# Energy-momentum tensor in Riemann-Cartan geometry

I am trying to derive the energy-momentum tensor of a Dirac field defined on a Riemann-Cartan background which is a space with a metric-compatible connection and non-zero torsion.

The action is

$$S = \int_M \mathrm{d}^{3+1}x |e| \frac{i}{2} ( \bar{\psi} \gamma^\mu D_\mu \psi - \overline{D_\mu \psi} \gamma^\mu \psi )$$

where $$D_\mu \psi= \partial_\mu \psi - \frac{1}{8} \omega_{\mu a b} [ \gamma^a, \gamma^b ] \psi$$. The energy-momentum tensor is defined as

$$T^a_\mu \propto \frac{1}{|e|} \frac{\delta S}{\delta e^\mu_a}$$

Essentially, I need to evaluate the variation of $$S$$ with respect to $$e$$. Now I am unsure how the spin connection $$\omega_{\mu a b}$$ varies under a variation of the tetrad $$e$$. I am aware of this answer here, but it is for a space with zero torsion. We can split the spin-connection up as

$$\omega_{\mu a b} = \tilde{\omega}_{\mu a b} + K_{\mu a b}$$

where $$\tilde{\omega}_{\mu a b}$$ the Levi-Civita connection and $$K_{\mu a b}$$ is the contortion tensor. $$\tilde{\omega}_{\mu a b}$$ depends on the dreibein and I know how this varies under $$e_a^\mu$$ but how does $$K_{\mu a b}$$ vary with respect to $$e^\mu_a$$?

First of all, your action $$S = \int_M \mathrm{d}^{3+1}x |e| \frac{i}{2} ( \bar{\psi} \gamma^\mu D_\mu \psi - \overline{D_\mu \psi} \gamma^\mu \psi )$$ is wrong. It should read $$S = \int_M \mathrm{d}^{3+1}x |e| \frac{i}{2} ( \bar{\psi} e^\mu_a\gamma^a D_\mu \psi - \overline{D_\mu \psi} e^\mu_a\gamma^a \psi ).$$
And as for deriving the energy-momentum tensor, you should variate against spin connection $$\omega_{\mu a b}$$ and the tetrad $$e$$, independently. Follow these steps:
• Use the variation against $$\omega_{\mu a b}$$ to yield the torsion tensor. Use this non-zero torsion tensor equations to calculate $$\omega_{\mu a b}$$ as a function of $$e$$.
• Use the variation against $$e$$ to arrive at the the energy-momentum tensor. And substitute the $$\omega_{\mu a b}$$ in the energy-momentum tensor equation with the corresponding function of $$e$$ obtained from the first step.