Wormholes

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity. They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other. The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

What is a wormhole? The geometry of space-time and how to twist it into a pretzel

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway? Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on where and when they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

A toy model

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.

But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.

Add time!!!!

Manifolds and coordinate patches

Definition of a manifold

1D: lines and loops

2D: planes, spheres, hyperbolic surfaces, topological & curved toruses (& note on topology, multiply connected manifolds)

3D: spaces

3+1D: space-time!

Common kinds of wormholes

Early work, Einstein-Rosen bridges, and their issues.

Spherically symmetric wormholes

Morris-Thorne wormholes

Thin shell wormholes

Including polyhedral wormholes.

Practical implications

Can I put it on a planet? (Mass, tidal forces, radiant flash from collapse accidents, "gravity" limit only consistent as tidal forces)

Spacecraft radiators -> wormhole spacecraft.

Credit

Author: Luke Campbell