Einstein field equations from covariant derivative of a general linear gauge transformation

Question

A general linear transformation is given by

\begin{align} \psi'(x) \to g \psi(x) g^{-1}, \end{align}

The gauge-covariant derivative associated with this transformation is

\begin{align} D_\mu \psi=\partial_\mu \psi -[iqA_\mu, \psi]. \end{align}

Finally, the field is given as

\begin{align} R_{\mu\nu}= [D_\mu,D_\nu], \end{align}

where, $R_{\mu\nu}$ is the Riemann tensor.

From here, what steps lead me to the Einstein field equation?

Answer 1

None. Nothing of this has anything to do with gravity. Your equation for "R" defines the gauge field curvature $F_{\mu\nu}$, not the Riemann curvature $R_{\alpha \beta\mu\nu}$.