Typo in d'Inverno?

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nrqed

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On page 172 he writes


[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?
 

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