  Motivated by studies of critical phenomena in the gravitational collapse of
vacuum gravitational waves we compare, at the linear level, two common
approaches to constructing gravitational-wave initial data. Specifically, we
construct analytical, linear Brill wave initial data and compare these with
Teukolsky waves in an attempt to understand the different numerical behavior
observed in dynamical (nonlinear) evolutions of these two different sets of
data. In general, the Brill waves indeed feature higher multipole moments than
the quadrupolar Teukolsky waves, which might have provided an explanation for
the differences observed in the dynamical evolution of the two types of waves.
However, we also find that, for a common choice of the Brill-wave seed
function, all higher-order moments vanish identically, rendering the (linear)
Brill initial data surprisingly similar to the Teukolsky data for a similarly
common choice of its seed function.
