  We develop a self-consistent, Gravitoelectromagnetic (GEM) formulation of a
slowly rotating, self-gravitating and dilute Bose-Einstein condensate (BEC),
intended for astrophysical applications in the context of dark matter halos.
GEM self-consistently incorporates the effects of frame dragging to lowest
order in $v/c$ via the Gravitomagnetic field. BEC dark matter has attracted
attention as an alternative to Cold dark matter (CDM) and Warm dark matter
(WDM) for some time now. The BEC is described by the Gross-Pitaevskii-Poisson
(GPP) equation with an arbitrary potential allowing for either attractive or
repulsive interactions. Owing to the difficulty in obtaining exact solutions to
the GEM equations of motion without drastic approximations, we employ the
variational method to examine the conditions under which rotating condensates,
stable against gravitational collapse, may form in models with attractive and
repulsive quartic interactions. We also describe the approximate dynamics of an
imploding and rotating condensate by employing a collective coordinate
description in terms of the condensate radius.
