haushofer

Science Advisor

- 2,239

- 569

## Main Question or Discussion Point

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:

[tex]

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}

[/tex]

We can always write the variation of the Lagrangian as

[tex]

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

[/tex]

Here, the E represents the equation of motion and the [tex]\Theta[/tex] 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:

[tex]

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}

[/tex]

[tex]

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)

[/tex]

[tex]

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}

[/tex]

The total variation reads

[tex]

\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)

[/tex]

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 [tex]\nabla_{\alpha}\Theta^{\alpha} [/tex]. So what I got is

[tex]

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)

[/tex]

and

[tex]

\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})

[/tex]

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

[tex]

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

[/tex]

if I define

[tex]

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

[/tex]

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

[tex]

\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})

[/tex]

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

[tex]

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

[/tex]

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.

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:

[tex]

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}

[/tex]

We can always write the variation of the Lagrangian as

[tex]

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

[/tex]

Here, the E represents the equation of motion and the [tex]\Theta[/tex] 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:

[tex]

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}

[/tex]

[tex]

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)

[/tex]

[tex]

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}

[/tex]

The total variation reads

[tex]

\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)

[/tex]

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 [tex]\nabla_{\alpha}\Theta^{\alpha} [/tex]. So what I got is

[tex]

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)

[/tex]

and

[tex]

\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})

[/tex]

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

[tex]

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

[/tex]

if I define

[tex]

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

[/tex]

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

[tex]

\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})

[/tex]

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

[tex]

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

[/tex]

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.

Last edited: