The Einstein equations can be written as (here I am following the notation of [Wald's book][1] on General Relativity)

\begin{equation}
    \partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}
-   \partial_{\mu}\Gamma^{\alpha}_{\nu\alpha}
+   \Gamma^{\alpha}_{\mu\nu}\Gamma^{\beta}_{\alpha\beta}
-   \Gamma^{\alpha}_{\nu\beta}\Gamma^{\beta}_{\alpha\mu}
=   T_{\mu\nu} 
\end{equation}

We can expand out the Christoffel symbols in terms of the metric $g_{\mu\nu}$; then the Einstein equations become a set of second order partial differential equations for the metric. With a good choice of coordinates (such as harmonic coordinates, $\Gamma^{\alpha}_{\mu\nu}g^{\mu\nu}=0$), one can show that the Einstein equations can be thought of as a set of second order hyperbolic (wave) equations for each metric component (again see for example Wald's book).

My question is: can one think of the Einstein equations as a set of transport equations for the Christoffel symbols? Has there been any work that has looked at the initial value problem for the Einstein equations as I wrote them above, as a set of partial differential equations for the Christoffel symbols?


  [1]: https://www.google.com/books/edition/General_Relativity/9S-hzg6-moYC?hl=en&gbpv=0