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 Arnowitt Deser Misner decomposition, and discuss several applications in simulations of compact object dynamics and cosmology.

由于在一般弯曲时空的时间和空间平移下不存在对称性，物质的能量和动量不像在平坦空间中那样守恒。这意味着，例如，通过表面的物质能量通量通常无法通过包含在封闭体积内的物质能量的等量增加来平衡——还有一个额外的“源”是由时空曲率作用产生的在这个问题上（反之亦然）。虽然必须选择时空的叶理，但可以计算该源项并协调随时间推移在任意体积中的通量和能量积累，从而使数量固有地依赖于坐标。尽管存在这种依赖性，但出于多种原因，这些量在数值相对论模拟中实际上是有用的。