Continuity equations for general matter: applications in numerical relativity

Understand

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.