Editing Einstein field equations (section)

{{Short description|Field-equations in general relativity}}
{{Redirect|Einstein equation|the equation E{{=}}mc<sup>2</sup>|Mass–energy equivalence}}
{{Use American English|date=January 2019}}
{{General relativity sidebar |equations}}
In the [[General relativity|general theory of relativity]], the '''Einstein field equations''' ('''EFE'''; also known as '''Einstein's equations''') relate the geometry of [[spacetime]] to the distribution of [[Matter#In general relativity and cosmology|matter]] within it.<ref name="ein">{{cite journal |last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[Annalen der Physik]] |volume=354 |issue=7 |pages=769 |year=1916 |url=http://www.alberteinstein.info/gallery/science.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |bibcode=1916AnP...354..769E |archive-url=https://web.archive.org/web/20120206225139/http://www.alberteinstein.info/gallery/gtext3.html |archive-date=2012-02-06}}</ref>

The equations were published by [[Albert Einstein]] in 1915 in the form of a [[Tensor|tensor equation]]<ref name=Ein1915>{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |date=November 25, 1915 |title=Die Feldgleichungen der Gravitation |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin |pages=844–847 |url=http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/6E3MAXK4/index.meta |access-date=2017-08-21}}</ref> which related the local ''{{vanchor|spacetime [[curvature]]|SPACETIME_CURVATURE}}'' (expressed by the [[Einstein tensor]]) with the local energy, [[momentum]] and stress within that spacetime (expressed by the [[stress–energy tensor]]).{{sfnp|Misner|Thorne|Wheeler|1973|p=916 [ch. 34]}}

Analogously to the way that [[electromagnetic field]]s are related to the distribution of [[Charge (physics)|charge]]s and [[Electric current|current]]s via [[Maxwell's equations]], the EFE relate the [[spacetime geometry]] to the distribution of mass–energy, momentum and stress, that is, they determine the [[Metric tensor (general relativity)|metric tensor]] of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear [[partial differential equation]]s when used in this way. The solutions of the EFE are the components of the metric tensor. The [[inertia]]l trajectories of particles and radiation ([[Geodesics in general relativity|geodesics]]) in the resulting geometry are then calculated using the [[geodesic equation]].

As well as implying local energy–momentum conservation, the EFE reduce to [[Newton's law of gravitation]] in the limit of a weak gravitational field and velocities that are much less than the [[speed of light]].<ref name="Carroll">{{cite book |last=Carroll |first=Sean |author-link=Sean M. Carroll |year=2004 |title=Spacetime and Geometry – An Introduction to General Relativity |pages=151–159 |publisher=Addison Wesley |isbn=0-8053-8732-3}}</ref>

Exact solutions for the EFE can only be found under simplifying assumptions such as [[Spacetime symmetries|symmetry]]. Special classes of [[Exact solutions in general relativity|exact solutions]] are most often studied since they model many gravitational phenomena, such as [[rotating black hole]]s and the [[Metric expansion of space|expanding universe]]. Further simplification is achieved in approximating the spacetime as having only small deviations from [[Minkowski space|flat spacetime]], leading to the [[Linearized gravity#Linearized Einstein field equations|linearized EFE]]. These equations are used to study phenomena such as [[gravitational waves]].