Relatively more or less dense "things" (to each other) will always generate positive curvature of spacetime. What you want is relatively less dense area than vacuum in another region of space, since vacuum is (for now) what we constitute as (somewhat) flat.
That's what Casimir effect tries to accomplish: define a small enough region of spacetime where some larger quantum fluctuations don't have enough space to happen, thus lowering its energy below vacuum.
Since the idea is to define small enough space-like interval in spacetime to curb some energies off, could defining small enough time-like interval help the same purpose, multiplying the former?