Tamaro said, “She must have decided that it wasn’t worth the risk.”
“Well, that was her right.” Tamara was annoyed now. Did he really think he could compare her to Massima? “Since she was only ever going to be a spectator, I don’t blame her for setting such a low threshold.”
“Do I have to beg you not to go?” he asked her. He sounded hurt now. “Have you even thought about what it would mean to me, if something happened to you?”
Tamara reached down and squeezed his shoulder reassuringly. “Of course I have. But I’ll be careful, I promise.” She tried to think back to what Ada had said, the way of putting it that had won over her own co. “We were born too late to share the thrill of the launch, and too early to take part in the return. If I turn down an opportunity like this, what’s my life for? Just waiting around until we have children?”
“Did I ever put pressure on you to have children?” Tamaro demanded indignantly.
“That’s not what I meant.”
“I’ve always been happy for you to work!” he said. “You won’t hear a word of complaint from me, just so long as you do something safe.”
Tamara struggled to be patient. “You’re not listening. I need to do this. Part of it’s the chance to help the chemists fix the fuel problem—and that in itself would be no small thing. But flying the Gnat is the perfect job for me: for my skills, my temperament, my passions. If I’d had to spend my life watching rocks like this pass by in the distance, I would have made the best of it. But this is a chance to do everything I’m capable of.”
Tamaro said coldly, “And you’d risk our children, for that?”
“Oh…” Tamara was angry now; she’d never imagined he’d resort to anything so cheap. “If I die out there, you’ll find yourself a nice widow soon enough. I know most of them have sold their own entitlements, but you’ll have mine, won’t you? You’ll be the definition of an irresistible co-stead.”
“You think this is a joke?” Tamaro was furious.
“How was I joking? It’s the truth: if I die, you’ll still get to be a father. So stop sulking about it, as if you have more at stake here than I do.”
He stepped away from her, visibly revolted. “I’m not fathering children with someone else,” he said. “The flesh of our mother is the flesh of my children; however long you might borrow it, it’s not yours. Least of all yours to endanger.”
Tamara buzzed with derision. “What age are you living in? I can’t even look at you, you buffoon!” She pushed past him onto the path and headed out of the clearing, half expecting him to start following her and haranguing her, but each time she stole a glance with her rear gaze he was still standing motionless where she’d left him.
When he vanished from sight behind a bend in the path, Tamara felt a strange, vertiginous thrill. Was she leaving him? At the very least, she wouldn’t be coming back to the farm until he sought her out and apologized. She could sleep in the office next to the observatory—in a bed without gravity, but she’d survive.
As she strode along between the dormant wheat-flowers, she began to feel a twinge of guilt. She wanted Tamaro to understand what the Gnat meant to her, but she didn’t want to bludgeon him into acquiescence. If he was afraid of losing his chance to be a father, the threat of desertion would be even more distressing than the prospect of her death: her children, not his, would inherit the family’s entitlement. What kind of fate was she prepared to force upon him? The choice between a lonely death and… what? Hiding the children he had with some widow? Stealing grain for them from his own crops, until the auditors finally caught him? He needed to grow up and accept her autonomy, but there were limits to how ruthless she was willing to be. She still loved him, she still wanted him to raise their children. Whatever they’d both said in the heat of the moment, she couldn’t imagine anything changing that.
Tamara thought of turning back and trying to effect a speedy reconciliation, but then she stiffened her resolve. It would be painful for both of them to pass the day with this quarrel hanging over them, but she had to let Tamaro feel the sting of it. Maybe their father would talk some sense into him. As often as he’d taken Tamaro’s side, Erminio knew how stubborn his daughter could be. If he’d overheard the morning’s conversation, what counsel could he offer his son but acceptance?
She came to the farm’s exit and seized the handle of the door in front of her without slowing her pace. The handle turned a fraction then stuck; she walked straight into the door, pinning her outstretched arm between the slab of calmstone and her advancing torso.
She cursed and stepped back, waited for the pain in her arm to subside, then tried the handle again. On the fourth attempt she understood: it wasn’t stuck. The door was locked.
The last time she’d seen the key, she’d been a child. Her father had shown her where it was kept in one of the store-holes, a tool to be wielded against fanciful threats that sounded like stories out of the sagas: rampaging arborines who’d escaped from the forest to conquer the Peerless, or rampaging mobs driven mad by hunger, coming to strip the grain from the fields.
It was possible, just barely, that Tamaro had run ahead of her by another route. But he would have had to cut through the fields, and he could not have done it silently.
So either he’d locked the door before she’d even risen that morning, before they’d exchanged a word, or Erminio really had been listening to them—and far from resolving to plead her case with Tamaro, he’d decided that the way to fix this problem was to keep her on the farm by force.
“You arrogant pieces of shit!” Tamara hoped that at least one of them was lurking nearby to overhear her.
Angry as she was, she was struck by one ground for amusement and relief: better that they try this stunt now than on the launch day. If they’d caught her by surprise at the crucial time it would not have been hard to keep her confined for a few bells. Once she’d failed to show up, Ada and Ivo would have had no choice but to leave without her, and her idiot family would have got exactly what they’d wanted. But apparently they couldn’t bear to defer the pleasure of punishing her.
Tamaro was coming down the path toward her.
“Where’s the key?” she demanded.
“Our father’s taken it.” He nodded toward the door, implying that Erminio was outside, beyond her reach.
“So what’s the plan?”
“I gave you chance after chance,” he said. “But you wouldn’t listen.” He didn’t sound angry; his voice was dull, resigned.
“What do you think is going to happen?” she asked. “Do you know how many people are expecting me to turn up for meetings in the next three days alone? Out of all those friends and colleagues, I promise you someone will come looking for me.”
“Not after they hear the happy news.”
Tamara stared at him. If Erminio was out there telling people that she’d given birth, this had gone beyond a private family matter. She couldn’t just forgive her captors and walk away, promising her silence, when the very fact of her survival would show them up as liars.
“I’ve burned all your holin,” Tamaro told her. “You know I’d never try to force myself on you, but what happens now is your choice.”
She searched his face, looking for a hint of uncertainty—if not in the rightness of his goals, in his chance of achieving them. But the man she’d loved since her first memory of life seemed convinced that there were only two ways this could end.
Either she’d agree to let him trigger her, and she’d give birth to his children—taking comfort in the knowledge that he’d promised himself to them.
Or she’d stay here, without holin, until her own body betrayed her. She’d give birth alone, and her sole victory would be to have cheated her jailer and her children alike of the bond that would have allowed them to thrive.
15
The hiss from the sunstone lamp rose in pitch to an almost comical squeak. Carla could hear the remaining pellets of fuel ricocheting around the crucible, smal
l enough now that the slightest asymmetry in the hot gas erupting from their surface turned them into tiny rockets. A moment later they’d burned away completely and the lamp was dark and silent.
Onesto walked over to the firestone lamp and turned it up, then went back to his desk.
The workshop looked drab in the ordinary light. Carla punctured the seal of the evacuated container, waited for the air to leak in, then tore away the seal and retrieved the mirror. After she’d inspected it herself she handed it to Patrizia, who surveyed it glumly.
It had been obvious for the last few days that the tarnishing wasn’t proceeding in the manner they’d predicted. The first tier had matched the reference card placed beside the mirror after a mere two chimes’ exposure; the second tier had taken two days. That alone showed that the time to create each photon couldn’t be the same in each case. But Patrizia’s idea that the time might be proportional to the period of the light couldn’t explain what they were seeing, either. For two near-identical hues on either side of the border between the second and third tiers, the period of the light was virtually the same—but while the fifth photon needed to complete the tarnishing reaction in the second tier had only taken two days to appear, after waiting more than twice as long for one more photon, the third tier remained pristine.
Carla sketched the results on her chest. “The photon theory can explain the frequencies where we switch from one tier to the next. But how do we make sense of the timing?”
“Maybe some energy leaks out of these valleys as heat,” Patrizia suggested. “Then it takes time to make up for that.”
“Make up for it how?”
“With a longer exposure.”
“But all you can do with a longer exposure is create more photons!” Carla protested. “And if the numbers of photons aren’t what I’ve drawn here, where does the five-to-four ratio in the frequencies come from?”
Patrizia hummed in self-reproach. “Of course. I’m not thinking straight.”
Carla saw Onesto glance up from his papers. He’d endured the jarring lighting for six days, and now he had to listen to the two of them stumbling around trying to make sense of their non-result. “I’m sorry if we’re disturbing you,” she said.
“You’re not disturbing me,” Onesto replied. “But to be honest, I haven’t been able to get much work done for the last two bells.”
“Why not?”
“Something’s been puzzling me about your theory,” he said, “and the more I see you puzzled yourselves, the more I’m tempted to break my silence. So if it’s not too discourteous, I hope you’ll let me speak my mind.”
Carla said, “Of course.”
Onesto approached. “Nereo posited a particle, the luxagen, to act as a source for Yalda’s light field. If I’ve understood what you’re saying, you’re now positing an entirely new particle that plays a very different role: traveling with the waves in the field itself, carrying their energy for them.”
Carla turned to Patrizia; the theory was hers to define and defend. Patrizia said, “That’s right.”
“Then why not complete the pattern?” Onesto suggested. “If you have reason to believe that the light field can manifest as a particle, why should Nereo’s own particle be different? Shouldn’t the luxagen be associated with waves in a field of its own?”
Patrizia looked confused, so Carla stepped in.
“There would be an appealing symmetry to that,” she said. “To every wave, its particle; to every particle, its wave. But I think it would complicate the theory unnecessarily. Without any evidence for a ‘luxagen field’, it’s hard to see what could be gained by including it.”
Onesto inclined his head politely. “Thank you for listening. I’ll leave you in peace now.”
He was halfway to his desk when Patrizia said, “You want us to treat the luxagen as a standing wave?”
Onesto turned. “I wasn’t thinking of anything so specific,” he admitted. “It just seems odd to treat the two particles so differently.”
At this response Patrizia’s confidence wavered, but then she persisted with her line of thought. “Suppose the luxagen follows the same kind of rules as the photon,” she said. “It has its own waves—and just like light waves, their frequency is proportional to the particle’s energy.”
Carla said, “All right. But…”
“If the luxagen is trapped in an energy valley,” Patrizia said, “its wave must be trapped as well. A trapped wave, a standing wave, can only take on certain shapes—each one with a different number of peaks.”
Carla felt the scowl vanish from her face. Unlike Patrizia’s last suggestion, this wasn’t hunger-addled nonsense. Onesto’s proposal had sounded naïve—but now Carla could see where her infuriating, erratically brilliant student was taking it.
For each shape it could adopt, the luxagen’s standing wave would oscillate with a specific frequency. The same kind of principle governed the harmonics of a drum: the geometry of the resonant modes of the drumhead dictated the particular sounds it could make, each one with its characteristic pitch.
But Patrizia’s rule linked frequencies to energies—so a trapped luxagen would only be able to possess certain energies. The energy closest to the top of the valley would set the gap that needed to be jumped in order for tarnishing to occur, and there would be no doing it by halves: a luxagen couldn’t accumulate five photons’ worth of energy and then wait around for a sixth. Once you reached the highest energy level there were no more resting places; it was an all-or-nothing trip. You either made the total number of photons you needed, all at once, and escaped the valley… or you didn’t.
As they talked it over, Patrizia sketched the general idea. Onesto looked on, pleased that his suggestion had proved helpful but a little daunted by the strange outcome.
“I still don’t understand the details of the timing,” Carla confessed, “but if you don’t get to make the photons separately, one by one, there’s no reason to expect the time it takes to be proportional to the number of photons.”
“Can we quantify any of this?” Patrizia asked.
Carla said, “We could try to write an equation for the luxagen wave. Whatever we know about the luxagen’s energy, we translate into the wave’s frequency; whatever we know about the luxagen’s momentum, we translate into the wave’s spatial frequency.”
The idea seemed straightforward, but they struck a problem almost at once. Taking the rate of change of an oscillating wave multiplied it by a factor proportional to its frequency, but also shifted the wave by a quarter-cycle: at every peak of the original wave the rate of change crossed zero, heading downwards, while at every such zero of the original wave the rate of change was at a minimum, the bottom of a trough. When Yalda had devised her light equation she had been able to go one step further: the second rate of change was shifted by another quarter-cycle, putting it a half-cycle away from the original—yielding the original wave turned upside down and multiplied by the frequency squared.
Multiples of the original wave were easy to combine. The geometrical relationship Yalda had sought to express—that the sum of the squares of the wave’s frequencies in all four dimensions was a constant—could be encoded in the wave equation simply by multiplying every term in that relationship by the strength of the wave, then re-expressing the squared frequencies as second rates of change.
But with a luxagen in a solid, the relationship between its energy and momentum included its potential energy, which depended on its position in the energy valley. It was impossible to write this relationship purely in terms of the energy squared—so it was impossible to talk only of frequencies squared. To go halfway and include the frequency itself meant taking the square root of the operation that turned the wave upside down—putting the square root of minus one into the wave equation.
“It looks as if we’re stuck with complex numbers,” Carla declared. “What does that mean? That our premises are wrong?”
Patrizia seemed to share her sense of tre
pidation, but she wasn’t ready to give up. “Let’s follow the mathematics,” she suggested. “We should see what the final answers are before deciding whether or not it all makes sense.”
To make the calculations easier they chose a field described by a single number—albeit complex—rather than a vector like light, with its different polarizations. They also assumed that the luxagen would be moving slowly. For a parabolic energy valley—the easiest idealization to work with—it was possible to solve the equation exactly.
As Patrizia had guessed from the start, there was a sequence of solutions with distinct shapes. Those shapes could be described with real numbers alone, though the wave’s variation over time swept out a circle in the complex number plane with a frequency corresponding to its energy.
Some solutions shared the same energy, though that was just a consequence of the idealized shape of the valley. Carla pushed on further and managed to calculate the effect of switching to a more realistic valley, closer to the kind that was actually produced by Nereo’s force in a solid.
For the parabolic case all the energies were governed by the natural frequency at which a luxagen—as a particle—would be expected to vibrate in such a valley. The gaps between the allowed energies all corresponded precisely to that frequency, while the lowest energy sat one and a half times higher above the valley floor. For a more realistic valley, all the energies were reduced slightly, and the perfect agreement between the multiple higher-energy solutions broke down, splitting the idealized single energy levels into closely spaced sets.
Onesto said, “Suppose the natural frequency for the valley is greater than the maximum frequency of light. That’s the assumption at the core of the original theory of solids. But what does it mean, in your terms?”
Carla thought for a moment. “It means the energy gap exceeds the mass of a photon—so creating a single photon can never give you enough energy to jump the gap.”