Incandescence

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.