Web Element: Math
Einstein field equations
[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]
[math]R_{\mu \nu} [/math] Ricci curvature tensor
where [math]R[[:Template:Sub]][/math] is the Ricci curvature tensor, Template:Mvar is the scalar curvature, Template:Mvar is the metric tensor, Template:Mvar is the cosmological constant, Template:Mvar is Newton's gravitational constant, Template:Mvar is the speed of light in vacuum, and Template:Mvar is the stress–energy tensor.
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler.Template:Sfnp The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
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 Template:Math, whereas Weinberg (1972)Template:Sfnp and Peacock (1994)Template:Sfnp are Template:Math, Peebles (1980)[1] and Efstathiou et al. (1990)[2] are Template:Math, Rindler (1977)Template:Citation needed, Atwater (1974)Template:Citation needed, Collins Martin & Squires (1989)[3] 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
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.
Taking the trace with respect to the metric of both sides of the EFE one gets
where Template:Mvar is the spacetime dimension. This expression can be rewritten as
If one adds Template:Math times this to the EFE, one gets the following equivalent "trace-reversed" form
For example, in Template:Math dimensions this reduces to
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).
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