The Eternal Flame by Greg Egan

energy valleys. But it was hard to see how either of these theories could predict a tiered pattern, rather than isolated bands of tarnish confined to a few special resonant frequencies.
    Patrizia’s account certainly explained the tiers—and the analogy she’d used between energy and frequency was so simple and elegant that Carla was ashamed not to have thought of it herself. But for all its virtues, the theory as a whole didn’t quite hang together.
    Carla broached the first problem as gently as she could. “You’re relying on some of the luxagens to get swept up by the light?”
    “Yes,” Patrizia agreed. “Pushed along faster and faster, until they’re moving at the speed of the wave itself.”
    “So how would that look, if we were moving alongside them? If we matched velocities with the luxagen, what would we see?”
    “It would appear motionless to us,” Patrizia replied, puzzled. Wasn’t that obvious?
    “And how would the light wave appear?”
    “Motionless too. Everything’s moving with the same speed.”
    “The light pulse is moving with the same speed,” Carla said, “the way you’ve envisioned this. But the history of a pulse is perpendicular in four-space to the wavefronts. So what are the wavefronts doing?”
    “Oh.” Patrizia lowered her gaze and slumped away from the guide rope. “They’re going backward. And so the luxagen wouldn’t be moving with the pulse—it would be trapped in an energy valley between two wavefronts, moving backward at a completely different speed.”
    Carla said, “Right. The motion of the pulse is not the motion of the wavefronts! It’s an easy mistake to make: I still get confused by the difference sometimes.”
    She sketched the situation Patrizia had described. “For the luxagen to end up moving with a constant velocity, it has to be sitting in an energy valley. In fact, I’d only expect that to happen with a very high intensity light source, otherwise the valley wouldn’t be deep enough. But if it did happen, the luxagen would be motionless with respect to the wavefronts, not the pulse.”

    “I understand,” Patrizia said sadly. She began to move away.
    Carla said, “Wait. There’s another problem here, and it’s worth discussing that as well.”
    Patrizia was mortified now. “Isn’t the first flaw bad enough?”
    “Bear with me for a moment,” Carla suggested. “Sometimes two errors actually cancel each other out.”
    “Only sign errors!” Romolo interjected. Carla raised a hand to hush him, then turned back to Patrizia. “You had four or five mobile luxagens able to cause tarnishing when they reached a threshold in their total energy,” she said. “But if they’re going to strike another luxagen trapped in an energy valley, knocking that luxagen free will require a transfer of kinetic energy equal to the depth of the valley. Kinetic energy goes down as true energy increases, so your original idea, with the mobile luxagens matching the speed of the light pulse, wouldn’t have worked. But if the luxagens are moving with the wavefronts, not the light pulse, everything is reversed: the higher the frequency of the light, the slower the light pulse, but the faster the wavefronts. So luxagens trapped between wavefronts will be moving faster and carrying more kinetic energy when the frequency of the light is higher.”
    Patrizia thought it through. “The trend ends up pointing the right way,” she said, “but the numbers don’t work out any more, do they? The frequencies at which four or five luxagens cross a threshold for kinetic energy won’t be in a five to four ratio.”
    “No,” Carla conceded. “And of course there are other problems that would have to be solved to make this work: you’d have to analyze the collisions in detail to see just how much kinetic energy was transferred from the mobile luxagens to the trapped ones, and also account for any radiation being produced. It’s hard to see how all those effects could