Lagrangian for the General Relativity

[MManuel Abad]

I've always found that the lagrangian for the Gravitational field is that from the Einstein-Hilbert action:

$<i>L</i>=R$ (R is the Ricci scalar; I'm not including the factor of $\sqrt{-g}$)

but when variational principles are applied, we get the vacuum field equations (obviously). I'd like someone to tell me which would be the FULL lagrangian (with matter coupled) for getting the Einstein's field equations.

 

[bcrowell]

You simply take the gravitational Lagrangian density and add in the Lagrangian density of the matter fields: $L=L_G+L_M$.

 

[Ben Niehoff]

For example, if you have electromagnetic fields, then the action is

$$S = \int \ast \mathcal{R} - \frac12 \int F \wedge \ast F$$

where $\ast$ is the Hodge dual and F is the electromagnetic 2-form (rescaled up to some factors of $2\pi$ which I don't remember...I use the above normalization in my research). Using the normalization I've given here and varying with respect to the inverse metric, you obtain

$$R_{\mu\nu} - \frac12 \mathcal{R} g_{\mu\nu} = \frac12 T_{\mu\nu}$$

where

$$T_{\mu\nu} = F_{\mu\rho} F_\nu{}^\rho - \frac14 g_{\mu\nu} F_{\rho\sigma} F^{\rho\sigma}$$

 

[MManuel Abad]

Wow, thankyou both, that was very useful! :)