Greg Egan - The Clockwork Rocket

She slid the mask away, expecting to see a tiny, perfect disk of light, but the whole field remained black. She’d bumped the telescope, and it was no longer pointed at Sitha at all.

Yalda found the star again, but it was hard to keep it centered for long without losing some of her dark adaptation. She tried switching to her left eye each time she had to pull away the mask to correct the tracking, then switching back to the right to resume shrinking the halo, but the two eyes were in cahoots, their pupils contracting in tandem even when only one of them was dazzled. Finally, she tried flipping onto her chest and letting one of her rear eyes take the glare. Amazingly, it worked: her front eyes retained their sensitivity.

When Sitha dropped out of reach once more, Yalda realized that for the last three chimes she’d been unable to make any improvements; she’d simply been trying out small adjustments and then undoing them. But the halo was very faint now, and it was unreasonable to expect it to vanish completely. She had gone as far as she could with Sitha.

And she had collected her first set of data.

Yalda woke early and set about translating the positions of the two dozen mirror pegs into a set of wavelength and velocity values. It was a complex calculation; it took her until late afternoon to complete it, double-checking every step. She plotted the points on a sheet of paper that she’d prepared with a grid; there were some tasks that were just too difficult to perform on her own skin.

The data curved down across the upper right corner of the plot: with increasing velocity, the wavelength fell. That general trend wasn’t news, but here at last was a hint at the detailed shape. Yalda contemplated some possibilities for the precise form of the mathematical relationship, but she knew that was premature. She needed to see if other stars gave her the same curve.

Tharak was next, almost as bright as Sitha, though its trail was less than half as long. Zento was faster, more distant. Yalda was learning what worked and what didn’t, acquiring an instinctive sense of the adjustments she needed to make to shrink the colored ellipses down to sharp white disks. On her sixth night of observations, she managed to peg two different stars, Juhla and Mina, before dawn.

Laboriously, she added each star’s points to the plot. That the light velocities she was sampling were clustered together was no great achievement; that simply reflected the fixed positions of the holes in which the adjustment pegs sat. But the corresponding wavelengths weren’t scattered too widely, either. Her method was yielding the same pattern, star after star.

As she ran out of bright targets, the observations became more difficult. After three nights of increasing frustration, Yalda gave up on Thero, unable to distinguish any change in its image despite wildly different settings for the pegs. She wondered if she’d grown sick from exhaustion: if she’d lost Thero’s trail, and was simply hallucinating blotches of light to fill the darkness.

She rested for two days: doing nothing but eating, sleeping, and taking short walks along the access path. Tullia had warned her not to push herself; nobody was immune to heat shock. After her trouble with the ascent she should have been more careful.

She tried a different star, Lepato. It took her all night, but her mind was clear now, and by dawn she’d shaped the mirror to conform to Lepato’s faint trail. Starlight was not as fragile or elusive as it seemed; with enough patience, you could even capture its likeness in stone and wood.

Yalda had been on Mount Peerless for a stint and seven days, and she had data for a dozen stars. It was time to try to make sense of what she’d gathered. Curled up on the observation bench that she’d moved into the office, she contemplated the sweep of the curve across her plot.

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.