Why does a Wormhole not look like an hourglass?

[Aayush Thakur]

TL;DR

In science fiction, we often see wormholes shaped like an hourglass. But what I have learned so far about exotic matter, negative mass density, wormholes and their stability conditions. I came to a conclusion that a wormhole should not look like an hourglass.

A region of negative energy density hints a little towards negative mass. So the throat of wormholes have very powerful attractive gravitational forces. So in order to stabilize a wormhole we think of using a region of negative energy to create a strong repulsive force that can keep the throat open. If we look at a wormhole from outside then it should look like a giant sphere, no hourglass shape. But if we visualise its geometry, like how it would curve the fabric and connect to different points, then its look something like a "fat or bloated cylinder". Something like ( ) not ) ( .

The hourglass shape leans more towards the throat being pinched due to attractive forces but since we have used negative energy region, there are repulsive forces that cause the throat to widen. This is just what I thought of.

Please correct me wherever I am wrong.

 

[Ibix]

Gravity isn't a force in general relativity, so it's a mistake to try to think in terms of balancing forces.

There are wormhole solutions to Einstein's equations. A simple eternal wormhole is the Ellis wormhole, which connects two more or less flat regions of spacetime. Note that they aren't exactly flat, that the spacetime is everywhere filled by exotic matter, and (IIRC) there is no gravitational effect from the wormhole - you can float near it without needing to orbit.

The picture on the wiki page is an equatorial cross-section. The wormhole throat is a circle here (and would be spherical in 3d). I'm not sure what you are attempting to describe - what you would see directly if you were looking would be a spherically symmetric shape, but the equatorial cross section might be described as a (funny shaped) hourglass and this is a reasonable embedding of it.

 

[Aayush Thakur]

Gravity isn't a force in general relativity, so it's a mistake to try to think in terms of balancing forces.

There are wormhole solutions to Einstein's equations. A simple eternal wormhole is the Ellis wormhole, which connects two more or less flat regions of spacetime. Note that they aren't exactly flat, that the spacetime is everywhere filled by exotic matter, and (IIRC) there is no gravitational effect from the wormhole - you can float near it without needing to orbit.

The picture on the wiki page is an equatorial cross-section. The wormhole throat is a circle here (and would be spherical in 3d). I'm not sure what you are attempting to describe - what you would see directly if you were looking would be a spherically symmetric shape, but the equatorial cross section might be described as a (funny shaped) hourglass and this is a reasonable embedding of it.

Thank you, Ibix. I think this was mainly a misunderstanding on my part due to informal language and visual intuition. I now see that thinking in terms of balancing forces isn’t appropriate in GR, and that the correct description involves geodesic focusing and defocusing.
My confusion was really about the visualization rather than the physics. I understand now that the throat is a minimal spherical surface in a spatial slice, and that any “hourglass” picture is just a coordinate-dependent embedding, not something one would physically observe.

 

[cianfa72]

I understand now that the throat is a minimal spherical surface in a spatial slice, and that any “hourglass” picture is just a coordinate-dependent embedding, not something one would physically observe.

So, do you mean that the whormhole throat would be visibile/look as a 2D spherical surface in a spacelike slice of the underlying spacetime solution of EFE ?

 

[PeterDonis]

what you would see directly if you were looking would be a spherically symmetric shape

Sort of. A spacelike slice of a wormhole is a very strange thing; it is not like the normal space you're used to.

In the normal space you're used to, you can take 3-dimensional space and consider it as a nested series of 2-spheres, with areas going from infinity down to zero.

But the wormhole space is a series of 2-spheres, where the areas go from infinity down to some finite positive value (the area of the throat 2-sphere), and then go back up to infinity on the other side.

The "hourglass" shape is simply what I just described in the previous sentence, with one spatial dimension taken out, so that the series of 2-spheres becomes a series of circles. Note that this means that the surface of the hourglass is the only part that is actually in the model--the interior of the hourglass is not part of the model and is not something you would experience. It's just an artifact of trying to visualize the space as embedded in ordinary 3-dimensional Euclidean space.

There is no easily intuitive way of visualizing what it would be like to be embedded in such a space.

 

[Ibix]

Sort of. A spacelike slice of a wormhole is a very strange thing; it is not like the normal space you're used to.

In the normal space you're used to, you can take 3-dimensional space and consider it as a nested series of 2-spheres, with areas going from infinity down to zero.

But the wormhole space is a series of 2-spheres, where the areas go from infinity down to some finite positive value (the area of the throat 2-sphere), and then go back up to infinity on the other side.

Yes. So if you looked at it (literally) from inside the spacetime, some null curves from/to your eye would cross the minimum radius surface and some wouldn't. Those that do cross reach infinity on the other side of the throat and the rest will reach it on this side. The bundle of rays that do cross the throat will be axially symmetric, and you will see a circular patch on the sky (probably surrounded by some gravitational lensing-type effects) that obscures your side's infinity with a view of the other infinity.

That's distinct from what the spacetime "looks like". I agree an hourglass isn't a daft description of the spatial slices.

 

[Aayush Thakur]

So, do you mean that the whormhole throat would be visibile/look as a 2D spherical surface in a spacelike slice of the underlying spacetime solution of EFE ?

Hey cianfa72, appreciate your question. Thank you PeterDonis for such a great explanation. Hope you got your answer with that cianfa72.

BTW, i am just a high school sophomore trying to learn about wormholes (theoretically only), i don't think i would have been able to give a detailed explanation like Peter's.

 

[Aayush Thakur]

Sort of. A spacelike slice of a wormhole is a very strange thing; it is not like the normal space you're used to.

In the normal space you're used to, you can take 3-dimensional space and consider it as a nested series of 2-spheres, with areas going from infinity down to zero.

But the wormhole space is a series of 2-spheres, where the areas go from infinity down to some finite positive value (the area of the throat 2-sphere), and then go back up to infinity on the other side.

The "hourglass" shape is simply what I just described in the previous sentence, with one spatial dimension taken out, so that the series of 2-spheres becomes a series of circles. Note that this means that the surface of the hourglass is the only part that is actually in the model--the interior of the hourglass is not part of the model and is not something you would experience. It's just an artifact of trying to visualize the space as embedded in ordinary 3-dimensional Euclidean space.

There is no easily intuitive way of visualizing what it would be like to be embedded in such a space.

Thank you PeterDonis. Just one tiny clarification i would like to confirm with you is when you said "one spatial dimension taken out", i think it could've been more precise by saying "spatial (angular) dimension". Though, the meaning is beautifully conveyed. But someone new to GR (like me), "spatial" alone could be misread as radial dimension.

 

[PeterDonis]

i think it could've been more precise by saying "spatial (angular) dimension".

Yes, that's correct; the hourglass visualization represents 2-spheres by circles, so it's taking out one angular (or rotational) dimension.

In more technical language, the topology of a spacelike slice of the wormhole spacetime is $S^2 \times R$ (whereas the topology of ordinary Euclidean 3-space is $R^3$). The "hourglass" is a representation of the topological space $S^1 \times R$.

 

[cianfa72]

Yes. So if you looked at it (literally) from inside the spacetime, some null curves from/to your eye would cross the minimum radius surface and some wouldn't. Those that do cross reach infinity on the other side of the throat and the rest will reach it on this side. The bundle of rays that do cross the throat will be axially symmetric, and you will see a circular patch on the sky (probably surrounded by some gravitational lensing-type effects) that obscures your side's infinity with a view of the other infinity.

I think you're actually assuming a stationary spacetime where the spacelike slices adapted to the time symmetry are the same (whormholes with the same geometry/topology) and you're literally looking at it "from the within" of the spacetime.

 

[Aayush Thakur]

I think you're actually assuming a stationary spacetime where the spacelike slices adapted to the time symmetry are the same (whormholes with the same geometry/topology) and you're literally looking at it "from the within" of the spacetime.

Yes you're right cianfa72. Spacelike slice taken here for the wormhole is symmetrical throughout its asymptotic geometry. They are assuming a stationary spacelike slice (like a screenshot in 3D world).

 

[cianfa72]

In more technical language, the topology of a spacelike slice of the wormhole spacetime is $S^2 \times R$ (whereas the topology of ordinary Euclidean 3-space is $R^3$). The "hourglass" is a representation of the topological space $S^1 \times R$.

By Whitney embedding theorem, n=6 as Euclidean space is enough to embed $S^2 \times R$ in. I'm not sure whether there is an embedding of it into $\mathbb E^4$ (note that such embedding is only a matter of topological/differential structures and in general it is not about the metric).

 

[cianfa72]

I was trying to visualize $\mathbb S^2 \times R$ as a "block" extending across all 3D euclidean space with a spherical hole inside it. You can take a series of nested spherical shells up to the boundary established by the "spherical hole" within it. Unfortunately, this model fails, since once you reach the smallest sphere, you can't go back to infinity on the other side (you're forced to retrace the sequence of nested spheres you have already encountered/passed through).