It seems that Einstein's field equation in a vacuum depends on Newton, to actually be solved.
where we assume in the weak gravity limit, it reduces to Poisson's equation.
Now, I suspect that the addition of the Stress-Energy Tensor fixes that, because it adds a source to the field, but my knowledge is limited here. Is that the case? I never thought about using the Stress-Energy Tensor because I'm working outside a massive body, so in a vacuum. But, is the addition of it (Stress-Energy Tensor) completing the equation, so it doesn't depend on any Newtonian limit?
Edit: The question was somewhat unclear. What I meant is:
Are EFE enough in order to derive the metric? Because the derivation of schwarzschild metric depends on it reducing to Newtonian limit, and this might be due to the equation not having an energy tensor. So if the energy tensor is added, EFE can derive schwarschild metric without any Newtonian limit taken into account.