Orthonormal frame in spacetime

An orthonormal frame $\{e_a\}$ satisfies $g(e_a,e_b)={\eta }_{ab}$. Any two such frames differ by a local Lorentz transformation.

The dual coframe obeys ${\theta }^a(e_b)={\delta }_b^a$ and reconstructs the metric as $g={\eta }_{ab}{\theta }^a\otimes {\theta }^b$.

Connection one-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. For the Levi-Civita connection they obey Cartan's torsion-free equation $d{\theta }^a+{\omega }^a{}_b\wedge {\theta }^b=0$ and metric compatibility ${\omega }_{ab}=-{\omega }_{ba}$.

The curvature two-forms are