I am new to GR and I am having trouble understanding how one goes back and forth between the metric $g_{\mu\nu}$ and the stress-energy tensor $T_{\mu\nu}$.
First, have a look at the following post. It provides an "easy" method for calculating $T_{\mu\nu}$ given a metric $g_{\mu\nu}$:
Stress-energy tensor explicitly in terms of the metric tensor
How does one reverse this formula? That is, given a stress-energy tensor $T_{\mu\nu}$ (or even more specifically, a distribution of mass-energy $T_{00}$), how do you compute $g_{\mu\nu}$? Of course it's technically possible to simply expand Einstein's equations by rewriting $R_{\mu\nu}$ in terms of $\Gamma^{\lambda}_{\mu\nu}$, which can then be expanded in terms of derivatives of $g_{\mu\nu}$, but the result would be a large, disgusting sum of gross PDEs. If it is possible to avoid that, I would love to know.
I understand that this is a bit unorthodox, since in most applications, it suffices to simply find $\Gamma^{\lambda}_{\mu\nu}$ from $R_{\mu\nu}$ in order to derive the equations of motion.
I want to know because, ultimately, I would like to create a Mathematica notebook in which the input would be an arbitrary geometry for the the mass-energy density and it would output the metric. It seems relatively straightforward but I cannot find anything online as a guide.