The Eternal Flame

12

Carla was reaching behind the textbooks for her stash of groundnuts when she heard someone moving on the ropes near the entrance to the classroom. She closed the cupboard quickly, embarrassed. She should have been strong enough to deal with her hunger without playing these stupid games.

Patrizia appeared in the doorway. “Do you have a moment, Carla? I need to ask you about something.”

“Of course.” Carla’s gut was squirming, cheated of the imaginary meal she’d promised it, but she kept her voice even and her face composed.

Patrizia dragged herself to the front of the room. “I know I made a fool of myself, with what I said about the tarnishing,” she began.

“That’s not true,” Carla insisted. “I asked for wild ideas, and you were brave enough to offer one. Just because it didn’t hold up doesn’t make you foolish.”

“Well, I’ve had another wild idea,” Patrizia admitted. “But this time, I was wondering if you’d hear it in private.”

“Of course.”

“I hope I’m not wasting your time,” Patrizia said. “Sometimes it’s so hard to concentrate that I start making stupid mistakes. Things I ought to know just… go into hiding.”

The misery in that last phrase was painful to hear, but Carla didn’t know what she could do about it. It wasn’t her place to tell the poor girl to put off the famine for another year or two—trading the risk that she’d face a much more arduous struggle, later, for a little more youthful energy and clarity when she really needed it.

“We all make mistakes,” she said. “Tell me your idea, I’ll be happy to hear it.”

Patrizia began haltingly. “The first part is just elementary mechanics, really. But I wanted to check it with you before I go any further.”

Carla did her best to hide her dismay. She’d been thinking of the groundnuts all through the lesson, but if she could survive her cravings for a whole bell she could remain polite for another few lapses.

She said, “Go ahead.”

“Suppose you have a motionless particle, and it’s struck by another particle that’s about three times as heavy,” Patrizia said. “I think their energy-momentum vectors before and after the collision would be something like this:”

“That looks fine to me,” Carla replied. The first diagram portrayed the history of the collision, while the second repositioned the same four vectors to make the geometrical rules that governed them visible at a glance. “You’re just using the triangle law, right? The sum of the two energy-momentum vectors has to be conserved, and their individual lengths are just the masses of the particles, which don’t change. So the vectors will form two sides of a triangle whose shape is left unchanged by the collision, and whose third side—the total energy-momentum—remains fixed.”

Patrizia seemed relieved, but still far from confident. “And all the possibilities for a collision like this can be found by rotating that triangle around its third side?”

“Yes.”

“That will give you the off-axis collisions as well? You just swing the triangle out…?” She sketched some examples, showing what happened to the particles’ momenta if they glanced off each other during the impact, scattering to either side of the original axis, bringing in another dimension of space.

“That’s all correct,” Carla assured her, letting a hint of impatience into her voice. Wherever Patrizia was taking this, she had the basics right, she could move on.

Patrizia said, “On the same assumptions, I calculated the final energy for the heavy particle, for a few different starting energies.” She opened a pocket, pulled out a sheet of paper and unrolled it.

Carla hesitated now. Though she’d surely made a similar plot once herself—as part of some long-forgotten exercise when she’d first been studying mechanics—this had gone past the obvious-at-a-glance stage. “The angle here is measuring how far the heavier particle ends up off-axis?” she asked.

“Yes,” Patrizia said. “The details of the collision itself—whether it’s glancing or head-on—would determine that angle, but I’m just trying to be clear about the final outcome, about the possible combinations of angles and energies allowed by the conservation laws. The really striking thing about these curves is the way the greatest angle of deflection always turns out the same! So long as the lighter particle starts off at rest, there’s a maximum angle at which the heavier particle can end up being knocked off course, and it only depends on the ratio of the masses—the energy of the collision doesn’t come into it.”

“Hmm.” Carla couldn’t recall ever being aware of that result, and she couldn’t see any simple geometrical reason why it had to be true, so she worked through the algebra on her chest. The claim turned out to be perfectly correct: the maximum angle of deflection was the same, regardless of the energy.

Carla’s impatience was tempered by curiosity now. Was Patrizia going to try to rescue her tarnishing theory by adding a second luxagen, three times heavier than the first?

Patrizia said, “I can put curves with exactly the same form as this through the data we measured in the light scattering experiment.” She dug out a second plot.

“All four curves use the same mass ratio, of about three to ten,” Patrizia explained. “You can find that straight away, from the maximum scattering angle! And then the only parameter left to determine is the vertical scale.”

Carla reached over and took the plot from her. With a judicious choice of just two numbers, Patrizia’s model had nailed every point. A pattern like this didn’t happen by chance. What these curves implied was that the light scattering off the luxagens was behaving exactly like a particle, about three times as heavy as the ones it was striking.

Except… this plot wasn’t showing the energy of a particle, it was showing the frequency of a wave. What they’d actually measured for that vertical axis had been the scattered light’s subsequent deflection through a prism, and then that had been converted to wavelengths and frequencies using the prism’s calibration against a light comb. So how did energy come into it? The energy in a light wave depended on its brightness—something they hadn’t even tried to measure.

“Tell me,” Carla asked, “what do you think’s going on here?”

Patrizia spoke tentatively. “Surely this means there’s some kind of particle, moving at the speed of the light itself? Not trapped in the wavefronts, like a luxagen would be, but actually traveling with the light.”

“And the luxagens we released from the mirrorstone scattered this particle?”

“Yes.”

“And then what?” Carla asked indignantly. “The light that had been pushing this mystery particle along decides to follow it? The laws of mechanics tell us how the particle alone should be moving after the collision… and the light wave accommodates that by adjusting its own speed, adjusting its frequency, to maintain the original relationship? Is the light supposed to be propelling this particle—or is the particle magically dragging the light around?”

Patrizia flinched. Carla hadn’t realized how sarcastic her tone had become. “I’m sorry,” she said. “I didn’t mean to be dismissive. I’m just confused. I don’t know how to make sense of this.”

Patrizia looked up and met her gaze; they both knew exactly what was making the conversation so difficult. She said, “I’ve been trying to think how we could explain the tarnishing experiment, making use of this result. Suppose there’s some reason why light waves have to be accompanied by these particles—let’s call them ‘luxites’, just to give them a name.”

Carla managed to stifle a derisive buzz. “Luxite” was the term that had been used by disciples of the ninth-age philosopher Meconio, the man who had first proposed—without a trace of evidence—that light was composed of “luminous corpuscles”. Giorgio had buried that notion with his double-slit experiment, and Nereo and Yalda had built a whole mountain of wave theory on top of the grave. Patrizia wasn’t to blame for Meconio’s failings, but the name carried too much baggage.

“Let’s call them ‘photons’,” Carla suggested. “Different root, same meaning.”

“If the light-makers are called luxagens, shouldn’t the particles that accompany light share the same root?”

“People might confuse the two,” Carla said. “This will be clearer, trust me.”

Patrizia nodded, indifferent. “The rule is, the photon moves at the speed of the light pulse,” she continued. “But for that to be true, to create light of a certain frequency means creating photons with a certain energy. So if a process generates a particular frequency of light, that imposes a peculiar constraint on the amount of energy involved: you can create one photon, or two, or three… but your choices are confined to whole numbers. You don’t get to make half a photon.”

“Wait!” Carla interjected. “What about the energy in the light wave itself? How is that connected to the energy of these particles?”

Patrizia gave an apologetic hum. “I’m not sure. For now, can we just say that it’s very small? That most of the energy in the light actually belongs to the photons?”

“It’s your theory,” Carla said. “Go ahead.”

Patrizia shifted anxiously on the guide rope. “Suppose the energy valleys for the luxagens in mirrorstone all have a certain depth. The luxagens would have a bit of thermal energy as well, raising them off the floor of the valley, but if that doesn’t vary too much there’ll still be a particular amount of energy that a luxagen would need to gain in order to climb out of the valley and into the void—leaving behind a tarnished surface.”

“That’s a reasonable starting point,” Carla agreed. Nereo’s theory implied that the luxagens should have climbed out of the valley on their own, eons ago—as their thermal vibrations generated light and ever more kinetic energy—but since nobody else had managed to solve the stability problem it would hardly be fair to expect Patrizia to deal with it.

“When you hit the mirrorstone with light of a single frequency,” Patrizia said, “the luxagens vibrate in time with the light, and create light of their own. But creating light means creating photons. Suppose a luxagen creates one photon; that will give it a certain amount of kinetic energy, but it might not be enough to get it out of the valley. Two might not be enough either, or three, but suppose four is sufficient. So once it’s made four photons, the luxagen escapes, the mirrorstone gets tarnished.”

Carla was following her now. “But if the light striking the mirrorstone has a lower frequency, that corresponds to a smaller energy for each photon, and there’ll come a point where it suddenly takes five to do the job. So that’s the first transition point we see in the tarnishing pattern: on one side four photons bridge the energy gap, on the other side you need five.”

Patrizia said, “Yes. All the nonsense I came up with before about the luxagens in the valleys being struck by different numbers of ‘wandering luxagens’ pushed around by the light… that’s all gone! The four and the five in the frequency ratio are just the numbers of photons that have to be created by the luxagens in order to escape from the valley.”

A stint ago, Carla would have called this new version twice as nonsensical as the first. If you could ping a rope as hard or as gently as you liked, making waves in it as strong or as weak as you liked, why should waves in the light field be so different, burdened with these strange restrictions and appurtenances? But if you were willing to treat the frequency of light as a surrogate for the energy of a particle moving at the same speed, Patrizia’s plot through the scattering data brought this hypothetical “photon” to life, showing it behaving in precisely the manner you’d expect when one particle collided with another.

Carla said, “Before we begin praising the genius of Meconio, can you think of any way we can test this idea?”

“I haven’t been able to come up with any wholly new experiment,” Patrizia admitted. “But there’s something in the original experiment that we haven’t measured yet.”

“Go on.”

“The time it takes for each part of the tarnishing pattern to appear.”

Carla could see the merit in looking at that more closely. “If it takes a certain amount of time to create each photon, then the extra time required for successive tiers to reach a given tarnishing density should be the same. We’d need to push it to a longer exposure, though, and get another tier at frequencies so low that it takes six photons to leave a mark.”

Patrizia said, “It might not take the same time to produce a photon at different frequencies. What if the light that’s driving the process has to go through a certain number of cycles?”

“Like… cranking the handle on those mechanical loaf-makers? It’s the number of turns, not the time you spend turning.” Carla had no idea what was required to crank out a photon, so there was no obvious way to decide between the two criteria. “The period of violet light is only one and a half times that of red light; we can make a long enough exposure to test both possibilities, and see if either of them fits the results.”

Patrizia emitted a chirp of delight. “So we’re really going to test this theory?”

“Of course,” Carla replied. “Isn’t that what we’re here for?”

When Patrizia had left, Carla took the groundnuts from the cupboard and went through her ritual. As she savored the odor, she realized that she’d rushed through the discussion too quickly, leaving too many problems unchallenged.

How could a luxagen “know” how long it had been exposed to light? Whether it was meant to be counting cycles of the light or simply recording the passage of time, what physical quantity could play the role of timer? Not the luxagen’s energy, or the jumps in the tarnishing pattern would have been smoothed away. The success of Patrizia’s theory relied on the axiom that you couldn’t make half a photon, but unless something was keeping track of the process—if it could not, in some sense, be half done—then why should it take any particular amount of time to create one of these particles?

The scattering curves were beautiful. The link between energy and frequency was beautiful. But the whole theory still made no sense.

Carla put the groundnuts away, wondering how she was going to persuade Assunto—who doubted the existence of particles of matter—to give her six times as much sunstone as before so she could now go hunting for particles of light.