“Right,” Carla said. “And if the valley is deep enough, those gaps could end up so large that you’d need to make six or seven photons to cross them.”
Patrizia turned to Carla. “Doesn’t that… solve the stability problem?”
Carla considered the question seriously. In the old way of looking at the problem, even if the walls of the energy valley were so steep that the luxagen rolled back and forth at a rate dozens of times greater than the maximum frequency of light, the tiniest deviation from a parabolic shape would introduce lower-frequency components into its motion—some of them low enough to produce light. And however feeble the radiation emitted that way, the luxagen would slowly gain energy and creep up the valley, until it finally escaped.
But that was in a world where energies could take on any value at all. In the new theory of luxagens as waves, a steep enough valley would have gaps between its energy levels that were insurmountable—and the inevitable imperfections in the shape of the valley would merely split some of those levels. As Onesto had pointed out, if the rungs of the original energy ladder were spaced sufficiently widely, adding a few extra rungs close to the originals wouldn’t suddenly make the whole thing traversable. The valley’s imperfection no longer undermined its stability.
“We still don’t know how long it takes to create a given number of photons,” Carla said cautiously. “But we do know that it takes much longer to make five than four, and a great deal longer to make six—even with the beam from a sunstone lamp to help. If we could understand what was going on there, I think we’d be getting close to explaining how some solids can be stable.”
Patrizia sketched the shapes of the first few luxagen waves on her own chest. “What happens if I add two of these solutions together—two waves with different energies? The sum will still solve the same equation… so what does the combined wave represent? Two luxagens, one with each energy?”
Carla said, “That doesn’t sound right. We found the wave equation by translating the energy-momentum relationship for a single particle. And what if I add two solutions in unequal proportions? Say, one quarter the first solution and three quarters the other?”
“Couldn’t that be… one particle with the first energy, and three with the second?” Patrizia didn’t sound too persuaded herself; she could probably see where this numbers game was heading.
“Irrational proportions, then,” Carla replied. “Multiply the second solution by the square root of two, then add it to the first. It’s still just one particle.”
Patrizia hummed with frustration. “You can multiply these waves by any number you like!” she said. “It doesn’t change their frequency, so it won’t change their energy—I mean the luxagen’s energy. Unless the wave has some energy of its own, separate from the particle’s energy, what does it actually mean if you double the size of the wave, or triple it?”
Carla was worried now. If the luxagen wave did have an energy of its own that depended on its amplitude, the discrete energy steps that were the theory’s great virtue would be erased. “What if we ignore the overall size of the wave?” she suggested. “Or better yet, we standardize the size of each solution, by some measure. Then we could still ask what it means to combine two solutions in a certain proportion. If we start with a wave with the lowest energy, and combine it with the next one, say at one part in twelve… what would that mean, physically? It can’t describe a particle with an energy lying one twelfth of the way between the two values.” That route would lead them back to continuous energies again, rendering the whole thing useless.
Patrizia spread her arms in a gesture of defeat; she’d run out of guesses.
“Part one energy, part another,” Carla muttered. “We could even have a luxagen that was part trapped in the valley, part free!”
Nothing was making sense any more. The exhilaration she’d felt when they’d found the energy levels had vanished now. Why should they take the luxagen equation seriously, if they couldn’t say what its solutions meant in all but a few special cases? If she tried to peddle this nonsense to Assunto as the answer to the stability problem, he’d have her teaching the wavelength-velocity relationship to three-year-olds for the rest of her life.
Then she heard her own words as if someone else had spoken them: Part trapped in the valley, part free. Two solutions you could combine, in any proportion. That proportion could be the missing timer—the means by which a luxagen in the tarnishing experiment kept track of how long it had been sitting in the light. Its energy couldn’t creep up over time… but the ratio between the two solutions could. The luxagen could start out as a trapped wave, but then gradually take on more and more of the free solution.
Carla didn’t know what this hunch was worth, but they had all the tools they’d need to test it. She said, “If we want to know how long it takes to get a luxagen out of the valley by blasting it with light… why don’t we just add the energy due to the light itself to the energy of the valley, and calculate exactly what that does to the luxagen wave over time?”
Patrizia quailed slightly. “That sounds like a long calculation.”
“Oh, it will be,” Carla promised her. “So before we even start, we should break for a meal.” She turned to Onesto. “Will you join us? Loaves for everyone, out of my entitlement. Let’s celebrate, replenish our strength—then start dragging some real predictions out of this equation.”
16
As she checked the link to the light recorder, Amanda leaned close to Carlo and whispered, “If this works, you should take it to the Variety Hall. They haven’t had an act that drew a crowd like this for years.”
There did seem to be about twice as many people gathered around the bench where they’d set up the signaling experiment than were usually present in the entire animal physiology workshop. Carlo didn’t know who’d invited them all, but he was feeling apprehensive enough without adding a layer of stage fright. He needed to keep both arms still or risk shifting the probes skewering his wrists, but he managed to roll his shoulders without the motion reaching below his elbows as he tried to unknot the tense flesh in his back.
Both probes had been aligned to pick up the signal to one finger of each hand. Amanda started the light recorder, then Carlo executed a sequence of moves with the chosen finger of his left hand, following the instructions on a sheet of paper clipped to the bench in front of him. Each individual action was simple enough, but they were arranged in an arbitrary progression that he could only adhere to by paying close attention, and he had deliberately refrained from any rehearsal. The eye-catching periodicity of his first, repetitive experiment had had its advantages, but this time he didn’t want his flesh to sense a pattern and pursue it on its own.
When this first stage of the performance was over, Amanda took the spool of paper out of the recorder, slipped it onto a shaft mounted on the bench, then wound the whole strip across onto another spool—the simplest way to inspect it without risking it getting tangled or damaged. To Carlo’s relief, there was a strong signal darkening the paper from start to finish; they wouldn’t need to dig around in his flesh any more to improve on it.
“Do you want to use this?” Amanda checked with him.
“Please.” Carlo wasn’t in great pain, but his body kept drawing his attention to the probes’ unnatural presence, refusing to let him feel at ease.
Amanda loaded the spool into the inverter, inspected the contact rollers for any grit or paper-fluff that could do mischief, then threaded the two leader tapes—from the recording itself, and from a second spool of unexposed light paper—together through the core of the mechanism and onto their respective receiving spools. Then she lit the lamp, closed the device, wound the spring, and engaged the drive. The spectators waited patiently as the machine whirred—better behaved than the usual crowd at any magic show.