Editing General relativity (section)

=== Global and quasi-local quantities ===
{{Main|Mass in general relativity}}
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.<ref>{{Harvnb|Misner|Thorne|Wheeler|1973|loc=§&nbsp;20.4}}</ref>

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" ([[ADM mass]])<ref>{{Harvnb|Arnowitt|Deser|Misner|1962}}</ref> or suitable symmetries ([[Komar mass]]).<ref>{{Harvnb|Komar|1959}}; for a pedagogical introduction, see {{Harvnb|Wald|1984|loc=sec. 11.2}}; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. {{Harvnb|Ashtekar|Magnon-Ashtekar|1979}}</ref> If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the [[Mass in general relativity#ADM and Bondi masses in asymptotically flat space-times|Bondi mass]] at null infinity.<ref>For a pedagogical introduction, see {{Harvnb|Wald|1984|loc=sec. 11.2}}</ref> Just as in [[Physics in the Classical Limit|classical physics]], it can be shown that these masses are positive.<ref>{{Harvnb|Wald|1984|p=295 and refs therein}}; this is important for questions of stability—if there were [[negative mass]] states, then flat, empty Minkowski space, which has mass zero, could evolve into these states</ref> Corresponding global definitions exist for momentum and angular momentum.<ref>{{Harvnb|Townsend|1997|loc=ch. 5}}</ref> There have also been a number of attempts to define ''quasi-local'' quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about [[isolated system]]s, such as a more precise formulation of the hoop conjecture.<ref>Such quasi-local mass–energy definitions are the [[Hawking energy]], [[Geroch energy]], or Penrose's quasi-local energy–momentum based on [[Twistor theory|twistor]] methods; cf. the review article {{Harvnb|Szabados|2004}}</ref>