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## Main Question or Discussion Point

I have always read that a wormhole will quickly collapse in on itself due to its own gravity, forming a black hole, unless it is held open by some exotic matter that has a negative energy density.

But couldn't there exist a wormhole with zero spacetime curvature? It would therefore have no gravity and it would not collapse.

Such a wormhole would have the topology of an uncurled 3-D torus. The topology could be described as a 3-D cylinder. Inside this wormhole, moving a short distance to your left or right or up or down would bring you back to where you started. Forward and backward would take you to the mouths of the wormhole. It has zero intrinsic curvature because all of Euclid's laws of flat 3-D geometry would be valid. (The sum of the angles of any triangle would still be 180 degrees)

The 2-D analogy is two flat parallel sheets of cardboard glued to the ends of a cardboard tube standing perpenducular to them. The surface of this tube has no intrinsic curvature; a piece of paper wrapped around it will not crinkle; a triangle drawn on it will have a sum of angles of 180 degrees.

General relativity certainly allows spacetime geometries with zero curvature. And GR does not say anything at all about the topology of spacetime. Could such a 0-curvature wormhole exist? Would it be stable, since it would have no gravity? Would it therefore be transversable?

But couldn't there exist a wormhole with zero spacetime curvature? It would therefore have no gravity and it would not collapse.

Such a wormhole would have the topology of an uncurled 3-D torus. The topology could be described as a 3-D cylinder. Inside this wormhole, moving a short distance to your left or right or up or down would bring you back to where you started. Forward and backward would take you to the mouths of the wormhole. It has zero intrinsic curvature because all of Euclid's laws of flat 3-D geometry would be valid. (The sum of the angles of any triangle would still be 180 degrees)

The 2-D analogy is two flat parallel sheets of cardboard glued to the ends of a cardboard tube standing perpenducular to them. The surface of this tube has no intrinsic curvature; a piece of paper wrapped around it will not crinkle; a triangle drawn on it will have a sum of angles of 180 degrees.

General relativity certainly allows spacetime geometries with zero curvature. And GR does not say anything at all about the topology of spacetime. Could such a 0-curvature wormhole exist? Would it be stable, since it would have no gravity? Would it therefore be transversable?