Gravitation Lagrangian in classical form

[Barnak]

I'm trying to express the classical gravitation Einstein-Hilbert lagrangian into some nice way, and I'm having a problem.

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) :

$S_{EH} = \int R \, \sqrt{-g} \; d^4 x$.

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

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

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

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

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

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

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

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

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

$\mathscr{L} = \frac{1}{2} g^{\mu \nu} (\, \partial_{\mu} \, \phi \,)(\, \partial_{\nu}\, \phi \,)$

 

[George Jones]

$H^{\mu \nu \lambda \kappa \rho \sigma} = \left( \, g^{\mu \nu} (\, g^{\lambda \rho} \, g^{\kappa \sigma} - g^{\lambda \kappa} \, g^{\rho \sigma}) + 2 \, g^{\nu \kappa} (\, g^{\mu \lambda} \, g^{\rho \sigma} - g^{\mu \rho} \, g^{\lambda \sigma}) \right)$

This looks like the same expression that Padmanabhan writes down in his book Gravitation: Foundations and Frontiers. It doesn't look like he does anything more with this.

 

[Barnak]

This looks like the same expression that Padmanabhan writes down in his book Gravitation: Foundations and Frontiers. It doesn't look like he does anything more with this.

Hmmm, that sounds very interesting. Could you say more about this ?

Can you write down exactly his version ?

 

[Barnak]

I've found a sample of Padmanabhan's book, from this place :

http://www.filestube.com/g/gravitation+foundations+and+frontiers

On page 243, he gives an expression for the gravitation lagrangian that is indeed exactly like mine, with the symbol $M$ instead of my $H$, and with a global sign difference (he's using the other sign convention for the metric).

So this is reassuring.

Sadly, his expression isn't made symetric in some of its indices, exactly like mine.

I'm pretty sure (from intuition) that this ugly tensor could be expressed in some nice way, but I don't know how.
I feel it's a combination of simpler tensors, that could reveal some symetries. Something like this (but it's yet unsatisfying) :

$H^{\mu \nu \lambda \kappa \rho \sigma} = g^{\mu \nu} (\, g^{\lambda \rho} \, g^{\kappa \sigma} - g^{\lambda \kappa} \, g^{\rho \sigma} ) + g^{\nu \kappa} (\, g^{\mu \lambda} \, g^{\rho \sigma} - g^{\mu \rho} \, g^{\lambda \sigma} ) + g^{\mu \rho} (\, g^{\nu \sigma} \, g^{\lambda \kappa} - g^{\nu \lambda} \, g^{\kappa \sigma} )$.

That tensor should be explicitely symetric for the indices $\lambda \kappa$, $\rho \sigma$ and also under the triple exchange $\mu \lambda \kappa \leftrightarrow \nu \rho \sigma$.

It could also be interpreted as a kind of "super-metric", or a kind of "metric" in the metric-space, but I'm not sure of this yet.

Any suggestions ?

 

[Barnak]

I think I've found a solution to my "problem".

In the simpler case of the Maxwell field, the lagrangian is this :

$\mathscr{L}_{\rm Maxwell} = -\, \frac{1}{4} \, F^{\mu \nu} F_{\mu \nu} \equiv \frac{1}{2} (\, g^{\mu \kappa} g^{\nu \lambda} - g^{\mu \nu} g^{\lambda \kappa})(\, \partial_{\mu} \, A_{\lambda})(\, \partial_{\nu} \, A_{\kappa})$.

So I define the following anti-symetric tensor :

$M^{\mu \nu \lambda \kappa} = g^{\mu \lambda} g^{\nu \kappa} - g^{\mu \kappa} g^{\nu \lambda}$,

which has the following properties :

$M^{\nu \mu \lambda \kappa} = -\, M^{\mu \nu \lambda \kappa}$,
$M^{\mu \nu \kappa \lambda} = -\, M^{\mu \nu \lambda \kappa}$,
$M^{\lambda \kappa \mu \nu} = M^{\mu \nu \lambda \kappa}$,
$M^{\mu \nu \lambda \kappa} + M^{\mu \lambda \kappa \nu} + M^{\mu \kappa \nu \lambda} = 0$.

That $M$ tensor is also the fundamental representation of the Lorentz generators, which satisfies some commutation relations. The Maxwell lagrangian can then be written like this :

$\mathscr{L}_{\rm Maxwell} \equiv \frac{1}{2} M^{\mu \lambda \kappa \nu} (\, \partial_{\mu} \, A_{\lambda})(\, \partial_{\nu} \, A_{\kappa})$.

So it should be "natural" to express the gravitation lagrangian using this $M$ tensor. Once all symmetrizations were made, I've found this :

$\mathscr{L}_{\rm EH} = \frac{1}{16} \, \mathcal{G}^{\mu \nu \lambda \kappa \rho \sigma} (\, \partial_{\mu} \, g_{\lambda \kappa})(\, \partial_{\nu} \, g_{\rho \sigma})$,

where

$\mathcal{G}^{\mu \nu \lambda \kappa \rho \sigma} = g^{\lambda \rho} M^{\mu \kappa \nu \sigma} + g^{\kappa \rho} M^{\mu \lambda \nu \sigma} + g^{\lambda \sigma} M^{\mu \kappa \nu \rho} + g^{\kappa \sigma} M^{\mu \lambda \nu \rho} + g^{\lambda \kappa} (\, M^{\nu \sigma \rho \mu} + M^{\nu \rho \sigma \mu}) + g^{\rho \sigma} (\, M^{\mu \kappa \lambda \nu} + M^{\mu \lambda \kappa \nu})$

This expression is fully symetric under the exchanges $\lambda \kappa \leftrightarrow \kappa \lambda$, $\rho \sigma \leftrightarrow \sigma \rho$ and the triple exchange $\mu \lambda \kappa \leftrightarrow \nu \rho \sigma$.

Now, I suspect that this last expression could be simplified a bit (I don't like the last terms in parenthesis). Any idea ?

 

[Barnak]

I made a mistake in the last expression. The correct tensor is this one, which is pretty heavy :

$\mathcal{G}^{\mu \nu \lambda \kappa \rho \sigma} = 2 \, g^{\mu \nu} (\, M^{\lambda \rho \sigma \kappa} + M^{\lambda \sigma \rho \kappa}) + g^{\mu \rho} (\, M^{\nu \lambda \sigma \kappa} + M^{\nu \kappa \sigma \lambda}) + g^{\mu \sigma} (\, M^{\nu \lambda \rho \kappa} + M^{\nu \kappa \rho \lambda}) + g^{\nu \lambda} (\, M^{\mu \rho \kappa \sigma} + M^{\mu \sigma \kappa \rho}) + g^{\nu \kappa} (\, M^{\mu \rho \lambda \sigma} + M^{\mu \sigma \lambda \rho})$.

EDIT : If I introduce a symetric tensor instead of $M$, the previous expression could be simplified a bit :

$Q^{\mu \nu \lambda \kappa} = g^{\mu \lambda} g^{\nu \kappa} + g^{\mu \kappa} g^{\nu \lambda}$, with the properties $Q^{\nu \mu \lambda \kappa} = Q^{\mu \nu \kappa \lambda} = Q^{\mu \nu \lambda \kappa}$, then :

$\mathcal{G}^{\mu \nu \lambda \kappa \rho \sigma} = 2 \, g^{\mu \nu} Q^{\lambda \kappa \rho \sigma} + 2 \, g^{\lambda \kappa} Q^{\mu \nu \rho \sigma} + 2 \, g^{\rho \sigma} Q^{\mu \nu \lambda \kappa} - g^{\mu \rho} Q^{\nu \sigma \lambda \kappa} - g^{\mu \sigma} Q^{\nu \rho \lambda \kappa} - g^{\nu \lambda} Q^{\mu \kappa \rho \sigma} - g^{\nu \kappa} Q^{\mu \lambda \rho \sigma} - 4 \, g^{\mu \nu} g^{\lambda \kappa} g^{\rho \sigma}$.

Hmmm, I must admit this is a mess !

I'm throwing all that stuff through the window