Einstein field equations
The Einstein field equations (EFE) may be written in the form:
=
where
is the Ricci curvature tensor,
the scalar curvature,
the metric tensor,
is the cosmological constant,
is Newton's gravitational constant,
the speed of light in vacuum, and
the stress–energy tensor.
The EFE is a tensor equation relating a set of symmetric 4×4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.
Despite the simple appearance of the equations they are, in fact, quite complicated. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor g_{mu
u}, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations.
One can write the EFE in a more compact form by defining the Einstein tensor
=
,
which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
=
.
Using geometrized units where G = c = 1, this can be rewritten as
=
.
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.
These equations, together with the geodesic equation, which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.
Sign convention
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
 \[6pt] {R^mu}_{a eta gamma} & = [S2] imes (Gamma^mu_{a gamma,eta}-Gamma^mu_{a eta,gamma}+Gamma^mu_{sigma eta}Gamma^sigma_{gamma a}-Gamma^mu_{sigma gamma}Gamma^sigma_{eta a}) \[6pt] G_{mu
u} & = [S3] imes {8 pi G over c^4} T_{mu
u} end{align})
The third sign above is related to the choice of convention for the Ricci tensor:
=
With these definitions Misner, Thorne, and Wheeler classify themselves as (+++),, whereas Weinberg (1972) is (+--),, Peebles (1980) and Efstathiou (1990) are (-++), while Peacock (1994), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are (-+-),.
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
=
.
The sign of the (very small) cosmological term would change in both these versions, if the +−−− metric sign convention is used rather than the MTW −+++ metric sign convention adopted here.
Equivalent formulations
Taking the trace of both sides of the EFE one gets
=
which simplifies to
.
If one adds
times this to the EFE, one gets the following equivalent "trace-reversed" form
=
 ,)
.
Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace
in the expression on the right with the Minkowski metric without significant loss of accuracy).
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