Teleparallelism

Remarks

The Weitzenböck connection is compatible with $g$ and has vanishing curvature, since the frame $e_a$ is parallel for it: parallel transport does not depend on the path. Consider the torsion scalar $$T = \frac14 T^{\rho\mu\nu}T_{\rho\mu\nu} + \frac12 T^{\rho\mu\nu}T_{\nu\mu\rho} - T^\rho{}_{\mu\rho}\,T^{\nu\mu}{}_\nu.$$ The teleparallel equivalent of general relativity takes $T \det(e^a{}_\mu)$ as Lagrangian. The scalar curvature of the Levi-Civita connection of $g$ equals $-T$ plus a total divergence, so the field equations are those of general relativity. Example: the constant tetrad $e^a{}_\mu = \delta^a_\mu$ on $\R^4$ gives $g = \eta$, $\Gamma = 0$ and $T^\lambda{}_{\mu\nu} = 0$. Non-example: the Levi-Civita connection of a curved metric, such as the Schwarzschild metric, is torsion-free but has nonzero curvature, so it is not a teleparallel connection. Sign conventions for $T^\lambda{}_{\mu\nu}$ and for the torsion scalar vary between authors.