Vary Metric w/ Respect to Veirbein: How To?

[Physicist97]

Hello! Given a metric in terms of the Veirbein, $g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}{\eta}_{ab}$ , how would you go about varying it with respect to $e^{a}_{\mu}$ ? I know that ${\delta}g_{\mu\nu}={\delta}e^{a}_{\mu}e^{b}_{\nu}{\eta}_{ab}+e^{a}_{\mu}{\delta}e^{b}_{\nu}{\eta}_{ab}$ , with ${\delta}{\eta}_{ab}=0$ . Then I would divide both sides by ${\delta}e^{a}_{\mu}$ , but that leaves me with the term ${\frac{{\delta}e^{b}_{\nu}}{{\delta}e^{a}_{\mu}}}$ . What would I do with that? Thanks for any help :)

 

[fzero]

Typically you'd assume that the components of the vierbein are independent variables, so that $\delta e^b_\nu / \delta e^a_\mu = \delta^b_a \delta^\mu_\nu$.

 

[Physicist97]

Thank you! I have another question, assuming there is torsion, how would you go about taking the variation of the christoffel symbol with respect to the veirbein, ${\delta}{\Gamma}^{\sigma}_{\mu\nu}/{\delta}e^{a}_{\tau}$ ? If it was torsion-free I would just take the christoffel symbols in terms of the metric, and do a very long, algebraically tedious calculation. But I can't seem to find an equation for the christoffel symbols when there is torsion.

 

[fzero]

Are you working from a particular reference? In Einstein-Cartan theory, I believe that the metric and torsion tensor are taken to be independent variables. In this theory, the covariant derivative is

$$ \nabla_\mu V_\nu = \partial_\mu V_\nu - ({\Gamma^\rho}_{\mu\nu} + {K^\rho}_{\mu\nu} )V_\rho,~~~(*)$$

where ${\gamma^\rho}_{\mu\nu}$ is the usual Christoffel symbol expressed in terms of the metric and ${K^\rho}_{\mu\nu}$ is the contorsion tensor, related to the torsion tensor ${T^\rho}_{\mu\nu}$ by

$$ K_{\rho\mu\nu} = \frac{1}{2} ( T_{\rho\mu\nu} - T_{\mu\nu\rho} +T_{\nu\rho\mu} ).$$

In this way, ${\Gamma^\rho}_{\mu\nu}= {\Gamma^\rho}_{\nu\mu}$ and all of the torsion is contained in ${T^\rho}_{\mu\nu}$. Because of the added term in (*) we say that the connection is no longer compatible with the metric.

I believe that you can also write this theory in terms of a vierbein and a spin connection ${\omega_\mu}^{ab}+{\gamma_\mu}^{ab}$. Here ${\omega_\mu}^{ab}$ is the parti of the spin connection that can be related to the Christoffel symbol (as in the theory without torsion), while ${\gamma_\mu}^{ab}$ contains the torsion via something like

$$ {\gamma_\mu}^{ab} = K_{\rho\sigma \mu} ( e^{a\rho} e^{b\sigma} - e^{a\sigma} e^{b\rho}).$$

Presumably you can take $e^a_\mu, {\omega_\mu}^{ab}$, and ${\gamma_\mu}^{ab}$ as independent variables.

I referred to this review by Shapiro to gather the formulas together, but I don't think he addresses the variational principle directly in this formalism.

 

[Physicist97]

Thank you again. No I'm not working from any reference. I'm trying to self study GR from Sean Carroll's online notes, and am working with Veirbeins, Spin Connection, etc. because i find them interesting, and to get practice with these things. Your explanation cleared a lot of things up, I appreciate the help.

 

[naima]

You wrote twice veirbein instead of vierbein. The word comes from Germany where vier = 4 (ein, zwei, drei,vier). Tetrad comes Greece (tetra = 4)
Greetings from France.