In the literature the
polar
perturbations are also called
even-parity
because they are characterized by their behavior under parity
operations as discussed earlier, and in the same way the
axial
perturbations are called
odd-parity
. We will stick to the polar/axial terminology since there is a
confusion with the definition of the parity operation, the reason
is that to most people, the words ``even'' and ``odd'' imply that
a mode transforms under
as
or
respectively (for
n
some integer). However only the polar modes with even
have even parity and only axial modes with even
have odd parity. If
is odd, then polar modes have odd parity and axial modes have
even parity. Another terminology is to call the polar
perturbations
spheroidal
and the axial ones
toroidal
. This definition is coming from the study of stellar pulsations
in Newtonian theory and represents the type of fluid motions that
each type of perturbation induces. Since we are dealing both with
stars and black holes we will stick to the polar/axial
terminology.