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?

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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).