Persamaan medan Einstein
아인슈타인 방정식
Ecuaciones del campo de Einstein
Уравнения Эйнштейна
Equacions de camp d'Einstein
Equazione di campo di Einstein
Einsteinen eremu-ekuazioak
アインシュタイン方程式
Einsteinovy rovnice gravitačního pole
Einsteinsche Feldgleichungen
Einsteins fältekvationer
爱因斯坦场方程
Einstein field equations
Einstein-vergelijking
Équation d'Einstein
Рівняння Ейнштейна
معادلات أينشتاين للمجال
Equações de campo de Einstein
Równanie Einsteina
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
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,
Mga ekwasyong field sa pangkalahatang relatibidad
equações de campo na relatividade geral
samenvatting van de algemene relativiteitstheorie van Einstein
partaj diferencialaj ekvacioj de la ĝenerala teorio de relativeco
ecuații de câmp în relativitatea generală
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
field equations in general relativity
équation de la relativité générale
Einstein'nın 1915 yılında yayımladığı denklem kümesi
equazione fondamentale della teoria gravitazionale della relatività generale
уравнения лежавшие в основе общей теории относительности
Beskrivelse af gravitationens fysiske fænomen
soubor rovnic
Beschreibung des physikalischen Phänomens der Gravitation
równanie pola grawitacyjnego
일반 상대성 이론의 기본 방정식
Derivation of Newton's law of gravity
Einstein equations
Derivation of local energy–momentum conservation