Stress-energy tensor given Einstein tensor Suppose that you want to solve the Einstein Field Equations $$R_{\mu\nu}-\frac12Rg_{\mu\nu}=\frac{8 \pi G}{c^4}T_{\mu\nu}$$ but already know all the components of the metric $g_{\mu\nu}$. If thats the case, this implies that we know the Ricci tensor $R_{\mu\nu}=g^{\rho\sigma}R_{\rho\mu\sigma\nu}$, which further implies that we know the scalar curvature $R=g^{\mu\nu}R_{\mu\nu}$ which would mean that we can find the Einstein tensor $G_{\mu\nu} \implies G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}$. Since we know the Einstein tensor does that this imply we can solve for the Stress-Energy tensor by $$R_{\mu\nu}-\frac12Rg_{\mu\nu}=\frac{8 \pi G}{c^4}T_{\mu\nu} \implies T_{\mu\nu}=\frac{c^4}{8 \pi G}({R_{\mu\nu}-\frac12Rg_{\mu\nu}})?$$ Is this even feasible?
 A: Yes, you can obtain the stress-energy tensor from the Einstein tensor. If you do not specify the stress tensor beforehand, any Lorentzian manifold will provide a solution to the Einstein equations. However, not necessarily it will be a physically interesting solution.
As a historical example, we can consider the Alcubierre Warp Drive, which has a geometry constructed to provide a way of allowing a spaceship to travel hyperfast within General Relativity (in a super short form, the spaceship is surrounded by a bubble of spacetime which moves faster than light, but the spaceship is not locally moving faster than light since it is at rest with respect to the bubble, so nothing is violated). The original paper by Alcubierre proposes a geometry and only then computes the stress tensor, which turns out to involve negative energy densities (and we currently do not know of any sort of classical matter with these sorts of properties).
The presence of negative energy densities, or matter going on acausal motion (moving faster than light), and so on is usually undesirable on General Relativity, so it is common to impose energy conditions to forbid these sorts of issues. Computing the stress tensor from the Einstein tensor might lead to tensors that violate these energy conditions, meaning that often these stress tensors might not be realizable by any sort of matter we currently know of.
In summary, yes, you can obtain the stress tensor given the Einstein tensor. However, this does not mean the stress tensor you obtain will be physically interesting or physically realizable.
