The Clockwork Rocket

by Greg Egan

  “For true energy to be conserved, the heights of the pairs of arrows have to be identical before and after the release. But the arrows after the release are tilted, because the springs are now in motion. So those later arrows need to be longer, in order to reach the same height. That means that each relaxed spring has a slightly larger mass than it had when it was compressed—and from the point of view of someone traveling alongside it, a larger true energy. Less potential energy means more true energy. Both the old energies are upside-down.”

  Giorgio let a hint of pained, Ludovico-esque weariness into his voice. “If kinetic and potential energy still agree, what can it actually mean to claim that they’re ‘upside-down’? Upside-down compared to what? When do we get to see any of this so-called true energy, to compare its direction with its alleged opposites?”

  “In light,” Yalda said. “We see the direction of true energy every time we create light.”

  She drew a simple diagram, line by line. “The chemists,” she said, “have been having a lot of trouble with their ladder of energies. If we’re to believe their calculations, the difference in chemical energy between fuel and the gas it becomes after burning isn’t anywhere near enough to account for the thermal energy of the gas. We kept telling them that they’d made a mistake, and that they should improve the accuracy of their measurements. But they were right, and we were wrong. The fuel itself doesn’t need to provide the energy to heat the gas… because that energy comes from the creation of light.

  “Light brings its own four-dimensional momentum into the equation. It’s the need to balance that that forces the gas particles to be moving so fast. We thought that when fuel was burned, the light and the heat that was created both came from the release of chemical energy—but the truth is nothing like that! Light energy and thermal energy are opposites: creating one is what gives us the other.

  “And we thought that when plants made food from soil, the light was merely an unintended by-product, a measure of inefficiency. But the energy in food isn’t extracted from the soil, and the light shining from a flower’s petals is not wasted energy escaping. Light energy and the chemical energy in food are opposites, too. If plants didn’t make light, they’d have no energy source at all.”

  Yalda paused to give Giorgio a chance to respond, but he remained silent. Whatever radical notions she was proposing for the foundations of physics, these claims about food and fuel were the most shocking: the least abstract, the most tangible.

  “Why can’t we cool our bodies by emitting light?” Yalda continued. “That’s what I asked myself on my way up Mount Peerless. But now it’s obvious! Emitting light can only give you more thermal energy than you started with. The very act of emitting too much light can make a living body as hot as burning sunstone.” Her grandfather’s frail body had never held enough energy to flatten a forest; rather, it had lost control of its production of light.

  Giorgio said, “If emitting light generates thermal energy… why can’t we cool down by absorbing light instead? Why isn’t sunlight as good as our beds for making us cooler?”

  Yalda was prepared for that. “Entropy. Light carries a certain amount of entropy—so if you absorb light, your entropy must increase. But if we cool down, our entropy decreases. What I think happens when sunlight strikes our body is that we don’t absorb it, we just scatter it. That way, we can simply take a share of its kinetic energy, and be warmed by it.”

  Giorgio stopped the interrogation to take stock. “Well, you’ll certainly please the chemists,” he said. “If you’re right about this, they’ll build a statue in your honor. And the biologists will be intrigued by your ideas on energetics, even if half of them think you’re insane. There’s even something to make Ludovico happy.”

  Yalda doubted that, though she knew what he meant. A wave traveling through any ordinary medium marked an increase in kinetic and potential energy, not true energy. If creating light required true energy, it could not be a ripple in some pre-existing medium; it had to be a whole new substance or entity that was created afresh in every flame. But if that brought the term “luminous corpuscles” to mind, in Yalda’s scheme light still had a wavelength—so Ludovico would call this arrogance and hypocrisy, not a triumph for his beloved Meconio.

  “Now a question from the mathematicians,” Giorgio said. “You’ve shown us equations for the geometry of wavefronts, but what about an equation for the wave itself—something analogous to the wave equation on a string?”

  “The geometry gives us that, easily,” Yalda replied. “For a simple wave, the sum of the squares of the frequencies in all four dimensions equals a constant. But we also know that the wave’s second rate of change in each direction will be the original wave multiplied by a negative factor proportional to the frequency squared.”

  She sketched some examples, showing how doubling the frequency of a wave quadrupled its second rate of change. The square of the frequency and the second rate of change were just two ways of talking about the same thing.

  “So if you sum the second rates of change of the wave along each of the four dimensions, and negate that sum, then you’ve got the original wave multiplied by a constant times the sum of the squares of the frequencies—which itself is a constant. And that’s the equation for a light wave: the sum of its second rates of change, negated, must equal a constant times the original wave.”

  Giorgio contemplated this for a pause or two, then responded with a sketch of his own.

  “An oscillation’s second rate of change is proportional to the opposite of the original wave,” he said. “But an exponential growth curve has a second rate of change proportional to the wave itself—there’s no negation.”

  “That’s true,” Yalda said. “But—”

  “If you construct a wave that oscillates rapidly as you move in one direction,” Giorgio said, “what’s to stop you from choosing a frequency in that direction so large that its square, alone, is greater than the number you’re aiming for as the sum of all four?”

  “But then you’ve overshot the total,” Yalda protested. “So you won’t be able to satisfy the equation.”

  “No? What if one of the other terms is negative?”

  “Oh.” Yalda knew where he was heading now. “If one of the oscillations has too large a frequency, you can still satisfy the equation—by replacing the oscillation in another direction with exponential growth.” The negated second rate of change, in that case, would be a negative multiple of the original wave, allowing the sum of all four terms to be brought back down to the target.

  Giorgio said, “So the question is: if light obeys the equation you’ve given us, how can it possibly be stable? Why doesn’t every tiny wrinkle in the wave explode at an exponential rate?”

  7

  As the crowd spilled out of the Variety Hall into the starlit square, Yalda’s feeling of delight lingered. She’d found the entire magic show enchanting, but the fact that she’d quickly guessed the trick behind the astonishing finale hadn’t detracted from the pleasure of the experience at all; instead, it had intensified it.

  She turned to Tullia. “That image of the hidden assistant, projected onto a curtain of smoke… if I ever get to teach the optics course, I’m going to steal that for my first demonstration!”

  “That part wasn’t bad,” Tullia conceded. “The pyrotechnics before the intermission were awfully tame, but that’s the new safety regulations for you. I suppose we should give the City Council its dues: letting off rockets inside the hall was never a good idea.”

  “Antonia should have come,” Yalda said, turning sideways to squeeze through a gap in the throng. “It might have cheered her up.”

  “Antonia doesn’t want to be cheered up,” Tullia replied. “She’s committed to sitting at home moping until she undergoes spontaneous fission.”

  “It must be hard for her to make a decision.” Yalda had found it difficult enough trying to side-step her own family’s expectations, but growing up with a co and the
n walking away from him would be something else entirely.

  “We could get her safely to another city in a couple of days, if she agreed to it,” Tullia said irritably. “But she’s got herself caught up in negotiations with her co—some complicated business involving four or five intermediaries. She thinks she can go back to him on her own terms.”

  “Maybe she can. Maybe she’s arranging that.”

  “Ha.”

  “Is it so bad that she wants him to raise her children?”

  “In principle, not at all,” Tullia replied. “The trouble is, he’s already proved himself incapable of taking her wishes seriously. If Antonia wanted to, she could spend five or six years living the free life in Red Towers or Jade City, and then find a nice co-stead who’d be grateful for the heirs.”

  Yalda said, “You make it sound so easy, it’s a wonder everyone isn’t doing it.”

  A streak of brilliant violet light appeared in the sky above the eastern horizon, spreading out rapidly from a fixed central point. The center itself remained dark, but as Yalda watched, the two dazzling threads emerging from it turned blue, then green, the new colors chasing the old in both directions. It was as if someone were dragging a giant star trail out from behind the edge of a mirror, exposing ever more of it while creating a perfect duplicate that seemed to be rushing in the opposite direction.

  Yalda was transfixed; Tullia was already counting out pauses as she threaded her way through the crowd, trying to sight the nearest clock tower and fix the time of the event precisely. The two of them had never actually made plans as to what they’d do if they witnessed a Hurtler, but they’d managed to get the division of labor exactly right without a single rehearsal. Motionless, Yalda could etch the position of everything she’d seen into her memory, holding on to an image of the line of light against the stars. Tullia wouldn’t have those details, but she’d soon have the crucial timing information that would render comparisons with reports from other cities twice as valuable.

  The center disgorged two red tails and faded to black; the pair of mirrored spectral worms, now fully birthed and separated, disappeared into opposite corners of the dusty haze that hung over Zeugma’s towers. Yalda had only ever seen the final part of this spectacle before: all those years ago, after the harvest, when the center must have been below the horizon for her. To date, seven reports of the same phenomenon had reached the university; the one she’d witnessed as a child was the third in that list. History and legend were full of shooting stars—some of them accompanied by all manner of implausible flourishes—but neither ancient astronomers nor the authors of the sagas had ever claimed to have seen anything like the Hurtlers.

  Yalda remained still, carefully gauging the angles between her memory of the Hurtler’s trajectory and the nearest bright stars. In her rear gaze she could see the young man glaring at her, shouting, but even if she’d been wandering the square aimlessly she would have done her best to pay him no attention.

  “Where’s your co?” he yelled again. Yalda marveled at his sheer boorishness; the most extraordinary event ever seen in the sky—unknown to the ages, but glimpsed by the luckiest people now once or twice in a lifetime—had just unfolded before his eyes, and all he could think to do was taunt her for her size, or her lack of a partner.

  The man bent down, picked up a broken piece of cobblestone, and flung it at her; it struck her on the side of the head. Yalda couldn’t stop herself; she turned toward him.

  He squealed triumphantly, “I said: where’s your co?”

  Yalda squatted and retrieved the stone from the ground, feeling the heft of it and the sharp edges. This made her far angrier than when it had collided with her skull, because she knew firsthand now what the thrower should have known, what should have dissuaded him. Unusually, the man was accompanied by his own co, as well as the expected group of delighted male friends.

  “Where’s your mother?” Yalda called back, and pitched the broken stone at him with all her strength.

  If her choice of words stunned him, it was the impact that brought him to his knees. She’d hit him squarely in the tympanum, more by luck than intent. He cried out in agony—which could only have made the pain in his organ of speech more intense—and his hum began warbling up and down, as the need to express his suffering fought with his attempts to curtail it.

  Some of his friends looked shocked; others were even more amused than before at this unexpected twist in the night’s entertainment. The man’s co bore an expression of horrified disbelief, as if she’d just seen a freight train run down an infant. Yalda felt a sudden pang of fear; she’d done the greater harm, and most potential witnesses around them were still paying more attention to the sky than the ground. Whatever they’d glimpsed in their own rear gaze, they might only be aware of half the story.

  Yalda hurried away from the scene of her imprudent revenge, and caught up with Tullia on the other side of the square.

  “You fixed the time?” Yalda asked her.

  “Yes.” Despite her presence of mind as the event had unfolded, Tullia now appeared a bit dazed. This was her first sighting, and it would have confirmed all the scarcely believable claims that until now she’d been free to doubt.

  “I’ve got the bearings,” Yalda said. “We should write up the observation now, and dispatch it tomorrow.”

  “Of course.” Tullia shook herself out of her stupor. “That was, what, three and a half pauses from violet to red?”

  “Sounds right.”

  “Which puts it far above the atmosphere, but still close; a fraction of the distance to the sun.”

  “A gross and a half severances or so,” Yalda confirmed.

  The people around them were still buzzing with excitement, but Yalda detected no real sense of how extraordinary the sight had been; it was as if they’d just witnessed an elegant fireworks display to cap off the magic show.

  “What if it had been closer?” Tullia asked. “What if it had hit the ground?”

  Yalda had never seriously considered the possibility of a collision; with barely more than half a dozen sightings in a generation, it struck her as a remote prospect. “I wouldn’t like to be standing at the point of impact,” she conceded.

  Tullia said, “I wouldn’t like to be on the same planet.”

  The current thinking about the Hurtlers was that something was colliding with the tenuous gas that wafted out from the sun to occupy the surrounding region. Just as an ordinary shooting star could burn up brightly in the atmosphere, even the sun’s thin exhaust might be enough to ignite a sufficiently rapid interloper.

  How rapid were the Hurtlers? If an object was moving so speedily that you might as well imagine its entire trajectory erupting with light all at once, then the closest part of that long straight line would appear to a watcher first in violet, the fastest color, with the other hues following. Each color would appear to fly out along two opposing, symmetrical trails, as the light arrived from pairs of equidistant locations ever farther from the watcher. Any measurable asymmetry in the color trails would imply a lower speed for the object itself—with light from earlier parts of the trajectory gaining a head start—but as yet nobody had observed such subtle effects with enough confidence even to be sure which way the Hurtlers were traveling.

  “If you can salvage my geometrical theory of time,” Yalda bargained, “the pay-off is that something so fast won’t be carrying as much kinetic energy.”

  “If I salvage your theory,” Tullia retorted, “nothing will even need kinetic energy to tear itself apart. Everything in the cosmos will be itching to turn into light and hot gas.”

  “Don’t blame me if there are no happy endings; I didn’t invent entropy. Darkness and cold dust… bright light and hot gas. Does it really matter which one we end up as?”

  They began making their way toward Tullia’s apartment, to put their observations onto paper.

  Tullia said, “You do realize that according to your theory, someone traveling along with the Hurtler
would think that half the light we just saw was coming in toward them, not going out?”

  Yalda made a quick sketch on her chest.

  “You’re right,” she said. “That’s eerie.” The arrow of time shared by the world and the solar atmosphere was so different from the Hurtler’s arrow of time that Tullia’s hypothetical traveler would have seen a part of the light-burst converging on them—violating the law of increasing entropy as surely as if a roomful of smoke had shrunk in on itself and turned back into fuel. Obviously entropy couldn’t increase along every direction in four-space at once, but it was unsettling to have an example of the bizarre disparity play out right in front of their eyes.

  Yalda brushed the complication aside; she was having enough trouble trying to make the exponential blow-up in the light equation go away. She was scheduled to deliver a summary of her theory to the school of natural sciences in less than two stints, but if she couldn’t offer Giorgio a plausible solution to the flaw he’d uncovered, he’d cancel the talk.

  When they entered the apartment, Antonia was seated on the floor with dye and paper beside her. A firestone lamp was sputtering on the shelf above, casting a forlorn shadow. She’d probably been composing another letter to Antonio, but when Yalda and Tullia approached to greet her, her skin was blank. Yalda wished she could have offered her advice or comfort, but what did a solo have to say about the choices she faced?

  “How was the magic show?” Antonia asked, forcedly cheerful.

  “Upstaged,” Tullia replied. She described the celestial mirror trick that had followed.

  “I heard some commotion from the street,” Antonia said. “I looked out the window, but it must have been over by then.”