  Due to the absence of symmetries under time and spatial translations in a
general curved spacetime, the energy and momentum of matter is not conserved as
it is in flat space. This means, for example, that the flux of matter energy
through a surface is in general not balanced by an equal increase in the energy
of the matter contained within the enclosed volume - there is an additional
"source" resulting from the curvature of spacetime acting on the matter (and
vice versa). One can calculate this source term and reconcile the flux and
energy accumulation over time in an arbitrary volume, although a foliation of
the spacetime must be chosen, making the quantities inherently coordinate
dependent. Despite this dependence, these quantities are practically useful in
numerical relativity simulations for a number of reasons. We provide
expressions for general matter sources in a form appropriate for implementation
in the ADM decomposition, and discuss several applications in simulations of
compact object dynamics and cosmology.
