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.