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$ \alpha = \beta $

The Einstein field equations (EFE) may be written in the form:

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

...

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):

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

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

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