[tex] G_{ab} - \Lambda g_{ab} = 8 \pi T_{ab}~~~(13.5) [/tex]

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

[tex] {\cal L} = \sqrt{-g} (R - 2 \Lambda) + {\cal L}_M [/tex]

[tex] G_{ab} - \Lambda g_{ab} = 8 \pi T_{ab}~~~(13.5) [/tex]

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

[tex] {\cal L} = \sqrt{-g} (R - 2 \Lambda) + {\cal L}_M [/tex]

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

In section 11.3 he shows that

[tex] \frac{\delta (R \sqrt{-g})}{\delta g_{ab}} = - \sqrt{-g} G^{ab} [/tex]

and

[tex] \frac{\delta ( \sqrt{-g})}{\delta g_{ab}} = \frac{1}{2} \sqrt{-g} g^{ab} [/tex]

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

[tex] \frac{\delta (R \sqrt{-g})}{\delta g^{ab}} = + \sqrt{-g} G_{ab} [/tex]

and

[tex] \frac{\delta ( \sqrt{-g})}{\delta g^{ab}} = - \frac{1}{2} \sqrt{-g} g_{ab} [/tex]

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?