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

Continuity equations for general matter: applications in numerical relativity