I am wondering if it is possible to describe general relativity such that its flat state is a Euclidean 4D metric:

$$ \eta^{\mu\nu}=\operatorname{diag}(+,+,+,+) $$

But with a "special" stress-energy tensor, called a non-zero vacuum state if you will such that it bends the metric into the Lorentz metric:

$$ \eta^{\mu\nu}=\operatorname{diag}(+,-,-,-) $$

What would be the structure of this stress-energy tensor?

Another way to ask my question is; what is the simplest stress-energy tensor which bends a $(+,+,+,+)$ flat metric into one with curvature $(+,-,-,-)$? (here we consider $(+,+,+,+)$ to be the flat state, and $(+,-,-,-)$ to be a "curved" state).

Suppose we use the signature of a Riemannian manifold

$$ \eta^{\mu\nu}=\operatorname{diag}(+,+,+,+) $$

as the starting point to describe a 4d Euclidean version of general relativity. Alternatively one may use a Wick rotation on time $t\to i \tau$ on normal general relativity to get this metric.

But instead of remaining in Euclidean GR, one then further flips the metric back to Lorentz $\eta^{\mu\nu}=\operatorname{diag}(+,-,-,-)$ but this time using the Stress-Energy tensor, instead of a wick rotation.

One will end up with the correct Lorentz signature $\eta^{\mu\nu}=\operatorname{diag}(+,-,-,-)$ but also with a non-zero stress-energy tensor.

My question is; what is the simplest stress-energy tensor which curves a $(+,+,+,+)$ flat metric into one with curvature $(+,-,-,-)$?