by Greg Egan
The velocity of light rose as its wavelength fell. Each quantity, then, might merely be proportional to the inverse of the other. If so, multiplying the two of them together would always yield the same result.
Yalda tested this idea for a dozen points across the spectrum. The product varied—by too much to be nothing but the jitters expected from imperfect data.
Still, if the relationship was more complex than her first naïve guess implied, that guess could still take her in the right direction. She drew a second plot, this time setting wavelength against the inverse of the velocity.
Her naïve guess would have required a perfectly straight line here—and chance errors alone would not have seen the points weave so systematically from one side of the line of best fit to the other.
In fact the data looked like a segment of a parabola or hyperbola, a quadratic of some kind. Yalda tried squaring the velocity as well as taking its inverse, but the plot was still plainly curved. She tried squaring the wavelength instead; that was no better.
Then she tried squaring both.
Yalda was too excited to remain still; she left the office and walked around the observatory grounds, wishing she had Tullia or Giorgio beside her to celebrate her discovery. A linear relationship between two squared quantities was neither too simple to believe, nor too messy and complex to be useful. Maybe it was just an approximation to the true relationship, but for now it would be enough of a challenge to take this result as given and see where it led.
Light was a very strange kind of wave. Under ordinary conditions, elastic waves in a string or pressure waves in a gas moved with a fixed velocity regardless of their wavelength. Exotic exceptions could be contrived—but with light, there was nothing exotic about it. The fact that its velocity varied wildly with its hue was the one thing everyone agreed upon: you only had to look up at the stars to be convinced of it.
One consequence of the varying velocity was that a pulse of light was not even expected to move in the same direction as the individual wavefronts within it. Bizarre as that sounded, it had been clear since Giorgio’s first tentative wavelength estimates. Every pulse of light, however apparently pure its color, would contain at least a small spread of different wavelengths. But since the different wavelengths moved at different speeds, the points where they all agreed and reinforced each other wouldn’t drift along merrily with the wavefronts themselves, as they did in a wave on a string. If the slippage in the velocity was great enough, they’d actually travel in the opposite direction.
Yalda summoned onto her skin a sketch she’d made in one of Giorgio’s lectures. With a few simple calculations, Giorgio had convinced her that if she could somehow watch a pulse of light in motion, she’d see the wavefronts within it sliding backward.
What did her own results add to that? She could now construct a more precise account of how these two different aspects of the light behaved. If she chose, say, a pulse of red light, she could plot its movement through space, along with the backsliding wavefronts from which it was built.
She returned to the office to consult her sheets of paper, then she sketched a new diagram on her chest.
As Yalda contemplated the picture, it struck her just how reminiscent it was of a beam of light and its accompanying wavefronts, shown at a single moment in time. The main difference was the annoying tilt between the wavefronts and the “beam”—here, a line showing the history of the pulse.
But what did that skewed angle actually signify? By changing to different units of measurement, she could stretch or squeeze the diagram as much as she liked. Nature had no idea what a pause or a flicker was; nothing real could depend on adhering to that traditional system of units. So she chose units of time that forced the pulse and the wavefronts to trace out lines at right angles to each other.
Where did that leave her? She had a right angle between some lines… and a linear relationship between two quantities squared.
She played around with the diagram for a couple of chimes, changing the units for distance as well as those for time in such a way that the separation between the wavefronts was simply declared equal to one. Well, why not? It wasn’t just time whose units were completely artificial; a scant had once been defined as the resting width of some self-important monarch’s thumb.
When she was done, a small right triangle sat inside a larger one in the same proportions. The hypotenuse of the larger triangle was a horizontal line that joined one wavefront to another, making its length simply equal to the wavelength of the light. The small triangle’s sides—corresponding to a distance the pulse traveled and a time in which it did so—had a ratio of lengths equal to the light’s velocity. The larger triangle shared that ratio, in such a manner that the length of one of its sides was the inverse velocity.
In her chosen units, then: the inverse velocity squared, plus one squared, was equal to the wavelength squared. That simple equation corresponded to the straight line that passed through the data she’d plotted. But now this relationship didn’t need to emerge from any hypothetical properties of the hypothetical medium whose vibrations manifested as light. The sum of the squares of the sides of a right triangle equaled the square of the hypotenuse. That was it: the entire wavelength-velocity relationship that she’d extracted from all those nights of painstaking observations had turned out to be nothing but a theorem from elementary geometry in disguise.
Except… that was nonsense. Geometry was concerned with figures in space, not lines that stretched across time as well. However suggestive these results were of geometry, that could only be an analogy, at best.
Albeit a mathematically perfect one. If she pretended that she really was doing geometry in a plane, she could simply rotate the whole physical structure of the red pulse—rigidly preserving the spacing of the wavefronts—and transform it into a faster, violet pulse.
The wavelength and velocity changed, of course, but those things were just measurements that depended on the way the stack of wavefronts was disposed, relative to the person doing the measuring. The two pulses, red and violet, were no more different in essence than a pulse of light traveling north and another traveling north-east.
The message from the stars was: light is light, always the same on its own terms. Qualities such as color, direction and speed were only meaningful distinctions once the light bumped into something else, against which it could be measured. In the void, it was simply light.
Yalda was feeling disoriented; she walked in a daze to the living quarters and lay down in the bed’s slippery white sand. None of her conclusions made sense; this was just heat shock talking. If she could hallucinate Thero for a whole night, she could lose her powers of reasoning for a day. She’d sleep off her sickness, and everything would be clear in the morning.
Yalda spent the next day re-checking her calculations. All the numbers she’d relied on were correct—and her geometrical constructions were so simple that a five-year-old could have confirmed that they were right.
The only thing she could still doubt was her interpretation. Her right triangle with its wavelength-long hypotenuse might actually be nothing but a useful mnemonic, an easy way of remembering the velocity-wavelength formula. Mathematics that echoed the rules of geometry could arise anywhere, with all the lines and angles that it implied really nothing more than abstractions.
So… light was a vibration in some exotic medium that just happened to possess qualities that perfectly mimicked all the would-be geometry she’d found in the equations? As well as contriving to support shear waves and pressure waves that traveled at exactly the same speed? Was there nothing this magical material couldn’t do?
The three polarizations of light traveled at the same speed, as if they were all the same kind of thing. Yalda brought one of her diagrams of pulses and wavefronts back onto her chest. The picture projected the three dimensions of space down to just one, but in reality each wavefront was a plane, and it traced out a three-dimensional set over time. Within that set there w
ould be three independent directions that were orthogonal to the path of the light pulse through the four dimensions that included time. The three polarizations could all be transverse waves—waves that pointed sideways, in that four-dimensional sense. There’d be no need for a miraculous coincidence to make all their velocities the same.
It was almost dusk. Yalda walked out of the building and sat at the top of the access path. Either she had lost her mind, or she had stumbled upon something that needed to be pursued much further.
She tinkered with the wavefront diagram on her chest. She’d been wondering about the significance of the inner triangle, the triangle whose hypotenuse was one. The ratio of its sides was the light’s velocity, but what exactly did the individual side lengths represent?
A simple argument with proportions established their values—which yielded a new triangular relationship, more elegant and symmetrical than the first: the sum of the squares of light’s frequencies in time and in space would equal one. Well, only her special choice of units set the sum equal to one, but the fact remained that the equivalent in cycles per scant or stride or saunter would still be independent of the color of the light.
That was really no different from saying that the true distance between furrows ploughed by a given plow did not depend on whether someone happened to walk across them askew. The wavefronts of light were all furrows from the same plow; the light’s speed, color, wavelength and frequency simply measured the angle at which you crossed the furrows.
But if light was going to play by these geometric rules, then everything it touched—every system that created or absorbed light, every substance that bent, scattered or distorted it—would have to function the same way. Ultimately, to keep the world consistent, any kind of physics that took place at one angle would have to work just as well if you picked it up and rotated it in four dimensions.
To accommodate light’s simplicity, half of science would need to be rewritten.
Yalda looked up; Sitha was starting to show against the fading gray sky. The colors were still weak, but the trail’s violet tip was as prominent as a skewer-worm’s barb.
“What have you done to me?” she said.
Then she remembered that there was no air between them, and she wrote the words across her chest instead.
6
“If time is exactly the same as space,” Giorgio asked Yalda, “why is it that I can walk to the Great Bridge, but I can’t walk to tomorrow?”
Yalda was distracted by a hubbub of exuberant buzzing and chirping from the adjoining room. In her absence, Giorgio’s co had given birth, and though the children were being cared for by their grandfather during the day, Giorgio couldn’t bear to be separated from them. He’d set up a nursery in the room beside his office.
Yalda focused on his question. “You’re already traveling toward tomorrow, along the most direct route possible. The shortest distance there is a straight line, and by standing still you’re following that line; you can’t do any better than that.”
“That makes sense,” Giorgio conceded. “But if I can’t do better, why can’t I do worse? Why can’t I dawdle and delay, and reach tomorrow later than expected? I can certainly do that if I walk to the Great Bridge.”
“And you can do it on your way to tomorrow,” Yalda replied. “If you cease standing still, if you wander around Zeugma, you will add some time to your journey. But because you can’t move very quickly, you can’t really manage much of a detour. The distance to tomorrow is vastly greater than the distance across Zeugma; the proportion by which you can increase it with any plausible peregrinations is unmeasurably small.”
Giorgio was amused, and she saw him slip out of his role for a moment to marvel openly at the sheer strangeness of these notions. Yalda knew she hadn’t convinced him that her ideas were correct, but he believed nonetheless that it was worth presenting them to the whole school of natural sciences: physicists, mathematicians, chemists and biologists. Before she spoke before so many colleagues, though, Giorgio wanted to be sure that she could defend her ideas against the inevitable barrage of objections, and he was doing his best to prepare her by anticipating every possible question and complaint.
“Exactly how far away is tomorrow?” he asked.
“As far as blue light can travel in a day.”
“Blue light? What’s so special about blue?”
“Absolutely nothing,” Yalda said firmly. “Violet is faster, and I believe there are even faster hues that we can’t perceive. But just as there’s a line in space that lies halfway between right and forward—marking equal progress in those two directions—there’s a line halfway between right and into the future. We perceive the light that reaches us at such an angle to be blue, and if we follow that light for a day, its progress marks out the equivalent distance.”
“I can’t compete with blue light,” Giorgio said, “so I can’t noticeably delay tomorrow. But why can’t I walk to yesterday?”
“For much the same reason,” Yalda replied. “Bending your path around until it’s turned backward would require an immense, sustained acceleration. In principle it ought to be possible, but it’s not something you should expect to be easy. You’re heading toward the future with a lot of inertia; you can nudge your trajectory a little with muscle power or a truck’s engine—but as you said, blue light isn’t easily outpaced.”
“But even if we only imagine it,” Giorgio persisted, “traveling toward the past would be very different from traveling toward the future. Traveling toward the future, we can shatter a stone into pieces with one blow; if we were traveling toward the past, the pieces would rise up and remake the whole before our eyes. Why is that distinction so clear… when directions in space such as north and south can barely be distinguished?”
“The same reason as we always suspected,” Yalda countered. “In the distant past, our part of the cosmos had much lower entropy; whether or not there was a single, primal world, things were certainly more orderly. The direction of increasing entropy looks radically different from the direction in which entropy decreases—but that’s not a fundamental property of space or time, it’s a happenstance of history.”
Giorgio wasn’t satisfied. “Time in either direction looks utterly different from any direction in space.”
“That’s because we’re surrounded by things that are moving almost entirely along that one axis,” Yalda said. “Not because physics decrees that they must move that way, but because they share a common history that has set them on that course. All the histories of all the worlds we can see form an almost straight bundle of lines through the four dimensions. The fastest star we know of is moving at barely one part in a gross of the speed of blue light. Living in a bundle of lines that are all so close to being parallel to each other, we shouldn’t be surprised that their common direction appears special to us.”
Giorgio changed his attack. “You say physics itself doesn’t decree that our histories are almost parallel. So according to your theory, an object could have a trajectory entirely orthogonal to our own?”
“Yes.”
“It could move with an infinite velocity?”
Yalda didn’t flinch. “Yes, that’s how we’d describe it.” It could cross what she and Giorgio thought of as a region of space in no time at all. “But that’s no stranger than saying that a vertical pole has an ‘infinite slope’: unlike a mountain road, it gets where it’s going vertically without bothering to go anywhere horizontally. An object that gets where it’s going without bothering to move across what we call time isn’t doing anything pathological; in reality, there’s nothing ‘infinite’ about it.”
“What about its kinetic energy?” Giorgio demanded. “Half its mass times its velocity squared?”
“That formula’s merely an approximation,” Yalda said. “You can’t use it for anything but small velocities.”
She summoned a diagram onto her skin. “If you want to know an object’s energy and momentum, draw an arrow whose length
is the object’s mass, and point it along the line of its history. If you think the object is motionless, the arrow will point straight along the time axis; if you think it’s moving, the arrow needs to be tilted accordingly.”
“The amount by which the height of the arrow is diminished—compared to the motionless version—is its kinetic energy. For small velocities that will match the old formula, but for higher velocities it will grow much more 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.”