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Sunday, September 2, 2012
The Orthogonal Universe
by Greg Egan
What would it be like to live in a universe with four dimensions that were all essentially the same?
The universe we inhabit has three dimensions of space and one of time, and though relativity has taught us that there is no absolute notion of time that is shared by everyone, the whole variety of directions in space-time that different people might call “the future” is entirely separate from the set of directions that different people might call “north”.
What would be the outcome if that distinction were erased, and there were four dimensions that were all as much alike as “north” and “east”? Such a universe is the setting for a trilogy of novels that I’m writing, with the overall title of Orthogonal. The first volume, The Clockwork Rocket, was published in 2011; the second, The Eternal Flame, has just been released by Night Shade books in the US, and will be out from Gollancz in the UK in October.
Since time as such is absent from the Orthogonal universe, a first guess might be that it would resemble a snapshot of the world we see around us at a single moment, albeit a snapshot with four dimensions of space rather than three. Worse, it would be a snapshot with no backstory: no sequence of prior events to organize and enrich the subjects caught in the flash. It would consist of nothing but scattered, isolated objects with no history or duration.
But it turns out that this guess would be wrong. What we see around us at a single moment takes the form it does because of the existence of our own fourth dimension. It doesn’t make sense to throw out that entire dimension, and then replace it by simply mimicking what we originally saw in the remaining three.
So forget about snapshots. A more methodical approach would be to take the equations that govern the behavior of matter and energy in our universe, and make the smallest possible changes to them needed to put all four dimensions on an equal footing. With the advent of relativity, the equations of modern physics already come very close to treating time and space even-handedly, often with the only difference being a plus sign appearing before a quantity involving distances and a minus sign before a similar quantity involving times. If we rewrite the equations slightly, turning those minus signs into plus signs, we can start to make solid predictions about the nature of a universe where all the dimensions are fundamentally the same.
Perhaps the most important thing to emerge is that, despite the absence of a special time dimension, objects still end up with a kind of persistence. In our own universe, we talk about the “world line” of an object: the path it traces out in the four dimensions of space-time as we follow its position over time. But even when we get rid of time, the equations tells us that there will still be world lines!
To see why this is true, the easiest place to start is to think about the kind of waves that we can expect the Orthogonal universe to contain. In our universe, both light and matter have fundamental wavelike properties, and it’s the geometry of these waves that underlies the way objects move.
The simplest wave imaginable in the Orthogonal universe consists of an endless series of equally spaced wavefronts, like an idealized version of the waves you might get by shaking a long flat board in calm water. In the figure below we only see two dimensions, but that’s enough to show everything that matters, because as we move along the other two dimensions nothing changes.
We’ve called one dimension “distance” and the other “time”, but as far as the geometry is concerned there’s no difference: there’s nothing special about the time dimension. We’ve also drawn a white arrow that runs at right angles to all of the wave fronts, and gives us an indication of the direction of the wave.
We’ll call the distance between wavefronts along the “distance” axis the wavelength of the waves, and the distance between wavefronts along the “time” axis the period of the waves.
Now, let’s compare this to another example, where we’ve taken exactly the same kind of waves but changed their direction slightly with respect to our “distance” and “time” axes.
The separation between these waves — measured directly from wavefront to wavefront — is exactly the same as before. But because the wave’s direction compared to our axes is different, the wavelength (measured along the “distance” axis) has grown shorter, while the period (measured along the “time” axis) has grown longer. We’ve drawn the waves with the longer wavelength in red and the waves with the shorter wavelength in violet, but the waves as such are actually identical in the two diagrams; all that’s changed is their relationship to our chosen axes.
What happens if we combine several waves with slightly different directions? The result will look like this:
We see a series of points where the wavefronts intersect and reinforce each other. These points all lie along a line that runs at right angles to the wavefronts themselves — and this is how a world line appears! If these waves described something like light, the line where they reinforced each other would be the world line of a pulse of light. If they were the quantum mechanical waves describing some kind of matter, the line would be the world line of an elementary particle.
Just as in our own universe, we can imagine successive three-dimensional slices that intersect these world lines at slightly different points for each slice, giving a picture of objects moving about. So even without an official time dimension, we naturally end up with concepts of change and motion that are very similar to those in our own universe.
What’s more, all the simple geometry that we learned in school that applies to distances in space now works just as well across all four dimensions. Everyone knows that the shortest distance between two points is a straight line, and that any detour or zig-zag only adds to the distance. In a universe with four dimensions of space, what a living creature would experience as “the passage of time” must really be a form of distance, so the shortest time between two events must also involve a straight line. Taking a detour when traveling from event to event can only add to the time experienced by the traveler — in stark contrast to the situation in our universe, where relativistic time dilation means that less time passes for space travelers.
It’s that very simple twist that underlies the plot of my Orthogonal trilogy. An alien civilization in the universe I’ve described are facing a catastrophe that threatens to destroy their planet in a matter of years, and their current technology is nowhere near sophisticated enough to avert it. But by sending a group of travelers on a long interstellar journey, those travelers will have a chance to spend several generations trying to develop a solution, even though the “straight line distance” to the catastrophe is just a few years.
Once the spacecraft ends up with its world line at right angles to that of the home world (orthogonal to it, hence the title of the trilogy), no time at all will be passing back home during that stage of the journey, no matter how long it lasts for the travelers. What sets the minimum time for the whole journey, as measured on the home world, is the time it takes for the spacecraft to achieve that orthogonal state — and as with a racing car trying to take a bend, the turning radius depends on the force that is applied to make the trajectory change direction.
The reversal of the usual kind of time dilation is only the beginning of a long list of strange phenomena. The speed we measure for any object or particle depends on the angle its world line makes with our own, and since any world line is as good as another in the Orthogonal universe, there can be no special speeds, such as the speed of light is special for us. Light will be able to travel at any velocity whatsoever, but the geometry of the waves means that the speed will be tied to its wavelength, and hence its color.
Light of a shorter wavelength — bluer light, in our own vocabulary — will travel faster than longer-wavelength light. Just compare the first two wavefront diagrams: when the direction of the wavefronts in the second diagram makes a larger angle with the time direction, indicating a greater velocity, the wavelength becomes shorter.
So when the aliens in this universe look up at the sky, every star’s light will be spread out into a tiny spectrum, with the violet light showing the star’s most recent position, while the red light shows the star’s position centuries earlier — since the slower red light will have taken longer to arrive.
It’s possible to map out the way this altered geometry leaves its mark on everything, from the microscopic structure of matter to the shape of the universe. More detail can be found on my web site, but if you don’t want to spoil too many surprises it’s best to read the novels first.
I want to thank Greg Egan for this wonderfully comprehensive post and I will note that The Clockwork Rocket has been my top sf and sff title of 2011 for its combination of the sense of wonder you glimpsed above and wonderful characters you get to care about, however alien they are from some points of view.
12:00 AM | Posted by Liviu | | Edit Post