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Then they say that the trace of this is [itex]-R+4 \Lambda=xT[/itex]

What?

Surely that's only possible if we have [itex]+\Lambda g_{ab}[/itex] as I have seen in every other text I've ever read?

However, when we derive the Freidmann equations from this we get a [itex]( \frac{\dot{a}}{a})^2=\frac{8 \pi G}{3} \rho -\frac{k}{a^2}+\frac{\Lambda}{3}[/itex]

So if we had [itex]-\Lambda[/itex] 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.