by Greg Egan
Fatima took Nino with her everywhere, introducing him to friends, fellow students and acquaintances without a trace of self-consciousness, as if he were a long-lost uncle who’d just arrived in their company by some mysterious alternative route. At first Yalda took this as some kind of unspoken reproach for her own reticence at the task, but then she realized that it was nothing of the kind. People put up with a very different attitude from Fatima, as Nino’s advocate, than they would have from the woman they blamed for the fact that he was still alive at all. Fatima was utterly partisan on her friend’s behalf, but there was no reason for anyone to think of her as self-serving.
Every day, Yalda tagged along as Fatima showed Nino the food halls, the workshops, the classrooms. He was getting reacquainted with places he hadn’t seen since before the launch, and roaming far enough from the axis to grow familiar with the changing centrifugal force. Some of the people they encountered were brusque, but no one started screaming threats or accusations. And even those who had no particular respect for Yalda, or Fatima, or for Frido’s oath of protection, might have been given pause by the realization that Yalda’s choice of co-stead was the bluntest possible assertion of a woman’s right to decide when, and with whom, she had children. With holin scarce, with pharmacology failing them, any purely cultural force in favor of autonomy was all the more precious.
Isidora and Sabino took turns teaching Yalda’s old class. Yalda sat in and listened, watching Nino struggling to extract some sense from all the arcane technicalities as Fatima whispered explanations to him. This was his world now, not the wheat fields, and whatever role he played in it he’d have to learn some of its language and customs.
Yalda made a bed for him in her apartment, and he accepted that intimacy without complaint or presumption. The first night he was with her she could barely sleep; she did not expect him to wake her and demand what she had offered him, but his presence made it impossible for her to forget the ending she had chosen for herself. Better that than to be taken by surprise, like Tullia. Her only other choice would have been to launch herself into the void again and wait for her cooling bag to run out of air, leaving her to cook in her own body heat. Because whatever she might have wished for in a moment of weakness, however strong the urge to renege might have become, the holin that could have bought her a year or two more was now irrevocably out of her hands.
Nino clutched the rope at the edge of the observation chamber and peered down at the countless tiny color trails fixed above the rocky slope.
“Those are the orthogonal stars?”
Yalda said, “Yes.”
He grimaced. “They look just like the stars back home. But now you’re saying that their worlds could kill us with a touch, if we so much as set foot on them?”
“That’s how it seems,” Yalda replied. “But then, who knows what will happen down the generations? We might even find a way to mine their rock, to render it harmless.”
Nino looked skeptical. He still found it difficult to accept that the Peerless had a future at all.
“Look at what we’ve survived already,” Yalda said. “Harder tests than any you gave us at the launch.”
“If those stars lie in the future,” he said, “why can’t you just search among them with your telescopes and see if they strike the world, or not?”
“Light from that part of their history can’t reach us here,” Yalda explained. “When we looked out at the ordinary stars, back home, we saw them as they were many years ago. The same is true of these stars—but ‘many years ago’ by our measure, now, means far from the world, far from any collision that might happen.”
“But if they continue as we see them—?”
“Then the world will end up in the thick of them,” Yalda said. “That much is clear.”
Nino was silent. Yalda said, “What we’re doing has the chance to help your children, far more than Acilio’s money ever could. Don’t you want to be a part of that?”
“It’s worth trying,” he conceded. “Better than rotting in that cell. And if you really can trust me with your own flesh—”
“Why wouldn’t I?” Yalda did her best to silence her doubts. “You’ve been a good father before. Just promise you won’t force the sagas down their throats.”
“I might tell them a couple of the old stories,” Nino said. “But the rest would be about the flying mountain whose people learned to stop time.”
He reached over and put his hand on Yalda’s shoulder. Nature dulled her fears, lulling her into a sense of rightness at the thought of what lay ahead. If she waited, if she asked for time to say her farewells, that would only make it harder. This was her last chance at the closest thing to freedom: her will, her actions, and the outcome in the world could all be in harmony.
Yalda said, “I want you to name our children Tullia and Tullio, Vita and Vito.” For all that she’d cared for Eusebio, if he was going to outlive her his name could look after itself. “If there’s a solo, call her Clara.”
Nino dipped his head in assent.
“Love them all, educate them all.”
“Of course,” Nino promised. “And you’ll be no stranger to them, Yalda. What I don’t know about you, your friends will tell them. Fatima will tell them a dozen stories of you a day.”
He’d meant to reassure her, but Yalda shivered with grief. A mountain could fly through the void, but she could not see her own children.
She fought against her sadness; if she succumbed to it now and stopped what they’d begun it would only be twice as painful the next time.
Yalda took hold of the ropes with three of her hands; with the fourth she drew Nino’s body closer. The color trails of the old stars were splayed out above them. His chest pressed against hers, innocently at first, but then their skin began to adhere. Yalda twitched, panic-stricken, picturing herself tearing free, but then she stifled her fear and let the process continue. When she looked down, a soft yellow glow could be seen passing through their conjoined flesh, its message older than writing.
Her eyelids grew heavy, and a sense of peace and reassurance suffused her thoughts. There was no need for words now. They were sharing light, and the light carried Nino’s promise to protect what she would become.
APPENDIX 1
UNITS AND MEASUREMENTS
APPENDIX 2
LIGHT AND COLORS
The names of colors are translated so that the progression from “red” to “violet” implies shorter wavelengths. In the Orthogonal universe this progression is accompanied by a decrease in the light’s frequency in time. In our own universe the opposite holds: shorter wavelengths correspond to higher frequencies.
The smallest possible wavelength of light, λmin, is about 231 piccolo-scants; this is for light with an infinite velocity, at the “ultraviolet limit”. The highest possible time frequency of light, νmax, is about 49 generoso-cycles per pause; this is for stationary light, at the “infrared limit”.
All the colors of light arise from the same pattern of wavefronts, rotated into different orientations in four-space.
In the diagram above, AB is the separation between the wavefronts in four-space, which is fixed regardless of the light’s color. AD is the light’s wavelength (the distance between wavefronts at a given moment) and BE is the light’s period (the time between wavefronts at a fixed location).
The right triangles ACB and ABD are similar triangles, because the angles at A are the same. It follows that AC/AB = AB/AD, and:
AC = (AB)2/AD
Also, the right triangles ACB and EAB are similar, because the angles at B are the same. It follows that BC/AB = AB/BE, and:
BC = (AB)2/BE
Pythagoras’s Theorem, applied to the right triangle ACB, gives us:
(AC)2 + (BC)2 = (AB)2
Combining these three results yields:
(AB)4/(AD)2 + (AB)4/(BE)2 = (AB)2
If we divide through by (AB)4 we have:
1/(AD)2 + 1/(BE)2 = 1/(AB)2
Since AD
is the light’s wavelength, 1/AD is its spatial frequency, κ, the number of waves in a unit distance. Since BE is the light’s period, 1/BE is its time frequency, ν, the number of cycles in a unit time. And since AB is the fixed separation between wavefronts, 1/AB is the maximum frequency of light, νmax, the frequency we get in the infrared limit when the period is AB.
So what we’ve established is that the sum of the squares of the light’s frequencies in space and in time is a constant:
κ2 + ν2 = νmax2
This result assumes that we’re measuring time and space in identical units. But in the table above we’re using traditional units that pre-date Yalda’s rotational physics. The data Yalda gathered on Mount Peerless showed that if we treat intervals of time as being equivalent to the distance blue light would travel in that time, the relationship between the spatial and time frequencies takes the simple form derived above. So the appropriate conversion factor from traditional units to “geometrical units” is the speed of blue light, vblue, and we have:
(vblue × κ)2 + ν2 = νmax2
The values in the table are expressed in a variety of units that have been chosen so that the figures all have just two or three digits. When we include a factor to harmonise the units, the relationship becomes:
(78/144 × κ)2 + ν2 = νmax2
Now, the velocity, v, of light of a particular color is simply the ratio between the distance the light travels and the time in which it does so. If we take the pulses of light in our first diagram, they travel a distance AC in a time BC, giving v = AC/BC. If we then use the relationships we’ve found between AC and AB and the spatial frequency κ, and between BC and BE and the time frequency ν, we have:
v = κ/ν
Again, we can only use this formula with traditional units after applying the appropriate conversion factor:
v = (vblue × κ)/ν
which, if we’re taking frequencies from the table above, becomes:
v = (78/144 × κ)/ν
The velocity we’ve been describing so far is a dimensionless quantity, related to the slope of a line tracing out the history of the light pulse on a space-time diagram. (The way we draw our diagrams, with the time axis vertical and the space axis horizontal, it’s actually the inverse of the slope.) Multiplying the dimensionless velocity by a further factor of 78, the speed of blue light in severances per pause, gives us the values in traditional units that appear in the table.
AFTERWORD
Much of what we know about the physics of our universe can be understood in terms of the fundamental symmetries of space-time. If you imagine any experiment that can be fully contained on a floating platform out in space, then orienting the platform in different directions or setting it in motion with different velocities will have no bearing on the outcome of the experiment. The particular directions in space and in time with which the platform is aligned make no difference.
However, in our universe the laws of physics distinguish very clearly between directions in space and directions in time. While you’re free to travel through space precisely due north if you wish, as you do so you will also be moving forward in time (as measured, say, by GMT). Expecting to be able to depart from Accra at 1:00:00.000 GMT and arrive at Greenwich to see the clocks showing exactly the same time—because you’d arranged to move “purely northwards” without any of that annoying progress through other people’s idea of time—is not just a tad optimistic, it’s physically impossible. “North” is a “space-like” direction (whatever else might be merely conventional about it), while “the future” is a “time-like” direction (however much it might differ from person to person traveling at relativistic speeds). No amount of relative motion can transform space-like into time-like or vice versa.
The underlying physics of Orthogonal comes from erasing that distinction between time and space—giving rise to an even more symmetrical geometry—and then applying a similar kind of reasoning to that which links the abstract geometry of space-time to the tangible physics of our own universe.
Does every last phenomenon described in the novel follow with perfect mathematical rigor from this process? Of course not! Centuries of effort by people far more able than I am has still not put the physics of our own universe on such a rigorous footing, and to reconstruct everything under different axioms—with no access to experimental results—would be a massive undertaking. So while I’ve tried to be guided throughout the novel by some well-established general principles, at times the finer details are simply guesswork.
That said, the most striking aspects of the Orthogonal universe—the fact that light in a vacuum will travel at different speeds depending on its wavelength; the fact that the energy in a particle’s mass will have the opposite sense to its kinetic energy; the fact that like charges will attract, close up, but then experience a force that oscillates with distance between attraction and repulsion; the existence of both positive and negative temperatures; and the fact that an interstellar journey will take longer for the travelers than for the people they left behind—are all straightforward consequences of the novel’s premise.
My initial thoughts about the Orthogonal universe were clarified by the discussion of the consequences of different numbers of space and time dimensions in Max Tegmark’s classic paper, “Is ‘the Theory of Everything’ Merely the Ultimate Ensemble Theory?” (Annals of Physics 270, pp 1-51, 1998; online at arxiv.org/abs/gr-qc/9704009). Tegmark classifies universes with no time dimension as “unpredictable” (p 34). However, he appears not to have considered cases where the underlying space-time is a compact manifold, making the universe finite. As discussed in the novel, finite universes with the right topologies can exhibit physical laws that support predictions—albeit imperfect ones if the data available spans less than the entire width of the universe. But this isn’t all that different from the situation under Newtonian physics, which also allows the possibility that an object with an arbitrarily high velocity might unexpectedly enter the region whose future you’re trying to predict.
Readers with a background in physics might be aware of a mathematical technique known as Wick rotation, in which equations that apply in our own universe are converted to a form with four spatial dimensions, as part of a strategy for solving the original equations. It’s worth stressing, however, that these “Wick-rotated” equations are not the same as those governing the physics in Orthogonal; there are some additional changes of sign that lead to very different solutions.
Supplementary material for this novel can be found at www.gregegan.net.