by Greg Egan
It erased the hourglass-shaped tunnel, leaving the two holes disconnected again, then pasted a narrow strip between the left-hand side of the top rim and the right-hand side of the bottom rim. As before, it extended the strip all the way around both circles, always connecting opposite sides of the rims, creating a pair of cones meeting at a point between the wormhole mouths. “This solution has positive mass. In fact, if GR held true at this scale, it would just be a pair of black holes sharing a singularity. Of course, even for the heaviest elementary particles the Schwarzchild radius is far smaller than the Planck-Wheeler length, so quantum uncertainty would disrupt any potential event horizons, and perhaps even smooth away the singularity as well. But I wanted to find a simple, geometrical model underlying that uncertainty.”
“So you expressed it by adding extra dimensions. If Einstein’s equations in four dimensions can’t pin down the structure of space-time on the smallest scale, then every ‘fixed point’ in the classical model must have some extra degrees of freedom.”
“Exactly.” The avatar gestured at the diagram, and it was subtly transformed: the translucent sheet became a mass of tiny bubbles, each one an identical perfect sphere. This was a heavily stylized view — rather like drawing a cylinder as a long line of adjoining circles — but Blanca understood the convention: every point in the diagram, though fixed in the two dimensions of the sheet, was now considered to be free to position itself anywhere on the surface of its own tiny sphere. “The extra space each point can occupy is called the ‘standard fiber’ of the model; it’s not long and fibrous, I know, but the term is a legacy of mathematical history, so we’re stuck with it. I started with a 2-sphere for the standard fiber; I only changed it to a 6-sphere when it became clear that six dimensions were needed to account for all the particles.”
The avatar created a fist-sized sphere floating above the main diagram, and covered it with a palette of colors that varied smoothly over the whole surface. “How does giving every point a 2-sphere to move in get around the singularity? Suppose we approach the center of the wormhole from a certain angle, and let the extra dimensions change like this.” The avatar drew a white line down the sphere from the north pole toward the equator, and a colored line appeared simultaneously on the main diagram: a path leading straight into the top cone of the wormhole. The path’s colors came from the line being sketched on the sphere; they signified the values of the two extra dimensions being assigned to each point.
As the line on the sphere crossed the equator, the path crossed between the two cones. “That would have been the singularity, but in a moment I’ll show you what’s become of it.” The avatar extended the meridian toward the south pole, and the path through the wormhole continued on through the lower cone, and emerged in the bottom region of ordinary space.
“Okay, that’s one geodesic. And in the classical version, all geodesics from one wormhole mouth to the other would converge on the singularity. But now ...” It drew a second meridian on the sphere, starting again from the north pole, but heading for a point on the equator 180 degrees away. This time, the colored path that appeared on the wormhole diagram approached the top mouth from the opposite side.
As before, when the meridian crossed the equator of the sphere, the path through the wormhole crossed between the two cones. Since the tips of the cones only touched at a single point, the second path had to pass through the same point as the first — but the avatar produced a magnifying glass and held it up to that point’s standard fiber for Blanca to see. The tiny sphere had two colored dots on opposite sides of its equator. The two paths never actually collided; the extra dimensions gave them room to avoid each other, even though they converged on the same point of ordinary space.
The avatar gestured at the diagram, and suddenly the whole surface was color-coded for the extra dimensions. Far from the wormhole mouths the space was uniformly white — indicating that the extra dimensions were unconstrained, and there was no way of knowing any point’s position on the standard fiber. Within each cone, though, the space gradually took on a definite hue — red in the top cone, violet in the bottom — and then, close to the meeting point, the color began to vary strikingly with the angle of approach: vivid green on one side of the top cone, sweeping round to magenta 180 degrees away — a pattern that emerged inverted on the cone below, before melding smoothly into the surrounding violet, which in turn faded to white. It was as if every radial path through the wormhole had been lifted “up” out of the plane of this two-dimensional space to a slightly different “height” as it approached, allowing them all to “cross over” at the center without fear of colliding. The only real difference was that the extra-dimensional equivalent of “height above the plane” had to occur in a space that looped back on itself, so that a line rotated through 360 degrees could change “height” smoothly all the way, and still end up exactly where it began.
Blanca gazed at the diagram, trying to see it from a fresh perspective despite the numbing familiarity of the concepts. “And a 6-sphere generates a whole family of particles, because there’s room to avoid the singularity in different ways. But you said you started with a 2-sphere. Do you mean later, when you were working with three-dimensional space?”
“No.” The avatar seemed somewhat bemused by the question. “I started exactly as you see here: with two-dimensional space, and a 2-sphere for the standard fiber.”
“But why a 2-sphere?” Blanca duplicated the diagram, but used a circle as the standard fiber instead of a sphere. Again, no two paths through the wormhole were the same color at the cross-over point; the main difference was that they took on different colors straight from the whiteness of the surrounding space, because there were no “north and south poles” now from which they could spread out. “In two-dimensional space, you only need one extra dimension to avoid the singularity.
“That’s true,” the avatar conceded. “But I used a two-dimensional standard fiber because this wormhole possesses two degrees of freedom. One keeps the geodesics from colliding at the center. The other keeps the two mouths of the wormhole itself apart. If I’d used a circle as the standard fiber, then the distance between the mouths would have been fixed at precisely zero — which would have been an absurd constraint, when the whole point of the model was to mimic quantum uncertainty.”
Blanca felt vis infotrope firing up, frustrated but ever hopeful. They’d reached the heart of the Distance Problem. The exaggerated size of the cones in the diagram was misleading; the gravitational curvature of ordinary space around an elementary particle was negligible, and contributed virtually nothing to the length of the wormhole. It was the way paths through the wormhole coiled around the extra dimensions of the standard fiber that allowed them to be slightly longer than they would have been if the two mouths had simply been glued together, rim to rim.
Or in reality, much more than slightly.
“Two degrees of freedom,” Blanca mused. “The width of the wormhole, and its length. But in your model, each dimension shares those two roles from the start — and if they don’t share them equally it gives nonsensical results.” Blanca had tried distorting the standard fiber to allow for longer wormholes, but that had been a disaster. Stretching the 6-sphere into a 6-ellipsoid of astronomical proportions allowed for hundred-billion-kilometer wormholes like the Forge had produced, but it also implied the existence of “electrons” shaped like pieces of string of astronomical length. And changing the topology of the standard fiber, rather than just its shape, would have destroyed the correspondence between wormhole mouths and particles.
The avatar responded, somewhat defensively, “Maybe I could have done it your way, starting with a circle to keep the geodesics apart. But then I would have had to introduce a second circle to keep the mouths apart — making the standard fiber a 2-torus. If I’d taken that approach, by the time I worked my way up to matching the particle symmetries I would have found myself lumbered with twelve dimensions: six for each purpose. Which would have work
ed just as well, but it would have been twice as extravagant. And after the debacle of string theory, it was hard enough selling anyone on six.”
“I can imagine.” Blanca responded automatically, before ve’d fully absorbed what the avatar had said. A moment later, it hit ver.
Twelve dimensions? Ve’d felt so besieged by the realist backlash that ve’d never even considered doing more than defending Kozuch’s six against the charge of “abstractionism.” Twice as extravagant? It certainly would have been in the twenty-first century, when no one knew how long wormholes really were.
But now?
Blanca shut down the avatar and began a fresh set of calculations. Kozuch herself had never said anything so explicit about higher-dimensional alternatives, but the avatar’s educated guess turned out to be perfectly correct. Just as a 2-torus was the result of expanding every point in a circle into another circle perpendicular to the first, turning every point in a 6-sphere into a 6-sphere in its own right created a 12-torus — and a 12-torus as the standard fiber solved everything. The symmetries of the particles, and the Planck-Wheeler size of their wormhole mouths, could arise from one set of six dimensions; the freedom of the wormholes to take on astronomical lengths could then arise from the remaining six.
If the 12-torus was much larger in the six “length” dimensions than the six “width” ones, the two scales became completely independent, the two roles entirely separate. In fact, the easiest way to picture the new model was to split up the whole four-plus-twelve-dimensional universe in much the same way as the ten-dimensional universe of the original Kozuch Theory — but with three levels, instead of two. The smallest six dimensions played the same role as ever: every point in four-dimensional space-time gained six sub-microscopic degrees of freedom. But the six larger dimensions made more sense if the roles were reversed: instead of a separate six-dimensional “macrosphere” for every point in the four-dimensional universe ... there was a separate four-dimensional universe for every point in a single, vast, six-dimensional macrosphere.
Blanca returned to the avatar’s wormhole diagram. It was easier to interpret now if the space was unfolded and laid flat; it could then be thought of as one slice of many through a small — and hence approximately flat — part of the macrosphere. One slice through a stack of universes. Blanca replaced the single microsphere at the center of the wormhole with a long chain of microspheres arcing from one mouth to the other, stringing together virtual wormholes from the vacuum of adjacent universes. An elementary particle would be stuck with a constant wormhole length, fixed at the moment of its creation, but a traversable wormhole would be free to tunnel its way into detours of arbitrary size. For the femtomouths produced in the Forge, the verdict was clear: they’d stolen enough vacuum from other universes — they’d snaked out far enough into the macrosphere’s extra dimensions — to equalize their lengths with the external distance between their mouths.
Of course, no one in C-Z would believe a word of this; it was abstractionism run riot. These hypothetical “adjacent universes” — let alone the “macrosphere” they comprised in their totality — would always be impossible to observe. Even if a wormhole could be made wide enough for a tiny robot to fly through, looking to the sides would reveal nothing but a distorted image of the robot itself, as light circled the wormhole’s cross-sectional sphere. The other universes, as ever, would remain 90 degrees away from any direction in which it was possible to look, or travel.
Still, the Distance Problem was solved, with a model that merely extended Renata Kozuch’s work, discarding none of her triumphs. Let them try bettering that in Earth C-Z! Neither ve nor Gabriel were running versions there — they’d left behind snapshots only to be run in the unlikely event that the whole Diaspora was wiped out — but Blanca thought it over and reluctantly dispatched a bulletin homeward, summarizing vis results. That was the correct protocol, after all. Never mind if the work was laughed at and forgotten; ve could argue the case in Fomalhaut C-Z, once there was someone awake worth arguing with.
Blanca watched the silver clouds circulating; there was a big quake coming soon, but ve’d lost interest in seismology. And although there were a thousand things yet to be explored in the extended Kozuch model — how the four-dimensional universes that played “standard fiber” to the macrosphere determined its own strange particle physics, for one — ve wanted to save something for Gabriel. They could map that real but unreachable world together, physicist and scape artist, mathematicians both.
Blanca shut down the glassy plain, the orange sky, the clouds. In the darkness, ve built a hierarchy of luminous spheres and set it spinning beside ver. Then ve instructed vis exoself to freeze ver until the moment they arrived at Fomalhaut.
Ve stared into the light, waiting to see the expression on Gabriel’s face when he heard the news.
* * *
Part Four
« ^ »
Yatima glanced hopefully at the star they’d called Weyl. If it wasn’t the last link in the chain, it had to be close. “Eight and a half centuries later, the Diaspora reached Swift. From there, you know as much as I do.”
Paolo said, “Forget Swift. What about Orpheus?”
“Orpheus?”
“Just because your clone didn’t wake there —”
Yatima laughed. “It’s got nothing to do with that. Do you think an ancient, space-faring civilization will want to hear about every last novelty we’ve encountered in our travels?”
Paolo was unswayed. “We wouldn’t be here, if it wasn’t for Orpheus. Orpheus changed everything.”
* * *
10
–
Diaspora
« ^ »
Carter-Zimmerman polis, Earth
55 721 234 801 846 CST
31 December 3999, 23:59:59.000 UT
Waiting to be cloned one thousand times and scattered across ten million cubic light years, Paolo Venetti relaxed in his favorite ceremonial bathtub: a tiered hexagonal pool set in a courtyard of black marble flecked with gold. Paolo wore full traditional anatomy, uncomfortable garb at first, but the warm currents flowing across his back and shoulders slowly eased him into a pleasant torpor. He could have reached the same state in an instant, by decree, but the occasion seemed to demand the complete ritual of verisimilitude, the ornate curlicued longhand of imitation physical cause and effect.
The sky above the courtyard was warm and blue, cloudless and sunless, isotropic. As the moment of Diaspora approached, a small gray lizard darted across the courtyard, claws scrabbling. It halted by the far edge of the pool, and Paolo marveled at the delicate pulse of its breathing, and watched the lizard watching him, until it moved again, disappearing into the surrounding vineyards. The scape was full of birds and insects, rodents and small reptiles — decorative in appearance, but also satisfying a more abstract aesthetic: softening the harsh radial symmetry of the lone observer; anchoring the simulation by perceiving it from a multitude of viewpoints. Ontological guy lines. No one had asked the lizards if they wanted to be cloned, though. They were coming along for the ride, like it or not.
Paolo waited calmly, prepared for every one of half a dozen possible fates.
* * *
11
–
Wang’s Carpets
« ^ »
Carter-Zimmerman polis, Orpheus orbit
65 494 173 543 415 CST
10 September 4309, 17:12:20.569 UT
An invisible bell chimed softly, three times. Paolo laughed, delighted.
One chime would have meant that he was still on Earth: an anticlimax, certainly — but there would have been advantages to compensate for that. Everyone who really mattered to him lived in Carter-Zimmerman, but not all of them had chosen to take part in the Diaspora to the same degree; his Earth-self would have lost no one. Helping to ensure that the thousand ships were safely dispatched would have been satisfying, too. And remaining a member of the Coalition, plugged into the entire global culture in real time, wo
uld have been an attraction in itself.
Two chimes would have meant that this clone of Carter-Zimmerman had reached a planetary system devoid of life. Paolo had run a sophisticated — but non-sentient — self-predictive model before deciding to wake under those conditions. Exploring a handful of alien worlds, however barren, had seemed likely to be an enriching experience for him, with the distinct advantage that the whole endeavor would be untrammeled by the kind of elaborate precautions necessary in the presence of alien life. C-Z’s population would have fallen by more than half, and many of his closest friends would have been absent, but he would have forged new friendships, he was sure.
Four chimes would have signaled the discovery of intelligent aliens. Five, a technological civilization. Six, spacefarers.
Three chimes, though, meant that the scout probes had detected unambiguous signs of life. That was reason enough for jubilation. Up until the moment of the pre-launch cloning — a subjective instant before the chimes had sounded — no reports of even the simplest alien life had reached Earth from the gleisners. There’d been no guarantee that any part of the C-Z Diaspora would find it.