I am trying to derive the Einstein field equations from this approach:
Let $\psi(t) = e^{-tM}$, where $M$ is a $4\times4$ matrix.
Then
$$ \frac{d}{dt}\psi(t) = -M\psi(t) $$
Let us now suppose that I change $d/dt$ to a covariant derivative of the general linear group $g$.
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.
It seems to me that I now have a relationship between The Riemann tensor and the stress-energy tensor (as M is usually interpreted to be the Hamiltonian in QM and this is the total energy of the system).
$$ D_t \psi(t) = - M \psi(t) $$
From here, what steps lead me to the Einstein field equation? I feel like I am almost there, but I need help completing the last few steps.