High School 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.