What is the physical meaning of non-commuting tetrads?

Question

I'm reading about the tetrad formalism in GR and one main difference between the coordinate and the tetrad frame is that coordinate derivatives commute $\partial_\mu \partial_\nu = \partial_\nu \partial_\mu$, while tetrad derivatives do not $\partial_m \partial_n \neq \partial_n \partial_m$.

Considering that partial derivatives are variations along a direction and tetrads are simply an orthonormal frame, what is the physical interpretation of the non-commutativity of tetrad directional derivatives?

Answer 1

I'm assuming by "tetrad derivative" you mean $\partial_m \phi \equiv e^\lambda_m \partial_\lambda \phi$, where $\partial_\lambda \phi$ is the partial derivative of a scalar field $\phi$ and $e^\lambda_m$ is the (inverse) tetrad.

Then the tetrad derivative is not a partial derivative. It is the partial derivative mapped to a locally inertial frame. Since this mapping generally depends on the manifold's coordinates, the commutator of tetrad derivatives will involve derivatives of the tetrad.

So for example, \begin{equation} \partial_m \partial_n \phi - \partial_n \partial_m \phi = \left[\partial_m e^\alpha_n - \partial_n e^\alpha_m \right] \partial _\alpha \phi \end{equation}