Solving Einstein Equation Confusion: My Notes vs Freidmann Equations

[latentcorpse]

My notes for a particular course say that $G_{ab} - \Lambda g_{ab}= x T_{ab}$ where $x=\frac{8 \pi G}{c^4}$
Then they say that the trace of this is $-R+4 \Lambda=xT$
What?
Surely that's only possible if we have $+\Lambda g_{ab}$ as I have seen in every other text I've ever read?

However, when we derive the Freidmann equations from this we get a $( \frac{\dot{a}}{a})^2=\frac{8 \pi G}{3} \rho -\frac{k}{a^2}+\frac{\Lambda}{3}$
So if we had $-\Lambda$ on the LHS as suggested in the notes then this will come over to the RHS and give a + as required by the Freidmann equations but now it is all inconsistent with the trace.

This is so confusing! Can anyone clear this up? Thanks.

 

[Je m'appelle]

Are you sure it isn't

$$G_{ab} + \Lambda g_{ab}= \frac{8 \pi G}{c^4} T_{ab}$$

in the original Equation, that is, +Λ instead of -Λ on the left-hand side?

In case your textbook/notes actually state -Λ, then there's a strong possibility of it being a typo, and in this case, just ignore it.

I'd also recommend you to notify your tutor/professor on this.