Editing General relativity (section)

=== Evolution equations ===
{{Main|Initial value formulation (general relativity)}}
Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the [[time evolution]] of the metric tensor. It must be combined with a [[coordinate condition]], which is analogous to [[gauge fixing]] in other field theories.<ref>{{Harvnb|Hawking|Ellis|1973|loc=sec. 7.1}}</ref>

To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the [[ADM formalism]].<ref>{{Harvnb|Arnowitt|Deser|Misner|1962}}; for a pedagogical introduction, see {{Harvnb|Misner|Thorne|Wheeler|1973|loc=§&nbsp;21.4–§&nbsp;21.7}}</ref> These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always [[existence theorem|exist]], and are uniquely defined, once suitable initial conditions have been specified.<ref>{{Harvnb|Fourès-Bruhat|1952}} and {{Harvnb|Bruhat|1962}}; for a pedagogical introduction, see {{Harvnb|Wald|1984|loc=ch. 10}}; an online review can be found in {{Harvnb|Reula|1998}}</ref> Such formulations of Einstein's field equations are the basis of numerical relativity.<ref>{{Harvnb|Gourgoulhon|2007}}; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see {{Harvnb|Lehner|2001}}</ref>