Sign error in d'Inverno's Lagrangian with cosmological constant

[nrqed]

On page 172 he writes


$$G_{ab} - \Lambda g_{ab} = 8 \pi T_{ab}~~~(13.5)$$

Using the results of section 11.3..the corresponding Lagrangian is

$${\cal L} = \sqrt{-g} (R - 2 \Lambda) + {\cal L}_M$$



But the sign of the Lambda term in the Lagrangian is wrong, it seems to me.


In section 11.3 he shows that

$$\frac{\delta (R \sqrt{-g})}{\delta g_{ab}} = - \sqrt{-g} G^{ab}$$

and


$$\frac{\delta ( \sqrt{-g})}{\delta g_{ab}} = \frac{1}{2} \sqrt{-g} g^{ab}$$



However, the signs are switched in both equations if we do the variation with respect to $$g^{ab}$$:

$$\frac{\delta (R \sqrt{-g})}{\delta g^{ab}} = + \sqrt{-g} G_{ab}$$

and


$$\frac{\delta ( \sqrt{-g})}{\delta g^{ab}} = - \frac{1}{2} \sqrt{-g} g_{ab}$$


So the Lagrangian he wrote does not lead to the equation he gave because the Lambda term will acquire a minus sign.


Can someone tell me if I am missing something?

 

[Greg Bernhardt]

@nrqed did you have any more insight on this topic?