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Does it emanate strictly from energy dense regions of space? What does that mean? Is it possible to, say, arrange clumps of matter in such a way as to create a virtual gravity well in space where there is no matter?

Say I somehow gather up a number of stars with total mass greater than the Chandrasekhar limit and set them into an orbit within a light year of each other such that they won't collide. I then wait a year at their geometric center. What would happen to me by the end of that year? Would some kind of virtual black hole suddenly appear out of nowhere?

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The Einstein equation says:

$$ {\bf G} = {\bf T} $$

where $\bf G$ is the Einstein tensor that describes the curvature, i.e. the gravity, while $\bf T$ is the stress-energy tensor. So the origin of gravity is the stress-energy tensor. This is typically dominated by mass, but includes less obvious contributions like pressure and momentum.

Actually solving the Einstein equation is impossible except in special cases, so we can't solve it for the collection of stars you describe. However numerical solutions have been calculated for systems like two black holes orbiting each other, and we'd expect several mutually orbiting black holes to be similar. Nothing weird happens with this arrangement of masses. The only classically unexpected phenomenon is that the system radiates gravitational waves so the black holes will gradually converge and eventually collide.

The nearest to what I think you're imagining would be a geon. This is a gravitational field that is sustained by its own energy so no mass is needed to create it. However no-one has (so far) managed to find a stable geon.

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