Incandescence

12

As the work team gathered in the Calculation Chamber, Roi caught sight of Neth and proclaimed hopefully, “Sixth time brings success!”

“Sixth?” Neth replied. “Surely this is the third?”

“It’s one task to frame a hypothesis, then another to test it,” Roi insisted. “So that’s six separate acts.”

Neth was too polite to object, and perhaps too serious to understand that Roi was only joking. If the proverb was worth anything, it certainly wasn’t worth taking literally. It did encourage persistence, though, and Roi had a feeling that their persistence was finally going to be rewarded.

Since Neth’s discovery that orbits around the Hub might become unstable, a dozen or so members of Zak’s original team had left to educate hatchlings into the secrets of weight and motion, and a dozen more had headed for the sardside, with the even more ambitious aim of recruiting a new team to build Bard’s tunnel. The task of those who remained was to find a geometry for space and time that satisfied Zak’s principle, in the hope of learning more about the dangers the Splinter would face in the future.

Tan had refined his ideas for characterizing geometry to the point where he could calculate the natural paths—the closest things to straight lines—on any curved surface. The vital step that remained, though, was to find the correct way to move from the geometry of space alone to a version that included time.

When Tan analyzed a path on a curved surface, he broke it up into a multitude of tiny, straight line segments of equal length. These small straight lines acted as markers for the direction of the curve. The geometry of the surface could then be embodied in a simple mathematical rule that Tan called a “connection”. The connection allowed you to take a direction at one point and shift it to another, nearby point, in a manner that respected the geometry of the surface. If a curve was a natural path, then when you broke it up into line segments and used the connection to shift them all one step forward, the shifted segments would coincide with the originals: shifting the first segment one step along the curve would give you the original direction of the second segment, and so on. If the curve was not a natural path, then the directions would fail to agree, and the resulting discrepancies would be a measure of how much the curve swerved unnecessarily, as opposed to merely following the geometry.

That the curves were broken into line segments of equal length was a crucial part of the recipe, because the analysis had to yield the same verdict if the surface was picked up and rotated, or if two people were viewing it from different angles. If you decreed that the curve should be broken up some other way, such as into segments that spanned equal horizontal distances, then different people would be left arguing over which direction was “horizontal”. Nobody would argue as to whether two successive segments were of equal length. With the connection respecting this rule—preserving the lengths of the segments as it moved them from point to point—everything worked smoothly, and everyone agreed on which paths constituted natural motion and which did not.

What happened, though, when you considered the path of a tossed stone, moving forward in time as well as through space? Anyone could draw a picture in which some chosen direction represented time, and the path of a moving object slanted across the skin, but how could people ever agree on the correct scale for such a diagram? Whether one heartbeat, one shift, or one lifetime passed from the top of the skin to the bottom was a completely arbitrary choice.

Nevertheless, suppose you settled on a scale. What would happen if you divided the path of a stone into segments of equal length? To Roi, who tossed a stone forward across the Null Chamber at one span per heartbeat, the path she drew would slant across the skin. If Zak happened already to be moving at the same pace in the same direction, the stone would be motionless to him, so he would draw a line that stretched solely in the time direction. Suppose that after five heartbeats, the stone hit an obstacle. Zak’s line would be “five heartbeats” long, whatever the scale of the picture made that. Roi’s line, though, would have to be longer: it would stretch five heartbeats in the time direction, but it would also cross five spans of space. The accounts of the two experimenters had to be compatible somehow, but they couldn’t expect to draw their separate diagrams and then measure the same path lengths.

What could they agree on? The simplest answer anyone in the team had been able to suggest was the time that had elapsed. If you marked off segments of the stone’s path representing equal intervals of elapsed time, everyone would agree how many segments there were from start to finish. If you looked for a connection that respected this scheme by never changing the amount of time spanned by a segment, then everything would have a chance to work smoothly.

This was what the team had tried first. They had hunted for a geometry of space and time whose connection left intervals of time unchanged, and which obeyed Zak’s principle.

In less than one shift, they had found one. In this geometry, everything was symmetrical about a special point, where the Hub could sit. The natural paths of the geometry included circular orbits around the Hub. The square of the period of each such orbit was proportional to the cube of its size. And the ratio between the garm-sard weight and the shomal-junub weight was precisely three. Close to the Hub, far from the Hub, always, everywhere, three.

It was the answer that Zak had guessed long ago, when he’d thought the Map of Weights might still hold true. It possessed an elegant simplicity, but it was impossible to reconcile with the measurements they had made. The current ratio of weights was two and a quarter; that had been confirmed a dozen times.

This failure had cast some doubt on the idea that natural motion could be described by the same kind of geometrical principles that applied to space alone. The team had considered looking for a completely new direction, but the consensus had been that they shouldn’t give up on Tan’s ideas so easily.

Was there any other rule that the connection could obey that might make sense? Could the idea of “constant length” that worked so well in space alone somehow be applied in the new context, in spite of the obvious problems?

It was Neth who had pointed out that if you drew a space-time diagram with an outrageously large scale for the time axis—thirty-six times thirty-six spans for one heartbeat, say—then the different points of view of people moving with mildly different velocities could be mimicked quite accurately by the very slight rotations of the picture that would be needed to make their own particular paths point purely in the time direction. The problem remained that if lengths on this diagram were taken as fixed, two people moving with different velocities would consider each other’s hearts to be beating faster than if their motion was the same, since a line that was “one heartbeat long” would span a smaller interval of time, and seem to pass more quickly, if it was slanted away from the time direction of the person who was measuring it. In reality, though, if the scale was large enough then the effect would be so tiny as to be impossible to measure. Who was to say that this wasn’t happening?

It was an audacious hypothesis, but nobody had any better ideas. The team had labored for five shifts to find a geometry in accord with it. Their success, when it came, had been a mixed blessing, but nevertheless it had convinced Roi that they were on the right track.

The second geometry, like the first, was symmetrical about one special point, and allowed for circular orbits. Far from the Hub, the periods of these orbits were approximated by the old square-cube rule, but for smaller orbits the approximation broke down, and the periods became longer than that rule implied.

As a consequence, the ratio of garm-sard weight to shomal-junub weight was no longer fixed at three. It started out close to three for orbits far from the Hub, which was promising; the problem was, as you approached the Hub the ratio became larger, not smaller. The ratio was greater than three, everywhere, and the two and a quarter they had measured was nowhere to be found in this geometry.

The team had spent a further six shifts checking and rechecking their results. A single error anywhere in their calculations might have thrown the orbital periods and the weight ratios in the wrong direction. There was no error, though. The geometry they had found followed Zak’s principle—that the sum of the true weights without spin was zero—and its connection respected Neth’s idea that different people’s space-time diagrams of moving objects should agree on the lengths of their paths. It was more beautiful, Roi thought, than the simpler geometry they’d found before; it certainly offered richer possibilities. But it did not describe the reality of the Splinter and the Hub.

As Roi had scrutinized the calculations, checking for some tiny, subtle mistake, an idea almost as outrageous as Neth’s original hypothesis had occurred to her. Among other possibilities, they were hunting for a sign error: an addition in place of a subtraction, or vice versa. A mistake like that could easily be the cause of the problem. If there was no sign error in the calculation, though, might there not be one in the hypothesis itself?

Neth had supposed that the length in space-time that everyone agreed on obeyed the same rules as a length in space alone. The square of a length in space was the sum of the squares of its components in three different directions: garm-sard, shomal-junub, rarb-sharq. Neth had simply added in the square of the time component, after it had been multiplied by the scale factor that converted time to distance.

Why add the square of the time, though? Such perfect symmetry suggested that time was exactly like space, that apart from units of measurement the two things were indistinguishable. It was clear to Roi that time was different: you could walk back and forth along the garm-sard axis as often as you liked, but you could hardly do the same between future and past. If the first scheme they’d used to deal with time had set it too much apart, declaring it absolute, universal and immutable, perhaps their second attempt had gone too far in the other direction.

As a compromise, what if they looked for geometries whose connection preserved a slightly different quantity than Neth had suggested: instead of summing the squares of all the components, what if they summed the spatial ones then subtracted the time?

The team had debated the merits of Roi’s proposal for more than half a shift. Many people had complained that it seemed arbitrary and ugly. Gul had pointed out that any object was motionless from its own perspective, so the “length” of its path for one heartbeat would be zero spans squared, minus one-heartbeat-converted-to-spans, squared: a negative number. But if, from another point of view, the object happened to be moving faster than the speed defined by Neth’s spans-per-heartbeat scale, then whoever saw it moving that quickly would ascribe a positive length to its path. How could these two facts be reconciled, when the path length had to be preserved?

“Perhaps,” Tan had suggested, “nothing can ever be seen to move faster than this speed.”

“Then what happens,” Gul had countered, “when I’m moving shomal at three quarters of this speed, compared to the rock of the Splinter, and you’re moving just as fast junub? How fast do you think I’m moving?”

Tan had retreated into calculations, then emerged with an answer. “We each measure the other to be traveling at twenty-four parts in twenty-five of the critical speed. You can’t simply add velocities in this scheme, the way you could in the first one.”

Reflecting on this, Gul had not abandoned his misgivings completely, but he’d mused, “Then in principle the critical speed might be observable. It would not just be some magic large number that we choose for convenience, to turn time into space and make the mathematics work.”

In the end, the team had agreed to test Roi’s scheme at the start of the next shift. If it failed, as the others had, then they would move away from Tan’s geometrical ideas and begin searching for an entirely new theory of motion.

Twenty-six people had gathered in the Calculation Chamber. Roi had looked in on Zak on her way; he’d offered her encouragement, but he’d been too tired to come and observe, let alone participate.

By consensus, Roi and Tan had been appointed lead calculators for this session. They would work independently of each other, while the remainder of the team, split in two, would act as their checkers. Only if both groups reached the same answer would it be trusted.

To save scratching out mathematical templates on skin, a wasteful and physically tiring process, Gul had devised an ingenious system for representing and manipulating templates by sliding stones around on a wire frame. It had taken Roi many shifts to master the system, but now she couldn’t imagine working any other way. When each frame full of templates was completed, she copied the last template to a new frame, then passed the full frame to the first of the checkers.

The team had calculated and recalculated the consequences of Neth’s idea many times, and the new templates had a very similar structure, so Roi made rapid progress, and each time she glanced around the chamber the checkers seemed to be keeping pace with her. The familiarity of the calculations also brought its perils, though; with the old version still fresh in their minds, the minor variations that Roi was introducing looked “wrong”, like small mistakes that needed correcting. Several times Roi caught herself nearly reverting to the old templates.

She reached a template describing a connection that respected the new definition of space-time length, and whose geometry was symmetrical about the Hub. That she had come this far without any new problems emerging was an encouraging sign, but as yet it told her nothing concrete, because everything was still expressed in terms of two unknown templates that remained to be found.

Roi used the connection to analyze the possible circular motions around the geometry’s central point. In space-time, circular motion became a helix, constantly advancing in time as it wound its way around the Hub. Only if the pitch of this helix was correct would the connection declare that it was natural motion: the path of a weightless, free-falling body.

Given the shape of a helix that constituted natural motion, she could find the period of any circular orbit. Since the geometry was symmetrical about the Hub, the period depended only on the size of the orbit, and two stones following two identically sized orbits inclined at a slight angle to each other would come together and move apart with exactly the same period as the orbit itself. In other words, she now knew the period of the shomal-junub cycle, and from that the shomal-junub weight.

Next, Roi calculated how the connection carried directions in space along the helix of the Splinter’s orbit. The speed at which the garm or sharq direction was turning—relative to the frame of the Rotator—gave the strength of the hidden spin weight which canceled the rarb-sharq weight.

For the third weight, she considered a point tied to the Splinter, but displaced from its center in the garm or sard direction. Such a point would follow a helix in space-time that would wind around the Hub with the same period as the center of the Splinter, but unlike the center its path would not constitute natural motion. The connection told her the weight it would feel; this included the spin weight, but she could subtract that from it easily enough to obtain the true garm-sard weight.

Roi summed the weights that arose in all three directions, and imposed Zak’s principle, requiring the sum to be zero.

This imposed a relationship between the two unknown templates, but it was still not enough to determine them individually. So Roi carried out a similar calculation for a different set of circumstances, summing the weights that would have been felt within the Splinter had it been plunging straight toward the Hub, rather than circling it. Again, Zak’s principle required the sum to be zero.

Now she could finally solve for the unknown templates, making all of her previously abstract results concrete.

The chamber was almost silent, save for the clicking of stones. Roi had been afraid that her “sign error” would simply cancel itself out, but it seemed to be propagating nicely, spreading its subtle changes throughout the calculations.

She found the template for the orbital periods. They obeyed the square-cube rule approximately, but closer to the Hub they now grew shorter than that simple rule implied, which was the opposite behavior to Neth’s scheme.

She calculated the ratio of the weights. It started out close to three, but on approaching the Hub it now grew smaller, not greater.

Roi passed the frame to Neth for checking and let the final template float in her mind. The ratio of the garm-sard weight to the shomal-junub weight was equal to three, minus six divided by the size of the orbit. That size was measured in units that couldn’t be determined, because among other things it depended on Neth’s unknown scale for converting time into distance. A small enough orbit would certainly have a ratio of two and a quarter, though; all you needed to do was to make its size “eight units”, whatever that came to in spans.

The same template made it equally clear that for an orbit of “six units”, the ratio would shrink further, to a value of two. The garm-sard weight, then, would be double the shomal-junub weight.

Zak’s principle, and the fact that the spin weight canceled the rarb-sharq weight exactly, made it easy to calculate another ratio: that of the garm-sard weight to the spin weight. When the first ratio fell to two, the second would rise to four. And when the second ratio hit four, as Neth had shown, orbits around the Hub would no longer be stable.

If the Splinter’s orbit was ever reduced to three-quarters of its present size, from that moment on the slightest garmwards nudge would send it spiraling into the Hub.

“No errors,” proclaimed Ruz, the last of Roi’s checkers. Shortly afterward, Kal gave the same verdict for Tan’s calculations. Their conclusions, reached separately, were brought together and compared.

“No discrepancies,” Ruz declared.

Everyone in the chamber, Roi knew, would understand what the results implied, but for a dozen heartbeats there was silence. Roi herself felt wrapped in a kind of protective skepticism: just because their esoteric calculations had finally been prodded into yielding some ratios that weren’t at odds with measurements in the real world, that didn’t make the whole theory correct.

Ruz was the first to speak. “Orbits of size six are unstable, but I’m puzzled by what happens even closer to the Hub. It looks as if orbits of size three or less are impossible, because the object would need to be moving faster than the critical speed.”

“At size two,” Gul added, “everything becomes peculiar. The direction in space that points in toward the Hub acquires a negative space-time length, as if it were a direction in time.”

Tan said, “There are many things here that we’ll need to study more carefully. And even if this geometry is correct, we haven’t shown that it’s unique. There might be other geometries that satisfy Zak’s principle and also give the kind of ratios we’ve seen.”

Roi was relieved; she couldn’t argue with any of Tan’s observations. They had made progress of a kind, but it was not yet proof of anything.

The shift wasn’t over but the calculations had been tiring, so some people started moving out of the chamber to eat. Roi joined them; she’d promised Zak that she’d tell him the outcome as soon as possible, and she could gather some food for him along the way.

As she was leaving, Neth approached her. “Sixth time successful,” she said. “Congratulations.”

“Congratulations yourself. It was your idea, I just twisted it slightly.”

Neth made a sound expressing modesty and suggesting that they desist from further mutual praise. “I’ve enjoyed working with you, Roi. I hope we meet again.”

“What do you mean?” Roi froze and stared at her, bewildered. How could another team have recruited one of their best workers, right in front of their eyes? “What have you become, a courier?”

“I’m not leaving the team,” Neth said. “Not as I understand it. I’m going to the sardside, to help with the tunnel.”

Roi chirped polite approval, but in truth she found this even more shocking. While surrounded by her colleagues and immersed in their distinctive brand of work, Neth had summoned up the strength for a solo, self-motivated defection; even if she chose to think of the tunnel builders as part of the same team, it was the theorists who had first captured her loyalty, and who reinforced it shift after shift. “You’re our best calculator,” Roi lamented. “The best at manipulating templates. The best at understanding what they mean.”

“That won’t be wasted with the tunnel builders,” Neth replied. “They’ll need mathematicians there, too.”

“Aren’t you still curious, though,” Roi pressed her, “to see where the theory takes us? To see our ideas refined? To see this geometry understood, to know if it’s unique, to map out all its implications?”

Neth hesitated. “Of course I’m still curious. And I hope that the next time we meet, you’ll have news about all of those things. But the tunnel is more important to me now. Twice, we’ve seen the possibility of danger. Twice, the mathematics has failed to rule it out. We haven’t proved anything, but what we’ve glimpsed in the distance is enough of a warning for me. I’m not prepared to wait for a disaster to show that we were right.”