Einstein field equations (EFE)

[math]R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} [/math]

where [math]R_{\mu \nu} [/math] is the Ricci curvature tensor, [math]R[/math] is the scalar curvature, [math]g_{\mu \nu} [/math] is the metric tensor (general relativity), [math]Λ [/math] is the cosmological constant, [math]G[/math] is Newton's gravitational constant, [math]c[/math] is the speed of light in vacuum, and [math]T_{\mu \nu} [/math] is the stress–energy tensor.

Sign convention

The above form of the EFE is the standard established by Gravitation by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

[math] \begin{align} g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt] {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt] G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu} \end{align} [/math]

The third sign above is related to the choice of convention for the Ricci tensor:

[math]R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} [/math]

With these definitions Misner, Thorne, and Wheeler classify themselves as Template:Math, whereas Weinberg (1972)Template:Sfnp and Peacock (1994)Template:Sfnp are Template:Math, Peebles (1980) and Efstathiou et al. (1990) are Template:Math, Rindler (1977)Template:Citation needed, Atwater (1974)Template:Citation needed, Collins Martin & Squires (1989) are Template:Math.

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

[math]R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.[/math]

The sign of the (very small) cosmological term would change in both these versions, if the Template:Math metric sign convention is used rather than the MTW Template:Math metric sign convention adopted here.

Equivalent formulations

Taking the trace with respect to the metric of both sides of the EFE one gets

[math]R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,[/math]

where Template:Mvar is the spacetime dimension. This expression can be rewritten as

[math]-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.[/math]

If one adds Template:Math times this to the EFE, one gets the following equivalent "trace-reversed" form

[math]R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,[/math]

For example, in Template:Math dimensions this reduces to

[math]R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,[/math]

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 Template:Mvar in the expression on the right with the Minkowski metric without significant loss of accuracy).