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The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on a wall in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the (− + + +) metric signature , the gravitational part of the action is given as [ 1 ]
Next, notice that only 10 of the original 14 equations are independent, because the continuity equation ; = is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant (in general, absent some symmetry, any choice of a curvilinear coordinate net on the same system would correspond to a numerically ...
The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled ...
The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows. Birkhoff's theorem has the consequence that any pulsating star that remains spherically symmetric does not generate gravitational waves , as the ...
Albert Einstein's discovery of the gravitational field equations of general relativity and David Hilbert's almost simultaneous derivation of the theory using an elegant variational principle, [B 1]: 170 during a period when the two corresponded frequently, has led to numerous historical analyses of their interaction.
In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter.
The Einstein field equation (EFE) describing the geometry of spacetime is given as = where is the Ricci tensor, is the Ricci scalar, is the energy–momentum tensor, = / is the Einstein gravitational constant, and is the spacetime metric tensor that represents the solutions of the equation.