# Vary Metric w/ Respect to Veirbein: How To?

• Physicist97
In summary, you would take the variation of the Christoffel symbol with respect to the veirbein, ##{\delta}{\Gamma}^{\sigma}_{\mu\nu}/{\delta}e^{a}_{\tau}##.
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 :)

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.

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.

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.

## 1. What is the Vary Metric with Respect to Veirbein?

The Vary Metric with Respect to Veirbein is a mathematical concept used in the field of general relativity. It is a measure of how the metric (a mathematical object that describes the geometry of spacetime) changes as the veirbein (a set of mathematical objects used to describe the local frame of reference) changes.

## 2. Why is it important to understand the Vary Metric with Respect to Veirbein?

Understanding the Vary Metric with Respect to Veirbein is important in the study of general relativity because it allows us to describe how the metric changes in different reference frames. This is crucial in understanding the effects of gravity and how it warps space and time.

## 3. How is the Vary Metric with Respect to Veirbein calculated?

The Vary Metric with Respect to Veirbein is calculated using a mathematical formula that involves the derivative of the metric with respect to the veirbein. This can be a complex calculation, but it allows us to quantify how the metric changes as the veirbein changes.

## 4. What are the applications of the Vary Metric with Respect to Veirbein?

The Vary Metric with Respect to Veirbein has various applications in the field of general relativity. It is used in the study of black holes, gravitational waves, and the overall structure of the universe. It also has implications in other areas of physics, such as quantum gravity.

## 5. Are there any practical uses for understanding the Vary Metric with Respect to Veirbein?

While the Vary Metric with Respect to Veirbein is primarily used in theoretical physics, it has practical applications as well. It is used in the development of advanced technologies, such as GPS systems and satellite communications, which rely on our understanding of general relativity to function accurately.

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