Mass in general relativity

Summary

In general relativity the energy of the gravitational field has no local density that is independent of coordinates. So the mass of a system cannot be found by integrating a density. Instead, mass is defined for isolated systems by the behaviour of the metric far away. The main definitions are the ADM energy at spatial infinity, the Bondi mass at null infinity and the Komar mass for stationary spacetimes. For stationary spacetimes they all agree. The positive mass theorem shows that the ADM and Bondi masses are non-negative under physical conditions.