Einstein field equations
label: Persamaan medan Einstein @in
label: 아인슈타인 방정식 @ko
label: Ecuaciones del campo de Einstein @es
label: Уравнения Эйнштейна @ru
label: Equacions de camp d'Einstein @ca
label: Equazione di campo di Einstein @it
label: Einsteinen eremu-ekuazioak @eu
label: アインシュタイン方程式 @ja
label: Einsteinovy rovnice gravitačního pole @cs
label: Einsteinsche Feldgleichungen @de
label: Einsteins fältekvationer @sv
label: 爱因斯坦场方程 @zh
label: Einstein-vergelijking @nl
label: Équation d'Einstein @fr
label: Рівняння Ейнштейна @uk
label: معادلات أينشتاين للمجال @ar
label: Equações de campo de Einstein @pt
label: Równanie Einsteina @pl
subject: Category:Equations of physics
subject: Category:Albert Einstein
subject: Category:General relativity
subject: Category:Partial differential equations
id: p/e035210
proof: Newtonian gravitation can be written as the theory of a scalar field, , which is the gravitational potential in joules per kilogram of the gravitational field , see Gauss's law for gravity where is the mass density. The orbit of a free-falling particle satisfies In tensor notation, these become In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form for some constant, , and the geodesic equation To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero and thus and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives where two factors of have been divided out. This will reduce to its Newtonian counterpart, provided Our assumptions force and the time derivatives to be zero. So this simplifies to which is satisfied by letting Turning to the Einstein equations, we only need the time-time component the low speed and static field assumptions imply that So and thus From the definition of the Ricci tensor Our simplifying assumptions make the squares of disappear together with the time derivatives Combining the above equations together which reduces to the Newtonian field equation provided which will occur if
proof: Contracting the differential Bianchi identity with gives, using the fact that the metric tensor is covariantly constant, i.e. , The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten: which is equivalent to using the definition of the Ricci tensor. Next, contract again with the metric to get The definitions of the Ricci curvature tensor and the scalar curvature then show that which can be rewritten as A final contraction with gives which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices, Using the EFE, this immediately gives,
description: Mga ekwasyong field sa pangkalahatang relatibidad @tl
description: equações de campo na relatividade geral @pt
description: samenvatting van de algemene relativiteitstheorie van Einstein @nl
description: partaj diferencialaj ekvacioj de la ĝenerala teorio de relativeco @eo
description: ecuații de câmp în relativitatea generală @ro
description: 万有引力・重力場を記述する場の方程式であり、アルベルト・アインシュタインによって導入された @ja
description: field equations in general relativity
description: équation de la relativité générale @fr
description: Einstein'nın 1915 yılında yayımladığı denklem kümesi @tr
description: equazione fondamentale della teoria gravitazionale della relatività generale @it
description: уравнения лежавшие в основе общей теории относительности @ru
description: Beskrivelse af gravitationens fysiske fænomen @da
description: soubor rovnic @cs
description: Beschreibung des physikalischen Phänomens der Gravitation @de
description: równanie pola grawitacyjnego @pl
description: 일반 상대성 이론의 기본 방정식 @ko
wiki page external link: https://archive.org/details/gravitationcosmo00stev_0
wiki page external link: http://scitation.aip.org/content/aip/magazine/physicstoday/article/68/11/10.1063/PT.3.2979
wiki page external link: https://archive.org/details/cosmologicalphys0000peac
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