Carla said, “Yes—but now go a step further and suppose that the same ratio holds for every wave and its corresponding particle, whether it’s a luxagen wave and a luxagen or a light wave and a photon. None of the physics makes sense unless this ratio is a universal constant; I think of it as ‘Patrizia’s constant’, because the whole idea started with her. It’s as if these particle masses really are the maximum frequencies of the corresponding waves… just measured in different units.”
Assunto looked pained for a moment, but then he said, “You mean like times and distances?”
“Perhaps.” Carla didn’t want to over-reach with the comparison: one was a fundamental truth about the cosmos that the Peerless itself had helped to prove beyond doubt; the other was an appealing, but still untested, speculation.
Assunto said, “So let’s take it for granted that we can turn any frequency into an energy, and vice versa. You have a luxagen trapped in some energy valley, and the corresponding wave equation has two solutions with definite frequencies.”
“Yes,” Carla replied. “What I’ve drawn for the two waves is their variation in space, but while maintaining that shape they’re oscillating in time, each one with its own pure frequency.”
“Then you add a light wave whose frequency matches the difference between the luxagen frequencies—and it drives the low-frequency luxagen wave up to the higher frequency?”
“Yes.”
“Well, that much makes sense,” Assunto said. “You can do something similar with waves on a string, if you vary the tension periodically at a frequency equal to the difference between the frequencies of two resonant modes.”
“What’s more surprising, though,” Carla said, “is the simple rule that this wave follows along the way. Where I’ve plotted the proportion of each of the waves, the arc that links the point that’s ‘purely wave one’ to the point that’s ‘purely wave two’ isn’t an artistic flourish: the dynamics really does follow a perfect circular arc. The sum of the squares of the two proportions remains equal to one, throughout the process.”
“I see.” Assunto was prepared to take her word for this, even if the significance of it escaped him.
Carla said, “Hold on to that thought.”
She produced the second sheet.
“I take it that this is the bad news.” Assunto examined the diagram. “The light never frees the luxagen? So… that’s the end of your theory of tarnishing?”
“Wait!” Carla pleaded. “When there are just two waves, two energy levels, you’d expect the dynamics to take you all the way from one pure wave to the other. Where else are you going to go? But here, there are a multitude of free waves whose frequencies are almost identical—what I’m showing on the vertical axis covers them all. So there are ways you can wander around in this space of possibilities—keeping the sum of the squares of the proportions equal to one, as before—without the trapped-wave proportion ever falling to zero.”
“Without it ever falling very far at all,” Assunto noted, pointing to the modest arc that showed the limits of the process. “Which I can well believe, given your assumptions. But why isn’t it fatal? How can this be a description of light knocking a luxagen out of its valley, if the wave barely changes no matter how long you expose it to the light?”
Carla braced herself. She had managed to convince Patrizia and Onesto that her hypothesis wasn’t entirely deranged, but Assunto would be the real test.
“The thing is,” she said, “there’s always more than one luxagen and a light wave to consider. There’s the whole slab of mirrorstone as well. We can sum up most of its influence in terms of a simple ‘energy valley’, but the reality is more complicated than that. With all the luxagen waves reaching part-way out of their own valleys, every luxagen is interacting with its neighbors—and to some degree with its neighbors’ neighbors, and so on.”
“So your model’s inadequate?” Assunto suggested.
“Yes,” Carla conceded. “But a model of the entire solid would just be intractable. The only way we can get anywhere is to try to find a rule of thumb that lets us extract useful predictions from the things we can model.”
Assunto was skeptical. “What kind of rule?”
“We start with two reasonable assumptions,” Carla said. “If a wave that is purely trapped interacts with the rest of the solid, it remains trapped. If a wave that is purely free interacts with the rest of the solid, it stays free.”
Assunto said, “I can live with that. But what happens to a mixture of the two?”
“I doubt we could ever predict that with certainty,” Carla admitted. “Not without knowing exactly what’s going on with every single luxagen in the solid. But maybe we can still predict what will happen on average. If we treat the square of the proportion of the wave that’s trapped as the probability that the luxagen will remain trapped when it interacts with the rest of the solid, everything makes sense—because the squared proportions always add up to one, just as the probabilities for any set of alternatives always add up to one. I know it sounds too simple to be true—but the mathematics seems to be offering us the perfect number to use as a probability when we can’t make an exact prediction.”
Assunto raised a hand for silence, and Carla let him think the whole thing over. Finally he said, “When, exactly, does this probability get turned into a fact? You have the luxagen wave changing shape under the influence of the light alone, but then at some point it’s supposed to interact with the rest of the solid, which finally determines its fate. But the probability keeps changing, as the wave changes shape. So what probability do you use?”
Carla said, “It makes no difference exactly when the interaction happens, so long as it happens often enough, and so long as the probability grows in direct proportion to the time. Suppose the probability is one in a gross after one pause, two in a gross after two pauses, and so on. If the rest of the solid only interacts with each luxagen once every pause, the rate of tarnishing will be one luxagen per gross per pause. But even if the interaction takes place far more frequently than that, each time it happens the probability will have risen to a much smaller value than it would have reached if the luxagen had been left undisturbed for longer. The two effects—the lower probability and the greater number of interactions—almost cancel each other out, and you end up with a simple exponential decay curve.”
She sketched the result.
Assunto was not impressed. “Almost every process looks linear on a short enough time scale, so whatever’s going on with the tarnishing the net result could end up looking like exponential decay. If I gave you the sunstone for one more experiment, and you came back to me with a curve like that, what would it prove? Nothing.”
“One curve would be meaningless,” Carla agreed. “But this is where the bad news finally redeems itself. When the energy gap is small enough for a light wave to bridge the two frequencies, the rate at which the probability grows is just proportional to the intensity of the light. But the tiers we found with the mirrorstone suggest that the energy gap is four times too big for that—and for lower frequencies of light, five times too big. In which case, the rate is no longer proportional to the intensity itself: it’s proportional to the fourth or fifth power.”
Assunto grasped the significance of this immediately. “So it’s a higher-order effect,” he said. “The light wave creates a small disturbance in the energy valley, and the effect of that isn’t perfectly linear—so a complete description would have to include ever-smaller terms that depend on the square of the wave, the cube, the fourth power…”
“And the fourth power of the wave,” Carla added, “contains a frequency four times higher than that of the wave itself. There is no light with a frequency high enough to bridge the energy gap in a stable solid—but the fourth power of the same disturbance oscillates four times faster.”
“So how are you proposing to test all this?” Assunto pressed her.
“In the past, I’ve wasted sunstone,” Carla ad
mitted. “There were things I could have measured that I didn’t even try to record. This time I’ll do it properly, once and for all. With a system of apertures and shutters, in a single run I can expose different parts of the same slab of mirrorstone to different intensities of light, for different lengths of time. The variation in the tarnishing over time should give us the exponential decay curves—and the variation with intensity should confirm the fourth-power rule in the first tier, and the fifth-power rule in the next. If we do find those power rules, surely that will be a sign that we’re on the right track.”
Assunto said, “Last time, the tiers were meant to mark the number of ‘photons’ each luxagen needed to make in order to break free.”
“They still do!” Carla replied. “These powers of the light’s intensity are the only way I know to calculate the tarnishing rates, but that doesn’t mean photons are out of the picture. When a luxagen changes its energy level, it still has to add a whole number of photons to the light: four or five, just as before.”
“But what drives the luxagen from one level to another in the first place?” Assunto answered the question himself. “Not a bombardment with particles, but the shaking of a wave.”
Carla couldn’t deny that. Patrizia’s interpretation of the scattering experiment in terms of colliding particles seemed irrefutable, but as yet there was no way to describe the tarnishing in the same language. They were still groping their way toward the truth, and the argument everyone had once thought settled in the days of Giorgio and Yalda was refusing to lie quietly in its grave.
Assunto said, “I’ll give you the sunstone for one more experiment, but that’s it. No more tinkering with the theory and trying again. If you don’t find the power rules you’ve predicted, you’ll have to accept that your ideas have been refuted and move on. Agreed?”
Carla had known that they were approaching this point, but to hear it put so starkly gave her pause. She could return to her collaborators and work through everything one more time: checking their calculations, revisiting their assumptions. Maybe they’d missed something crucial that would lead them to change their predictions—or something that could sweep away the lingering confusion and provide them with a surer bet.
But in less than two stints, Assunto would be answering to an entirely new Council, and there was no guarantee that he’d still have the power to offer her any sunstone at all. If they ended up losing the chance to perform this last experiment, there were no calculations that could tell them whether or not they’d been wasting their time. They needed to know the result itself, even more than they needed to be right.
Carla said, “Agreed.”
19
Carlo abandoned the voles a bell earlier than usual to join the celebration in the hall below the main physics workshop. The corridors along the way were lined with posters for Silvano’s next election rally, promising voters the chance to MAKE YOUR CHILDREN PROUD.
Carla had urged him to invite all his colleagues and their families, but as far as he could see only Amanda and her co had turned up. The whole chamber was festooned with chains of small lamps, and—rather cruelly for the women, Carlo thought—there were baskets of seasoned loaves attached to every cross-rope, putting out an aroma that made it hard even for a moderately well-fed man to focus on anything else.
Patrizia, Carla’s young student, clung to a rope near the center of the hall, fending off an endless barrage of congratulations. “It took the three of us to get this far,” she kept saying. “And I didn’t solve the stability problem, that was Carla.” Her modesty appeared entirely sincere, but when Carlo moved among the clusters of physicists orbiting this star all he heard was talk of the urgent need to start applying “Patrizia’s principle” to some new problem or other.
He tried not to begrudge the girl her share of acclaim, but it undercut his sense that he ought to join in the rejoicing out of simple loyalty. What was there to celebrate, really, in this minuscule advance in the theory of solids? It had made Carla happy, and no doubt it would have some kind of payoff eventually, but what urgent need had it fulfilled? The ancestors would be oblivious to however long it took to find the cure to their woes. The travelers didn’t have that luxury.
Carla caught up with him. “Are you enjoying yourself?” she asked.
“Of course.”
“You don’t seem to be talking to anyone.”
Carlo said, “I get all the luxagen-speak I need from you.”
She feigned a punch at his shoulder. “Actually, it’s not all physicists here. Don’t you want to meet the woman who’ll be flying the Gnat?”
“That astronomer who found the Object?”
Carla emitted an exasperated hum. “Where have you been hiding for the last stint? Tamara gave birth. This is her replacement, Ada.”
Reluctantly, Carlo followed her across the hall. Ada was surrounded by her own circle of admirers, but they parted for Carla and she made the introductions.
“You’re a biologist, aren’t you?” Ada asked Carlo.
“That’s right.” There was an awkward silence, and Carlo realized that he was expected to say something more about his work, but he knew how that was likely to end. Everyone had heard the story of his amputation, and he was tired of being the butt of that joke.
Ada said, “Maybe you can answer this for me. Why should lizard skin be sensitive to infrared light?”
Carlo was about to deny that any such thing was true, when he realized what she was talking about: one of the chemists had extracted a component of the skin that fluoresced in visible light when it was illuminated with IR. “I’m not sure that it’s actually sensitive, in that the animal would know when it’s being exposed to infrared. As far as I’m aware it’s just a fluke, a chemical property with no biological significance.”
“Fair enough,” Ada said. “I was just curious, it seemed so strange.”
Carlo wasn’t really in the mood for small talk, but he didn’t want to embarrass his co. What did he know about this woman? “You must have been surprised when your colleague stepped down,” he ventured.
“It wasn’t that formal,” Ada replied. “She didn’t resign, we just got word from her family.”
“Ah.” That was shocking in its own way, but it made a lot more sense. No one in their right mind would give up the chance to fly the Gnat, but it wasn’t unheard of for couples with other plans entirely to wake in the night and let instinct take over.
“I wanted to see the children,” Ada said sadly. “But her co’s a farmer, and they’re quarantined with blight.”
“Quarantined?” Carlo had no reason to doubt her word, but he was taken aback. “I worked with wheat myself, not long ago. Wheat blight’s not usually that hard to control.”
“Her father said it was something new,” Ada explained.
Carlo felt a twinge of anxiety; he’d met half a dozen of his agronomist friends a few days earlier, and they hadn’t mentioned a new strain of blight. Had his defection so offended them that they were shutting him out of the loop? Or maybe they’d just been too busy teasing him about his mutinous fingers.
“Well, good luck with the journey,” he said. He started to back away along the rope when he caught Ada casting a quizzical glance at Carla, as if she’d expected something more from the exchange. Carlo paused, wondering which further nicety would be most appropriate: congratulations on her promotion, or commiserations on the fate of her friend.
Carla said, “Ada’s offered me a place on the Gnat.”
Carlo turned to Ada; her expression made it clear that this was the subject she’d been waiting to discuss. “I thought that was all down to the lottery,” he said.
“When the winner pulled out we asked the Council to reconsider,” Ada explained. “They agreed to let us choose a new crew member on the basis of their expertise. Tamara had talked about picking another chemist—but orthogonal matter isn’t something that chemists have actually worked with. Since Carla seems to have solved Yalda’s First Pro
blem… I thought she might stand the best chance of also solving the Third.”
Carlo felt sick. Carla seemed excited, but he could tell that she was fearful too. A moment ago he’d told himself that no sane person could give up a chance like this, but his perspective had undergone a wrenching shift.
“She didn’t solve the stability problem overnight,” he said. “Do you really expect a once-in-a-generation breakthrough to be repeated on demand? Under pressure, in that tiny vehicle…?”
Ada raised a hand reassuringly. “That’s not what I was thinking at all. I don’t expect the mysteries of orthogonal matter to be resolved on the spot. I just want someone with us who’s familiar with the new ideas, and who’ll have a chance of applying them if the opportunity arises. Ivo’s a brilliant chemist with a vast amount of experience, but there’s no point telling him to start thinking of luxagens as waves. And frankly, there’s no point telling me either; I have no idea what it implies.”
Carla said, “We’ll have a few days to decide. But Ada wants to take the final crew list to the new Council for approval at their first meeting, so this is the time to ask her any questions.”
“Right.” Carlo struggled to clear his head. The mere thought of his co inside the Gnat as it receded to invisibility was painful enough, but now he had to face up to the purpose of the mission: capturing a mountain-sized mass of fuel by setting it alight. Orthogonal rocks that no one understood sprouting flame wasn’t the worst-case scenario—it was the whole plan.
He looked to Carla again. As anxious as she was, it was plain that this was what she wanted. And after all her work with the tarnishing experiments, all the false starts and blind alleys, all the grief Assunto had given her… didn’t she have the right to this moment of glory? He wasn’t going to tell her to be content that she’d done her bit for the ancestors.