# Different signatures of the metric in Einstein field equations

Throughout the GR lectures, we have always used (- , + , + , +) signature for the metric tensor but in some chapters it was switched to (+ , - , - , -) and immediately after that Einstein field equations were derived

$$R^{\mu \nu}- \frac{1}{2}g^{\mu \nu}R + \Lambda g^{\mu \nu} = \kappa T^{\mu \nu}$$

where $$\Lambda = \frac{8 \pi G}{c^2}$$. Yet without the cosmological constant, EFE in covariant form was

$$R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R = -\kappa T_{\mu \nu}$$

This part is a bit mysterious to me, because when I carry out the derivation I don't get the minus sign on the R.H.S of the equation. Additionally, according to wikipedia

$$R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} \hspace{5mm} or \hspace{5mm} R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R - \Lambda g_{\mu \nu} = -\kappa T_{\mu \nu}$$

Last equation is fine since they represent EFE with different signatures. However, I wonder if there is a way to tell which signature is used in above equations.

• It's nonsense to switch it, unless you are Robert Wald, you have written a classic text on GR and you know that spinors are mostly treated in the literature in the +---- (i.e.particle and QFT convention). It is the only accepted exception. Apr 29, 2021 at 23:38
• See my answer, but note that the last two equations you wrote weren't aren't for different signatures, they're for different conventions for the Ricci tensor (the cosmological constant term would change though). I also agree with DanielC's comment that changing signs in any standard lectures on GR does not seem wise. Apr 29, 2021 at 23:41