Editing Einstein field equations (section)

=== Sign convention ===
The above form of the EFE is the standard established by [[Gravitation (book)|Misner, Thorne, and Wheeler]] (MTW).{{sfnp|Misner|Thorne|Wheeler|1973|p=501ff}} The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):
<math display="block">\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 \kappa T_{\mu \nu}
\end{align}</math>

The third sign above is related to the choice of convention for the Ricci tensor:
<math display="block">R_{\mu \nu} = [S2] \times [S3] \times {R^\alpha}_{\mu\alpha\nu} </math>

With these definitions [[Gravitation (book)|Misner, Thorne, and Wheeler]] classify themselves as {{math|(+ + +)}}, whereas Weinberg (1972){{sfnp|Weinberg|1972}} is {{math|(+ − −)}}, Peebles (1980)<ref>{{cite book |last=Peebles |first=Phillip James Edwin |title=The Large-scale Structure of the Universe |publisher=Princeton University Press |year=1980 |isbn=0-691-08239-1 }}</ref> and Efstathiou et al. (1990)<ref>{{cite journal |last1=Efstathiou |first1=G. |first2=W. J. |last2=Sutherland |first3=S. J. |last3=Maddox |s2cid=12988317 |title=The cosmological constant and cold dark matter |journal=[[Nature (journal)|Nature]] |volume=348 |issue=6303 |year=1990 |pages=705 |doi=10.1038/348705a0 |bibcode=1990Natur.348..705E }}</ref> are {{math|(− + +)}},  Rindler (1977),{{citation needed|date=October 2014}} Atwater (1974),{{citation needed|date=October 2014}} Collins Martin & Squires (1989)<ref>{{cite book |last1=Collins |first1=P. D. B. |last2=Martin |first2=A. D. |last3=Squires |first3=E. J. |year=1989 |title=Particle Physics and Cosmology |location=New York |publisher=Wiley |isbn=0-471-60088-1 }}</ref> and Peacock (1999){{sfnp|Peacock|1999}} are {{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 display="block">R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\kappa T_{\mu \nu}.</math>

The sign of the cosmological term would change in both these versions if the {{math|(+ − − −)}} metric [[sign convention]] is used rather than the MTW {{math|(− + + +)}} metric sign convention adopted here.