I'm trying to express the classical gravitation Einstein-Hilbert lagrangian into some nice way, and I'm having a problem.(adsbygoogle = window.adsbygoogle || []).push({});

It is well known that the Einstein-Hilbert action is the following (I don't write the constant in front of the integral, to simplify things) :

[itex]S_{EH} = \int R \, \sqrt{-g} \; d^4 x[/itex].

After removing the hypersurface term, the lagrangian is this (this is a well known result) :

[itex]\mathscr{L}_{EH} = g^{\mu \nu} \, (\, \Gamma_{\lambda \kappa}^{\lambda} \; \Gamma_{\mu \nu}^{\kappa} - \Gamma_{\mu \kappa}^{\lambda} \; \Gamma_{\lambda \nu}^{\kappa})[/itex].

When I substitute the definition of the Christoffel symbols and simplify things, I get this :

[itex]\mathscr{L}_{EH} = \frac{1}{4} H^{\mu \nu \lambda \kappa \rho \sigma} (\, \partial_{\mu} \, g_{\lambda \kappa})(\, \partial_{\nu} \, g_{\rho \sigma})[/itex],

where I defined this horrible thing (this is the source of my problem) :

[itex]H^{\mu \nu \lambda \kappa \rho \sigma} = \left( \, g^{\mu \nu} (\, g^{\lambda \rho} \, g^{\kappa \sigma} - g^{\lambda \kappa} \, g^{\rho \sigma}) + 2 \, g^{\nu \kappa} (\, g^{\mu \lambda} \, g^{\rho \sigma} - g^{\mu \rho} \, g^{\lambda \sigma}) \right)[/itex].

This expression is ugly : it isn't symetric in [itex]\lambda \kappa[/itex] and [itex]\rho \sigma[/itex]. Of course, I can make it symetric, but then the [itex]\mu \nu[/itex] indices are getting in the way and make things more complicated.

Is there a "natural" way to define a proper [itex]H^{\mu \nu \lambda \kappa \rho \sigma}[/itex] ? Any thoughts on this ?

The lagrangian above is nice because it is very similar to the real scalar field lagrangian without mass :

[itex]\mathscr{L} = \frac{1}{2} g^{\mu \nu} (\, \partial_{\mu} \, \phi \,)(\, \partial_{\nu}\, \phi \,)[/itex]

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# Gravitation Lagrangian in classical form

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