Mass in general relativity

The concept of mass in general relativity (GR) is more subtle to define than the concept of mass in special relativity. In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined. The reason for this subtlety is that the energy and momentum in the gravitational field cannot be unambiguously localized. (See Chapter 20 of .) So, rigorous definitions of the mass in general relativity are not local, as in classical mechanics or special relativity, but make reference to the asymptotic nature of the spacetime. A well defined notion of the mass exists for asymptotically flat spacetimes and for asymptotically Anti-de Sitter space. However, these definitions must be used with care in other settings. In special relativity, the rest mass of a particle can be defined unambiguously in terms of its energy and momentum as described in the article on mass in special relativity. Generalizing the notion of the energy and momentum to general relativity, however, is subtle. The main reason for this is that that gravitational field itself contributes to the energy and momentum. However, the "gravitational field energy" is not a part of the energy–momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). While in certain situations it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the stress–energy–momentum pseudotensor, this separation is not true for all observers, and there is no general definition for obtaining it.