The Gauss Bonnet Lagrangian and the conserved current

[haushofer]

Hi,

I have a computational question about the Hilbert action with an added Gauss-Bonnet term. The Lagrangian L ( seen here as a scalar density) then looks like this:

$$L = {\cal L} \sqrt{g} = \Bigl(R+ R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} - 4R_{\mu\nu}R^{\mu\nu} + R^{2}\Bigr)\sqrt{g}$$

We can always write the variation of the Lagrangian as

$$\delta L &=& E^{\mu\nu} \delta g_{\mu\nu} + \nabla_{\rho}\Theta^{\rho}$$

Here, the E represents the equation of motion and the $$\Theta$$ is the conserved current if E=0. The idea now is to regard coordinate transformations and derive the conserved current. So my question is: how do we calculate this current for the particular Lagrangian given above? I used the following identities:

$$R^{\mu\nu\rho\sigma}\delta R_{\mu\nu\rho\sigma} &=& g^{\lambda\alpha}R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\alpha}\delta g_{\sigma\lambda} + 2R^{\mu\nu\rho\sigma}\nabla_{\mu}\nabla_{\rho}\delta g_{\sigma\nu}$$

$$R^{\mu\nu}\delta R_{\mu\nu} &=& g^{\alpha\beta}R_{\alpha\mu\beta\nu}\Bigl(\nabla_{\lambda}\delta\Gamma^{\lambda}_{\mu\nu} - \nabla_{\nu}\delta\Gamma^{\lambda}_{\mu\lambda}\Bigr)$$

$$R\delta R &=& R\na_{\alpha}\Bigl(g^{\mu\nu}\delta\Gamma^{\alpha}_{\mu\nu} - g^{\mu\alpha}\delta\Gamma^{\lambda}_{\mu\lambda}\Bigr) - R\ R^{\mu\nu}\delta g_{\mu\nu}$$

The total variation reads

$$\delta L = \Bigl(2R^{\mu\nu\rho\sigma}\delta R_{\mu\nu\rho\sigma} - 8 R^{\mu\nu}\delta R_{\mu\nu}\ + 2R\delta R \ + \frac{1}{2} {\cal L}\ g^{\mu\nu}\delta g_{\mu\nu}\Bigr)$$

So what I did was to plug in those variations, collect the terms proportional to the variation in the metric and called this E, and collect the terms proportional to derivatives of the variations of the metric and this should be equal to $$\nabla_{\alpha}\Theta^{\alpha}$$. So what I got is

$$E^{\alpha\beta} =\sqrt{g}\Bigl(2R^{\mu\nu\rho\alpha}R_{\mu\nu\rho}^{\ \ \ \ \beta} - (2R+1)R^{\alpha\beta} + \frac{1}{2} {\cal L}\ g^{\mu\nu}\Bigr)$$

and

$$\nabla_{\alpha}\Theta^{\alpha} = 4R^{\mu\nu\rho\sigma}\nabla_{\mu}\nabla_{\rho}\delta g_{\sigma\nu} - 8 R^{\mu\nu}(\nabla_{\alpha}\delta\Gamma^{\alpha}_{\mu\nu} - \nabla_{\nu}\delta\Gamma^{\alpha}_{\mu\alpha}) + (2R+1)g^{\mu\nu}(\nabla_{\alpha}\delta\Gamma^{\alpha}_{\mu\nu} - \nabla_{\nu}\delta\Gamma^{\alpha}_{\mu\alpha})$$

In the case of the Hilbert action, I know how to deal with the term

$$g^{\mu\nu}(\nabla_{\alpha}\delta\Gamma^{\alpha}_{\mu\nu} - \nabla_{\nu}\delta\Gamma^{\alpha}_{\mu\alpha}) \equiv \nabla_{\alpha}X^{\alpha}$$

if I define

$$X^{\alpha} \equiv \Bigl(g^{\mu\nu}\delta\Gamma^{\alpha}_{\mu\nu} - g^{\mu\alpha}\delta\Gamma^{\lambda}_{\mu\lambda}\Bigr)$$

So these kind of terms I can handle, and as far I can see I'm left with showing that

$$\nabla_{\alpha}\tilde{\Theta}^{\alpha} = 4R^{\mu\nu\rho\sigma}\nabla_{\mu}\nabla_{\rho}\delta g_{\sigma\nu} - 2 X^{\alpha}\nabla_{\alpha}R - 8 R^{\mu\nu}(\nabla_{\alpha}\delta\Gamma^{\alpha}_{\mu\nu} -\nabla_{\nu}\delta\Gamma^{\alpha}_{\mu\alpha})$$

after some rewriting with the product rule.

Can anyone comment on this, show me where/if I made some errors, and how to proceed? Should I plug in the explicit variation

$$\delta_{\xi} g_{\mu\nu} = 2 \nabla_{(\mu}\xi_{\nu)}$$

for the variation in the metric, or do I need to do something else? I couldn't find any useful articles about this, so I hope someone here can help me ! Regards,

Haushofer.

 

[haushofer]

A little kick :)

 

[shoehorn]

It's not at all clear what you want to do. Are you asking how to derive the field equations from a Gauss-Bonnet Lagrangian?

 

[haushofer]

Well, I want to vary the Lagrangian, to obtain one part that comprehends my equations of motion for the metric, and one part that comprehends my divergence of $$\Theta$$.

The Gauss-Bonnet term is called a topological invariant, so I have the suspicion that adding the Gauss-Bonnet term to my Hilbert action doesn't add anything to my field equations ( the variation of the metric is then considered to be the homeomorphism which leaves the Gauss-Bonnet term invariant ), and that the variation of the Gauss-Bonnet term is a total derivative. Is that correct? But that's not quite clear from my considerations given above. I have the feeling that a nifty trick is needed because my calculations are quite messy.

 

[shoehorn]

I'm unsure of what benefit the divergence terms will be to you. It's straightforward to vary the action with respect to the metric to give you terms that are proportional to the metric, plus terms that involve variations of the curvature tensors with respect to the metric. Partial integration (and accepting, say, that your manifold has no boundary) then gives you the standard Gauss-Bonnet equations for the gravitational field.

It's been a while since I've done the calculations. Give me a bit and I'll run through them for you; hopefully I'll be able to post something back this evening.

 

[haushofer]

That would be very nice, thanks !