by Greg Egan
“I told her that they were like her, with the same kind of bodies. She said that was obvious now, but she didn’t ask anything more.”
Theo said, “Giving her the whole story just as you were leaving would have been worse.”
“I know,” Ada conceded. She was silent for a while, then she said, “If I’d stayed, I would have tried to find a way to tell her, and then hope she’d be able to forgive me. But I couldn’t do that and walk away.”
Seth could feel the northward current tugging on the boat. “Let’s hope this river’s long and fast,” he said. They’d have less than a day to get far enough back along the slope to escape the heat of the sun. “They’ll all be waiting for us, wondering where we’ve been: Raina and Amina, Sarah and Judith, Andrei and Nicholas.”
Darkness rushed forward across the glimmering surface of the water, then the boat descended into the chasm.
Afterword
A Note on the Translation
If you could listen to the speech sounds used by the characters in this novel, not only would you hear no words in your own native language, you would not hear any of the proper names employed in the story, such as “Seth,” “Theo,” “Baharabad,” or “Sedington.” The actual sounds that played the same roles as these words would not consist of anything a human would recognize as a string of phonemes, so it would be pointless to attempt to render them phonetically. Like every other word, they need to be translated.
Words such as “smile,” “laugh,” “groan,” and so on are used to indicate the nature of the emotions that elicit these acts, rather than any anatomical or phonetic similarities to human utterances and gestures.
While numbers as such are a universal concept, several kinds of measurement have been converted into familiar units, where it makes sense to do so. Angles are translated into degrees, not because the people of this world would have chosen to divide a circle into three hundred and sixty parts, but because “ninety degrees” immediately tells the reader what proportion of a circle is involved, which is all that matters. Similarly, periods of time less than a day are translated as seconds, minutes, and hours, because these words convey the appropriate proportions, regardless of the particular scheme of diurnal subdivisions actually employed.
The word “year” is not used at all, since the world of the novel neither orbits its sun nor experiences cyclic seasons. The names of our own, chronological seasons have been repurposed as names for different solar latitudes, which determine the angle at which the sun crosses the sky, and hence the average temperature—albeit in a contrary fashion to the way this works for us. The region where the sun passes directly overhead at noon is colder than any other, hence “the midwinter circle.” Odd as this sounds, it would be odder still to use “winter” for the zones of scorching heat to the north and south, where the sun never rises as high.
Colors have generally been translated on the basis of the kinds of common objects that bear them. I have written other novels with non-human characters where the most crucial thing a color conveyed was the wavelength of light observed in a star trail or a laboratory experiment, but in this case there was no need to tie the names of colors to any particular physical quantity. For most purposes, to a human the word “green” means “the color of certain kinds of foliage,” “red” means “the color of one’s own kind of blood,” and so on, and the translation here aims to preserve those kinds of associations wherever possible.
Similarly, “water” here is a ubiquitous liquid upon which life depends, and “air” is a substance (in this case, a liquid not a gas) that covers the world. They are not—and could not be—precisely the things that bear these names in our own world, since there are no atoms of hydrogen, oxygen, or nitrogen in the universe of the novel, but the translation reflects the roles they play. “Steam” is what “water” becomes when it is heated and turns invisible; the fact that it has dissolved and dispersed into the more abundant liquid, rather than evaporated into a gas, is a distinction worth keeping in mind, but not one that merits a neologism.
A Note on the Geometry
The events of this novel take place in a universe where the usual three dimensions of space and one of time have been replaced by two dimensions of space, and two of time. Instead of 3+1, we have 2+2.
What does it mean to have two dimensions of time? It does not mean that any one observer’s experience of time becomes two-dimensional. Seth has a linear stream of consciousness, with a sense of history in which events follow one after another, rather than being spread out over a two-dimensional plane. The presence of a second dimension of time offers more directions in space-time with the potential to be someone’s arrow of time—but each arrow still points in a single direction.
Perpendicular to an observer’s arrow of time are the three dimensions that they experience as “space”—and this is where the difference between 3+1 and 2+2 is most apparent. “Space” with 2+1 dimensions is like a pared-down version of the full 3+1 dimensions of our own universe, with the behavior of ordinary objects displaying the kinds of exotic effects that we would only see in relativistic physics.
For example, any distance that Seth measures obeys an altered version of Pythagoras’s Theorem, in which the square of the overall distance is found by summing the squares of two ordinary distances (such as east-west and up-down), then subtracting the square of the north-south distance, rather than adding it. If the result is positive, it is the square of an ordinary distance. If the result is negative, it is the opposite of the square of an “axial” distance—a distance whose square is always subtracted in this kind of calculation, just like a distance measured directly along the north-south axis.
A similar rule governs the notion of “spacelike” and “timelike” separations between events in our own, (3+1)-dimensional universe. Events that happen at different places at the same moment are said to have a spacelike separation, while events occurring at the same place at two different times have a timelike separation. You might wonder if relativity could blur this distinction by asking, “The same according to whom?” But assuming you can’t travel faster than light, your locations at noon and at one p.m. can never be my idea of two different places at the same moment in time, whatever our relative state of motion.
The corresponding prohibition in (2+1)-dimensional space is that an object that lies along an axial direction can never be rotated in such a way that it ends up aligned in an ordinary direction, or vice versa. If a stick starts out pointing due north, turning it to the east will cause the distance it spans in both the north-south and the east-west directions to increase, but the difference of the squares of the two numbers will remain constant. For example, the individual measurements might change from five meters north-south and zero meters east-west, to thirteen meters north-south and twelve meters east-west, because thirteen squared minus twelve squared is equal to five squared. If this sounds odd, compare it to the situation in our world, where a five-meter stick could end up spanning three meters north-south and four meters east-west, because three squared plus four squared equals five squared. That’s just Pythagoras’s Theorem, with the usual sum of squares rather than the difference. But in Seth’s world, the stick’s north-south span will always be greater than its east-west span, so it can never end up pointing due east.
What if the stick starts out pointing due east? East-west is just one of two ordinary dimensions, so this kind of stick can end up perpendicular to its original direction: say, pointing straight up. But any attempt to rotate it toward the north or the south must still follow the rule that the difference of squares is unchanged, so the stick can never end up pointing more to the north than in the other directions.
In our universe, if we fix one end of a rod to a pivot then swing it around freely, the other end traces out the surface of a sphere. But in the universe of the novel, the same kind of rod will trace out either a one-sheeted hyperboloid—an infinite saddle-shaped surface that wraps around the north-south axis—or an infinite bo
wl-shaped hyperboloid that faces north or south.
Between these hyperboloids sit a pair of cones, facing north and south: these are surfaces where the difference of squares of the distances from the pivot is exactly zero. Within these cones, the difference will always be negative.
It is the nature of light in our universe—and by analogy, what we choose to call light in the universe of the novel—that any portion of its world line through the full, four-dimensional space-time has a difference of squares that is precisely zero. If a beam of light were to follow a trajectory through (2+1)-dimensional space for which the difference of squares was negative, the final quantity pertaining to the world line would have yet another square (the square of the elapsed time) subtracted from it . . . which could never bring it up to zero. So light can never travel within the two cones, and no one can see (by light) in these directions.
However, there is nothing to prevent material objects, or vibrations within a material medium, from traveling either inside or outside these “dark cones.” The rule that applies to the world lines of material objects is that the difference of squares must be negative: this is obviously true for the case of an object standing still, when there is zero change in position, minus the square of the elapsed time. And for any trajectory within the dark cones, the overall difference of squares will again be negative.
Outside the dark cones, the trajectory through space starts out with a positive difference of squares. To bring this down to a negative value, the elapsed time for the object must be greater than the length of the trajectory itself. In other words, the speed of the object must be less than 1, in units where the speed of light is also 1. So in these ordinary directions, as in our own universe, nothing can travel faster than light—but within the dark cones, where light itself can’t travel, no speed limit applies.
More details can be found at www.gregegan.net.
