Different signatures of the metric in Einstein field equations

Question

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.

Answer 1

For both signatures, the terms in the EFE (except the cosmological constant) have the same sign (see the answers here for why Einstein GR and metric signature). So without more information, no, it is not possible to tell just based on the EFE. The reason for different signs in the EFE is based on other conventions, e.g. for defining the Riemann tensor and the Ricci tensor. But just looking at the EFE, I don't think it's possible to know what conventions someone is using.

The Wikipidea page you linked does talk about the various different sign conventions, and this is discussed in more detail in this blog post here, which you may find useful (it references a long list of textbooks and who uses what).