial perturbation degree of freedom is described by a two dimensional massive
vector action, and that the polar perturbation degree of freedom is described
by a two dimensional dilaton massive gravity action. Varying the dimensionally
reduced actions, we rederive covariant and gauge-invariant master equations for
the axial and polar degrees of freedom. Thus, the two dimensional massive
vector and massive gravity actions we derive by dimensionally reducing the
perturbed Einstein-Hilbert action describe the dynamics of a well studied
physical system: the metric perturbations of a static black hole. The $2+2$
formalism we present can be generalized to $m+n$ dimensional spacetime
splittings, which may be useful in more generic situations, such as expanding
metric perturbations in higher dimensional gravity. We provide a self-contained
presentation of $m+n$ formalism for vacuum spacetime splittings.
