by Greg Egan
If you looked through a wormhole, what would you see?
There are many possible answers to that question, ranging from “Whatever’s on the other side,” to “Nobody has even proved that wormholes exist!” So let me narrow it down a bit. In my story “The Slipway,” [in the July/August issue] astronomers notice a small circle in the sky that appears to contain hundreds of stars that weren’t there before. One potential explanation for this baffling phenomenon is that they’re looking at a newly created wormhole, and are peering through it into another region of space. To test this idea, they will need to ask what they might expect to see, if they really are looking at an example of the kind of wormhole that has been studied by physicists for decades. In real life, no one has ever seen a wormhole, but as a theoretical possibility they have a long history.
In the 1980s, Kip Thorne famously developed a model for a traversable wormhole, prompted by Carl Sagan, who needed some means for the protagonist of his novel Contact to take an interstellar journey. In fact, the very same model had already been explored in 1973 by two other physicists (working independently of each other), Homer Ellis and K. A. Bronnikov. This kind of wormhole has two spherical mouths, arbitrarily far from each other in space, that are connected by an internal bridge whose length is unrelated to the ordinary distance between the mouths. To keep the wormhole from pinching closed, the particular kind of curved spacetime it contains needs to be maintained by the presence of an exotic form of matter, with a negative energy density.
We will sidestep the issue of whether matter like that could ever really be found, or manufactured, even by an advanced civilization, and concentrate on the question: what would such a thing look like? If we assume the material component is either transparent or has been concentrated in a way that keeps it from blocking the view (perhaps restricted to a slender framework of girders, like the struts in a geodesic dome), then of course we wouldn’t expect to see the wormhole itself. We would see through it, to the other side. But although “Whatever’s on the other side” remains a valid answer, we can sharpen the question further. Suppose the wormhole leads to another location in our own Universe, sufficiently close that the background of distant galaxies is more or less the same as it is for us. Given that we can observe galaxies billions of light-years away, this is not a huge restriction: even if the wormhole led to the Andromeda Galaxy, two and a half million light-years from the Milky Way, then if we were able to look past the local stars into deep space, the backdrop would be essentially the same from either vantage.
But we want to know how this backdrop will appear when we compare our ordinary view of it to the view we see through the wormhole. Will it be like staring at a spherical TV screen, showing a broadcast from Andromeda? Like gazing into the crystal ball from The Wizard of Oz?
The wormholes studied by Ellis, Bronnikov, and Thorne are spherically symmetrical. This means that a light ray that plunges straight toward one mouth of the wormhole, aimed at the center of the apparent sphere, must emerge from the other mouth in the same manner, only it will be traveling away from the sphere, not toward it. This sounds a bit like the way light would reflect off a perfect mirror-ball—an actual polished sphere, not the disco kind with lots of small, flat faces—except for the fact that the “reflected” light does not come back to us, but emerges, as if reflected, at the far end of the wormhole.
We can draw a picture of these light rays, showing how they would enter one mouth of the wormhole and leave the other.
Here we are supposing that we simply have two regions of perfectly flat space that have been cut and joined to each other, so as soon as you cross into one sphere, you immediately emerge from the other. In fact, the solutions to the equations of General Relativity discovered by Ellis and the others would make things a little more complex than this, predicting a curved “throat” inside the wormhole and a curved region in the space around each mouth. But we will gloss over those details, and just treat the wormhole as a kind of splice.
Light that is initially converging toward one mouth of the wormhole ends up diverging from the other mouth. This effect, where rays that are coming together are spread apart, is also produced by concave lenses, the kind that are in glasses that correct for short-sightedness. So we would expect looking through a spherical wormhole to be a bit like looking through a thick, concave lens.
So far, so good, but before we can determine what the cosmic backdrop will look like through a wormhole, there is a missing ingredient we need to address. We have drawn a whole lot of light rays entering one mouth of the wormhole, then emerging from the other mouth. But which ray, exactly, matches up with which?
Should the rays entering at each position on the first sphere emerge from the corresponding position at the other mouth? Since we’re assuming that the wormhole mouths are both in the same universe, and indeed are close enough that any local observers could orient themselves by a shared set of beacons chosen from the distant galaxies, this proposal isn’t meaningless. But if we made this choice, how would things turn out?
In the animation above, the incoming and outgoing rays have been color-coded so we can follow them through the wormhole. (The hues here are for identification purposes only, and are not meant to imply any kind of gravitational red shift or blue shift.) As well as the collection of light rays, we have shown two identical objects passing through the wormhole: two cutouts of the letter “R” with their front side painted black and the opposite side painted red. One of them is spinning, to make the different colors of the two sides visible, and to show that, when they start out, the black side reads like a normal “R,” while the red side is reversed.
You can see that if we join up the two mouths of the wormhole this way, the objects we send through it will emerge as mirror images of themselves! Our cutout letters, after passing through the wormhole, have reversed black sides, while it is their red sides that read like the normal letter.
This might just sound like an amusing novelty: travel to Andromeda, and your left hand becomes your right. But there are strong indications from particle physics that if we could produce a geometrical transformation like this, it would also convert matter entering the wormhole into anti-matter when it emerged. That would not be so much fun. We don’t really know how wormholes could form, but if they were constructed by technological means this would amount to a major design flaw, and if they were created by some natural process, it would certainly lead to spectacular cosmic fireworks, as all the interstellar hydrogen drifting through was converted to anti-hydrogen.
So, is this alarming consequence inevitable?
Luckily, the answer is no. Our naive idea of simply matching up the corresponding points on the wormhole mouths has led to a situation where their “mirror-like” behavior has led to an actual mirror-inversion of every traveler. But as anyone who has played around with mirrors knows, the easiest way to turn a mirror image back into the original is with another mirror.
Of course we don’t want to throw a second wormhole into the mix. Rather, we want to match up the points on the two spheres, geometrically, as if the correspondence were produced by a reflection in some mirror. Then the reflection associated with the way the two mouths are matched up to each other, and the reflection associated with the way light “bounces off” one mouth and emerges from the other, will cancel each other out.
A reflection in which mirror, though? It would be nice to make a choice here that wasn’t completely arbitrary. One possibility would be to pick a plane for the mirror that lies midway between the two wormhole mouths. The result would then be something like this:
If you examine the two patterns of matching colors in this second animation, you’ll see that they are indeed the mirror image of each other, with the mirror in question the gray line that bisects the picture. And this abstract “reflection” really does cancel out the effect on objects passing through the wormhole. Our cutout letters emerge with the black side reading normally.
One other choice we could make is to identify every point on one wormhole mouth with the diametrically opposite point on the other mouth: its antipode. This is not a pure reflection: rather, it arises as a combination of a reflection and a rotation, but it is, in a sense, the least arbitrary choice of all, since it does not require us even to choose a particular mirror plane. What would this look like?
Here, if you concentrate on the non-spinning “R” that enters the first wormhole mouth from the left, you’ll see it emerge from the other wormhole mouth at the diametrically opposite point, with its red side showing now instead of its black side. But since this swapping of the sides is accompanied by a mirror reversal of the shape, the two effects cancel out, and as a three-dimensional object, the cutout that emerges is geometrically identical to the one that went in.
Now we are finally in a position to match up the light rays across a whole sphere, and see what the end result would look like.
The animation above shows the view as we circumnavigate a wormhole that is matched up with the other mouth via a reflection in the plane midway between the two. (Instead of a backdrop of galaxies, we have drawn a backdrop of text, which is easier to compare between the views.) When we are looking in the direction of the distant wormhole mouth (the “m” in “my vow”), we see light that enters its far side and emerges on the near side of our local wormhole mouth, with its direction unchanged. If we move 180 degrees around the wormhole and look back the other way (the “q” in “quartz”), for similar reasons the light will also reach us with its direction unaltered. But if we look in any direction parallel to the plane of the imaginary mirror that we used to identify the wormholes (such as the “j” in “judge”), the light that reaches us through the wormhole will come from the opposite point in the sky to the direction we are looking.
If, instead, the two wormhole mouths are identified by matching every point to its antipode, the light from the distant galaxies that comes to us through the wormhole (at least at the very center of the view) will show us exactly the same galaxy that we’d see if we were looking in the same direction and the wormhole was absent. However, the view as a whole is rotated around the center by 180 degrees.
So, the upshot is that we can’t predict exactly how the cosmic backdrop would appear through a wormhole, without knowing precisely how the two mouths of the wormhole match up with each other. But we can predict a range of possible views that correspond to the different ways the wormhole could be joined up, given the restriction that it doesn’t turn everyone who steps through it into antimatter.
As for the particular wormhole in “The Slipway,” you’ll need to read the story to discover just what view it gives rise to. But I hope you’ve enjoyed this quick tour of the geometry that governs what we might, or might not, see, if a circle in the sky ever starts showing us stars that were not there the night before.