The Clockwork Rocket

slowly. The object’s momentum is the distance across space that the arrow spans; again, that agrees with the old formula if the object is moving slowly.”

Giorgio pretended that he hadn’t seen the picture before. “What’s this ‘true energy’?”

“The natural measure of energy is the height of the arrow in the time direction,” Yalda explained. “That way, energy is related to time in the same way momentum is related to space. Kinetic energy is a derived, secondary quantity.”

“But ‘true energy’ becomes less when you tip the arrow over,” Giorgio noted. “So when something moves… you’re now declaring that its energy is decreased?”

Yalda said, “Yes. Nothing else makes sense.”

Giorgio’s eyes widened in admiration at her effrontery. “So your theory turns the last three ages’ worth of science on its head. I suppose you’re also claiming that potential energy is upside-down in the same fashion?”

“Of course! We defined it to agree with kinetic energy, so it has the same relation to true energy.” Yalda summoned a picture of two springs accompanied by appropriate mass-length arrows: their four-dimensional momenta. “When the springs are compressed and motionless, we say they have a high potential energy. Now release them, let them fly apart, and see how things add up.”

“For true energy to be conserved, the heights of the pairs of arrows have to be identical before and after the release. But the arrows after the release are tilted, because the springs are now in motion. So those later arrows need to be longer, in order to reach the same height. That means that each relaxed spring has a slightly larger mass than it had when it was compressed—and from the point of view of someone traveling alongside it, a larger true energy. Less potential energy means more true energy. Both the old energies are upside-down.”

Giorgio let a hint of pained, Ludovico-esque weariness into his voice. “If kinetic and potential energy still agree, what can it actually mean to claim that they’re ‘upside-down’? Upside-down compared to what? When do we get to see any of this so-called true energy, to compare its direction with its alleged opposites?”

“In light,” Yalda said. “We see the direction of true energy every time we create light.”

She drew a simple diagram, line by line. “The chemists,” she said, “have been having a lot of trouble with their ladder of energies. If we’re to believe their calculations, the difference in chemical energy between fuel and the gas it becomes after burning isn’t anywhere near enough to account for the thermal energy of the gas. We kept telling them that they’d made a mistake, and that they should improve the accuracy of their measurements. But they were right, and we were wrong. The fuel itself doesn’t need to provide the energy to heat the gas… because that energy comes from the creation of light.

“Light brings its own four-dimensional momentum into the equation. It’s the need to balance that that forces the gas particles to be moving so fast. We thought that when fuel was burned, the light and the heat that was created both came from the release of chemical energy—but the truth is nothing like that! Light energy and thermal energy are opposites: creating one is what gives us the other.

“And we thought that when plants made food from soil, the light was merely an unintended by-product, a measure of inefficiency. But the energy in food isn’t extracted from the soil, and the light shining from a flower’s petals is not wasted energy escaping. Light energy and the chemical energy in food are opposites, too. If plants didn’t make light, they’d have no energy source at all.”

Yalda paused to give Giorgio a chance to respond, but he remained silent. Whatever radical notions she was proposing for the foundations of physics, these claims about food and fuel were the most shocking: the least abstract, the most tangible.

“Why can’t we cool our bodies by emitting light?” Yalda continued. “That’s what I asked myself on my way up Mount Peerless. But now it’s obvious! Emitting light can only give you more thermal energy than you started with. The very act of emitting too much light can make a living body as hot as burning sunstone.” Her grandfather’s frail body had never held enough energy to flatten a forest; rather, it had lost control of its production of light.

Giorgio said, “If emitting light generates thermal energy… why can’t we cool down by absorbing light instead? Why isn’t sunlight as good as our beds for making us cooler?”

Yalda was prepared for that. “Entropy. Light carries a certain amount of entropy—so if you absorb light, your entropy must increase. But if we cool down, our entropy decreases. What I think happens when sunlight strikes our body is that we don’t absorb it, we just scatter it. That way, we can simply take a share of its kinetic energy, and be warmed by it.”

Giorgio stopped the interrogation to take stock. “Well, you’ll certainly please the chemists,” he said. “If you’re right about this, they’ll build a statue in your honor. And the biologists will be intrigued by your ideas on energetics, even if half of them think you’re insane. There’s even something to make Ludovico happy.”

Yalda doubted that, though she knew what he meant. A wave traveling through any ordinary medium marked an increase in kinetic and potential energy, not true energy. If creating light required true energy, it could not be a ripple in some pre-existing medium; it had to be a whole new substance or entity that was created afresh in every flame. But if that brought the term “luminous corpuscles” to mind, in Yalda’s scheme light still had a wavelength—so Ludovico would call this arrogance and hypocrisy, not a triumph for his beloved Meconio.

“Now a question from the mathematicians,” Giorgio said. “You’ve shown us equations for the geometry of wavefronts, but what about an equation for the wave itself—something analogous to the wave equation on a string?”

“The geometry gives us that, easily,” Yalda replied. “For a simple wave, the sum of the squares of the frequencies in all four dimensions equals a constant. But we also know that the wave’s second rate of change in each direction will be the original wave multiplied by a negative factor proportional to the frequency squared.”

She sketched some examples, showing how doubling the frequency of a wave quadrupled its second rate of change. The square of the frequency and the second rate of change were just two ways of talking about the same thing.

“So if you sum the second rates of change of the wave along each of the four dimensions, and negate that sum, then you’ve got the original wave multiplied by a constant times the sum of the squares of the frequencies— which itself is a constant. And that’s the equation for a light wave: the sum of its second rates of change, negated, must equal a constant times the original wave.”

Giorgio contemplated this for a pause or two, then responded with a sketch of his own.

“An oscillation’s second rate of change is proportional to the opposite of the original wave,” he said. “But an exponential growth curve has a second rate of change proportional to the wave itself—there’s no negation.”

“That’s true,” Yalda said. “But—”

“If you construct a wave that oscillates rapidly as you move in one direction,” Giorgio said, “what’s to stop you from choosing a frequency in that direction so large that its square, alone, is greater than the number you’re aiming for as the sum of all four?”

“But then you’ve overshot the total,” Yalda protested. “So you won’t be able to satisfy the equation.”

“No? What if one of the other terms is negative?”