by Greg Egan
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 yield
ing 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."
13
Toward the center of the Nuclear Stellar Disk the density of stars began a precipitous climb. Within a cluster two hundred light years wide a billion stars sped along a complex tangle of orbits, and the deeper into this swarm you dived the more crowded and violent it became. To Rakesh, it brought to mind the image of a nest of furious ants caught in a steep subsidence, kept from falling into the depths only by the sheer energy of their motion.
At the bottom of the pit lay Goudal-e-Markaz: a black hole with the mass of three million suns, the one place from which you could fall no further. It wasn't easy to reach this nadir: the hole's zone of capture was barely fifty million kilometers wide, and it was rare for a star to lose so much of its angular momentum that it could execute a head-on dive into oblivion.
However, a bull's-eye hit was not the only route to destruction. Once every hundred millennia or so a star would come close enough to Goudal-e-Markaz for tidal forces to disrupt it catastrophically. As it dived toward the hole, the star would be stretched along its orbit at the same time as it was squeezed in the orthogonal directions, a distended streak of nuclear fire growing ever hotter and more compressed. In some encounters the star would merely be torn apart and the debris sprayed across a range of orbits, but if the tidal compression was strong enough to trigger a burst of new fusion reactions, as the star swung away from the hole and the pressure was released it could explode with the force of a hundred supernovas. The remnants of these explosions could still be seen thousands of years later, tenuous but energetic shells of gas spreading out into the galactic nucleus.
Ordinary supernovas were even more common, of course, and the central cluster was littered with their remnants: white dwarfs, stellar mass black holes, and neutron stars. The Aloof's map showed no less than fifteen million neutron stars. That was a daunting census, and the chaotic dynamics of the region made it impossible to rule out more than a few per cent as potential culprits in the death of the Arkmakers' world.
Standing in the control room of Lahl's Promise, looking out into the blaze of stars that hid their quarry, Rakesh asked Parantham, "Would you be willing to visit fifteen million neutron stars, one by one, until we found a living Ark?"
She replied without hesitation, "Absolutely."
For a moment Rakesh considered calling her bluff, but he was sure his own will would crack long before hers. When the creators of de novos chose their traits, they were prone to excesses that rarely appeared in even the most technologically augmented versions of inheritance. No gene for keenness could ever compete with Parantham's fiat-driven personality.
He said, "I think it's time we built a decent telescope."
She nodded assent, without betraying the slightest hint of relief that he hadn't been serious about inspecting each candidate in person. "Where?"
"Here would be as good a place as any," Rakesh suggested. "At least there's plenty of raw material." They could try to select an observation point even closer to the galactic center, but their chances of finding a closer star that still clung on to a substantial asteroid belt weren't good.
"That's fitting," Parantham said. "To use a little of the Arkmakers' world, in order to find their new home. I don't think they'd begrudge us that."
Rakesh felt his now habitual twinge of discomfort, at the thought that they might be risking an act of desecration. But there seemed to be nobody around to be offended, apart from the Aloof, who he was sure were looking over their shoulders constantly, ready to veto any unacceptable behavior.
From this distance, a six-hundred-meter Ark somewhere in the central cluster would in principle be resolvable with an optical telescope four million kilometers wide, but to obtain a clear spectrum that would unambiguously identify the wall material it would be prudent to aim a bit higher.
"Ten million kilometers?" he suggested.
"That sounds about right."
Rakesh plucked some standard designs from the library and tweaked them for their specific goals and the local conditions. They could mine the rubble of the Arkmakers' world for the raw materials, then stream the refined feedstock into an orbit clear of dust and micrometeors, where the mirror segments would be constructed. With sunlight and the stellar winds as their main energy sources the project would proceed at a leisurely pace, taking more than a year to complete. Still, that was nothing to the time they'd already spent in transit within the bulge, let alone the time it would have taken to hop from neutron star to neutron star.
Rakesh initiated the process, sending twelve delivery modules into the asteroid belt to sow the rocks with engineering spores.
"No veto from the Aloof," he noted. "They won't allow spores in from the outside, but they don't mind us spreading these ones around."
"Which either means that they trust us," Parantham replied — making it clear from her tone what she thought of that proposition—"or they're watching us so closely that they know exactly what these spores will and won't do."
Rakesh mused
, "What if we tried to build a node here? Conforming to the Amalgam's standards? We'd have our very own short cut through the bulge."
Parantham responded cautiously, "I'd say we're too far from the edge to establish a reliable link with the Amalgam's network."
"Perhaps," Rakesh conceded. "That's not the point, though. What if we tried it?"
Parantham said, "I think the moment we so much as formed a serious intention to contravene the rules they've spent the last million years enforcing, our hosts would turn our bodies into dust and feed it to Goudal-e-Markaz. We're here on their sufferance. We shouldn't even think about pushing our luck." She smiled. "Do you dream, Rakesh?"
"Yes."
"Then draw up a list of topics for your dream censor. We wouldn't want the Aloof getting the wrong idea."
Rakesh followed one of the mirror spiders, hovering beside it as it drank from the feedstock streaming down from the asteroid belt. Though the milky flow resembled a liquid, it was actually composed of tiny granules, each one consisting of a core of volatiles wrapped in a distinctive mineral shell that both protected it and labeled its contents. The spider's acute vision and nimble mouthparts allowed it to extract exactly what it needed, leaving the rest for other users further downstream.
Once the sac of its belly was full, the spider launched itself back toward its web with a barely visible burst of ions. It had already constructed a rigid, skeletal frame for the mirror segment it was building. Each segment was a kilometer wide, but even when the planned ten billion segments were completed they would only fill one ten-thousandth of the eighty trillion square kilometers spanned by the telescope's total width. Seen as a whole, the mirror would be mostly empty space, but the individual segments lost none of their light-gathering power by being spread out like this, and it increased the resolution of the instrument by a factor of a hundred.