# Wormhole with zero curvature

kochanskij
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?

## Answers and Replies

For starters, a flat space-time is not the same thing as a flat submanifold of codimension 1 embedded in space-time in the extrinsic sense i.e. just because a space-time is flat doesn't mean codimension 1 submanifolds embedded in said space-time must have vanishing extrinsic curvature. Moreover, a flat space-time is necessarily locally isometric to ##\mathbb{R}^{4}## so this certainly imposes constraints on the choices.

Mentor
Such a wormhole would have the topology of an uncurled 3-D torus.

If the spacetime curvature is zero, this doesn't describe a wormhole. It describes an entire universe with 3-D torus spatial topology. Here's how you can tell: if the spacetime curvature is zero everywhere, what distinguishes the wormhole "mouths" from any other point?

kochanskij
The zero-curvature wormhole would indeed be an entire separate tiny universe with 3-D torus topology if one direction of the torus were not uncurled and attached to two flat 3-D spaces. The wormhole topology is really that of a 3-D cylinder.

All of Euclid's laws of geometry would apply everywhere, so all this space is Euclidean (0-curvature). The distinguishing property of the "mouth" is the size of the dimensions perpendicular to the axis of the wormhole. Inside the wormhole, the circumference of "space" in the left-right direction and in the up-down direction would be very small. You would see what looked like copies of yourself very nearby since you would be looking around the whole circumference of this "space". As you exit the mouth, the size of these two directions would jump to infinity. The apparent copies of yourself would quickly seem to get very far away until they were infinitely far. You would then be out of the wormhole and in another flat infinte 3-D space.

Mentor
The zero-curvature wormhole would indeed be an entire separate tiny universe with 3-D torus topology if one direction of the torus were not uncurled and attached to two flat 3-D spaces.

How can you tell where the "attachment" points are if the curvature is zero everywhere? See further comments below.

All of Euclid's laws of geometry would apply everywhere, so all this space is Euclidean (0-curvature).

I'm a bit unclear on what you're suggesting: are you suggesting that the spacetime is completely flat (i.e., Minkowski geometry everywhere)? Or are you just suggesting that it's *spatially* flat (i.e., Euclidean spatial slices, but possibly curvature in the time dimension)?

The distinguishing property of the "mouth" is the size of the dimensions perpendicular to the axis of the wormhole. Inside the wormhole, the circumference of "space" in the left-right direction and in the up-down direction would be very small. You would see what looked like copies of yourself very nearby since you would be looking around the whole circumference of this "space". As you exit the mouth, the size of these two directions would jump to infinity.

I don't think this is possible with zero curvature: for the "circumference of space" in any direction to change, there must be curvature.

The_Duck
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.

I think the circular "creases" along which you glue the tube to the flat sheets are regions of infinite curvature. So your space is flat at most points, but at the cost of squeezing all the curvature into the some singularities at either end of the tube.

For example, I can parallel transport a vector around a closed path that crosses the crease twice such that when the vector returns to its origin position, it has rotated. Therefore the space must have some curvature somewhere. Since everywhere else is flat, the curvature is in the creases.

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kochanskij
I was suggesting that all spacetime inside, outside, and around the wormhole was flat Minkowski geometry. Neither space nor time would have any curvature. Only the topology and circumference of spacetime would be different inside the wormhole.

I believe that you (PeterDonis) might be right with your comment that in order for the size of space or its topology to change, there must be curvature. This would mean that there would be curvature, and gravity, at the mouths of the wormhole.
Can anyone agree or disagree with this? Can anyone prove or disprove it? I'm going to give this point further study. Thank you for your insightful comment, although you just collapsed my wormhole!