Mass in general relativity

In general relativity, the notion of mass is more subtle than in classical mechanics or special relativity. The theory itself does not single out a preferred local definition of mass, and several distinct notions of mass have been introduced that are applicable under different geometric or physical circumstances. This subtlety stems from the fact that the energy and momentum of the gravitational field cannot be unambiguously localized.

As a consequence, mass in general relativity cannot be defined as a local quantity in the same sense as in Newtonian physics, since there is no invariant local energy density for the gravitational field. Instead, notions of mass are recovered indirectly from global or boundary data, which typically requires additional geometric structure, such as symmetries or assumptions about the asymptotic behavior of spacetime. Well-defined notions of total mass exist in important classes of spacetimes, such as asymptotically flat spacetimes and asymptotically anti-de Sitter spacetimes, where suitable boundary conditions allow for a meaningful definition. Other settings require different or more refined constructions.