What is Greg Egan saying in here?

"Multiply Connected Space

The ordinary space we inhabit is usually assumed to be simply connected. This means, roughly, that in empty space if you take a string that connects two points, A and B, and someone else does the same, then by moving the strings around – without detaching them from their fixed endpoints – and stretching or shortening them as necessary, you could make the two of them coincide along their entire lengths. In a certain sense, if we don’t care about the exact shapes of different paths, there is only one way to get from A to B.

Infinite Euclidean space is simply connected. So is the surface of a sphere. But the surface of a torus, or doughnut, is not: if you wrap one string around the torus the long way once, and wrap another one twice, then even if they start and end at the same points, you will never be able to change the number of times they wrap around the hole, so they can never be made identical.

A space that is connected (in the sense that there is always a path between any two points), but not simply connected, is known as a multiply connected space, because there are multiple, fundamentally different ways to get from A to B.

If we choose A and B to be the same point, then we can always get from A to A by doing nothing: following a “path” of zero length that goes nowhere but A. Then the difference between a simply connected space and a multiply connected space boils down to whether or not there are any other ways of getting from A to A: any loops through the space that can’t be contracted down to a single point.

In The Book of All Skies, the protagonist lives on a world that has formed in the presence of two structures known as the Hoops, which are disks that connect different “skies”: different expanses of otherwise ordinary space. If we ignore one of the Hoops and just consider the effects of a single one, the multiply connected space it gives access to comprises a certain number N of copies of ordinary space, numbered 1, 2, 3, ..., N, which have been spliced together as follows: if you pass through the Hoop in space i in one direction, you emerge from the Hoop in space i+1, and if you pass through the Hoop in space i in the other direction, you emerge from the Hoop in space i–1. Whenever this recipe suggests you would go higher than space N or lower than space 1, you simply wrap around to space 1 or space N respectively.

To give a simple illustration, in the image on the right there are just three copies of ordinary space, with the Hoop that joins them in black. The precise way they are connected is illustrated by the coloured curves, which together comprise a single loop. If you join up these curves by matching the colour at their endpoints, you can see that moving clockwise around the red/green curve at the bottom takes you to the green/purple curve in the middle, and from there to the purple/red curve at the top, which connects back to the start of the red/green curve at the bottom.

This loop can’t be shrunk down to a point, because if you tried to shrink it, it would have to cut across the edges of the Hoop, which are taken to be singularities that are not actually part of the space.

The purpose of this page is to discuss the way that Newtonian gravity can be adapted to a multiply connected space like this. Of course, General Relativity is a more accurate theory of gravity, and that theory is necessary to fully understand all the possibilities for spaces with exotic topologies. But if we treat the Hoops as we have described them as a fait accompli that splice together regions of (approximately) flat space, containing bodies of a similar size and composition to the planets in our own solar system, then Newtonian gravity is a reasonable approximation for how we would expect gravity to work under those circumstances.

Now, in much of what follows, we will actually be talking about the way that electrostatics – the study of electric fields due to motionless electric charges – would work in the same kind of flat, multiply connected space. The underlying mathematics is essentially the same: in ordinary space, the gravitational attraction between particles with mass, and the electrostatic attraction between oppositely charged particles, both follow an inverse square law, with the force of attraction inversely proportional to the square of the distance between the particles. But why bother switching to electrostatics as a proxy for gravity, if the mathematics is identical, and gravity is what we are really interested in?

There are two reasons to do this. The first is that some of the mathematical tricks we will use make the most intuitive sense if we describe them in terms of familiar electrical phenomena, like conductive metals and oppositely charged particles. The gravitational analogues of these things are either wildly exotic, or flat-out physically impossible, which would make it distracting to talk in those terms, even though the final results we end up with still apply to the gravitational field.

The other reason is that the theory of electrostatics in multiply connected spaces actually has real-world applications in ordinary space, and those applications motivated most of the mathematics that we’ll use to solve the problems that interest us. And even in the imaginary world of the novel, where the gravitational field in a multiply connected space is accessible to observation and experiment, some of the electrostatic analogues are still extremely useful, since they are accessible to simpler, smaller-scale experiments.

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