by Greg Egan
The exhibit said, “These diagrams show the pure gravitational potential, which always produces an attractive force.” A disembodied hand appeared and placed a small test particle at the edge of each well, with predictable consequences: both particles fell straight in. “Starting from rest, a collision is unavoidable. But if there’s any sideways motion, that alters the dynamics completely.” The hand placed a particle on the rim of the first well, but this time gave it a flick that sent it into an elliptical orbit around the central weight.
“The best way to see what’s really going on is to follow the body along its orbit.” The surface’s grid pattern began to spin, tracking the particle, and as it did the shape of the well changed dramatically: the center of the funnel inverted into a tall, steep spike, raising the weight above the surrounding surface. “In a rotating reference frame, the centrifugal force for a given amount of angular momentum acts like an inverse-cube repulsion.” Inverse-cube conquered inverse-square for small distances, so centrifugal force won out over gravity near the center; the star or planet from the bottom of the well was now high on a summit. The outer region of the funnel continued to slope down, though, so there was a circular trench around the spike where this initial fall in the surface reversed into a climb.
The patches of floor on which they were standing began to circle the table, tilting as necessary to keep them from overbalancing. Orlando groaned at the gimmick, but seemed amused in spite of himself. They caught up with the rotating reference frame, leaving the particle apparently moving only along a fixed, radial line. It rolled back and forth in the trench, cradled and confined by this hollow in the energy surface, the extremes of its elliptical orbit now revealed as nothing more than the farthest points it could reach as it tried to climb either the central spike or the gentler slope of the outer wall.
When the ride stopped, the exhibit offered them three chances to flick a particle into orbit around the second gravity well. Orlando accepted. The first two particles he launched spiraled down to a collision, and the third went skidding off the rim of the table. He muttered something about wishing he was deaf, dumb, and blind.
The exhibit transformed the surface to show the effect of centrifugal force. The inverse-fourth-power attraction of gravity was stronger than inverse-cube repulsion near the center, so even when the reference frame began to spin, the well remained a well. But further away, centrifugal force took over and turned the downward slope of the approach into an ascent. And where the ascent reversed and the surface plunged, in place of the first well’s circular trench there was a circular ridge. Compared to the three-dimensional universe, the entire potential energy surface was upside-down.
The exhibit spun them around with the reference frame. Then, its disembodied hand moving with them, it placed a particle on the outer slope of the ridge; unsurprisingly, it fell directly away from the center. A second particle, placed on the inner slope, fell straight into the well.
“No stable orbits.” Orlando picked up the particle that was rolling away and tried to balance it precisely on the ridge, but he couldn’t position it accurately enough. Paolo saw a flash of fear in his eyes, but he said wryly, “At least that means no Lacertas. Everything that’s going to fall together would have done it long ago.”
They walked on to the next exhibit, a model of the macrosphere’s cosmological evolution. As matter clumped together under mutual gravitational attraction from the initial quantum fluctuations of the early macrosphere, rotational motion either cut in at some point and blew the condensing gas cloud apart, or the process “crossed over the ridge” and the collapse continued unchecked. Star systems, galaxies, clusters and superclusters, all stabilized by orbital motion, were impossible here. But the fractal distribution of the primordial inhomogeneities meant that the end products of the collapse process had a wide spectrum of masses. Ninety percent of matter ended up in giant black holes, but countless smaller bodies were predicted to form, sufficiently isolated to survive for long periods, including hundreds of trillions with a stability and energy output comparable to stars.
Orlando turned to Paolo. “Stars without planets. So where will the Transmuters be?”
“Orbiting a star, maybe. They could stabilize an orbit with light sails.”
“Built out of what? There’ll be no asteroids to mine. Maybe they created a lot of raw materials with the singularity when they first crossed through, but for anything new they’d have to mine the star itself.”
“That’s not impossible. Or they could live on the surface, if they chose. That’s where any native life is expected to be found.”
Orlando glanced back at the model, which included something like a Hertzsprung-Russell diagram, plotting the evolving distribution of stellar temperatures and luminosities. “I wouldn’t have thought many stars would be cool enough. Except for brown dwarves, and they’d freeze completely in no time at all.”
“You can’t really compare temperatures. We’re used to nuclear reactions being orders of magnitude hotter than chemical ones, making them inimical to biology. But in the macrosphere they both involve similar amounts of energy.”
“Why?” Orlando’s gestalt still betrayed a sense of unease, but he was clearly hooked now.
Paolo gestured at an exhibit further along, beneath a rotating banner reading PARTICLE PHYSICS.
The macrosphere’s four-dimensional standard fiber yielded a much smaller set of fundamental particles than the ordinary universe’s six-dimensional one. In place of six flavors of quarks and six flavors of leptons there was just one of each, plus their antiparticles. There were gluons, gravitons, and photons, but no W or Z bosons, since they mediated the process of quarks changing flavor. Three quarks or three antiquarks together formed a charged “nucleon” or “antinucleon,” similar to an ordinary proton or antiproton, and the sole lepton and its antiparticle were much like an electron and positron, but there was no combination of quarks analogous to a neutron.
Orlando scrutinized the table of particles. “The lepton is still much lighter than the nucleon, the photon still has zero rest mass, and the gluons still act like gluons ... so what shifts the chemical energy closer to the nuclear?”
“You saw what happened with the gravity wells.”
“What’s that got to do with it? Ah. Same thing happens in an atom? Electrostatic attraction also goes from inverse-square to inverse-fourth, so there are no stable orbits?”
“That’s right.”
“Hang on.” Orlando screwed his eyes shut, no doubt dredging ancient memories of his flesher education. “Doesn’t the uncertainty principle keep electrons from crashing into the nucleus? Even if there’s no angular momentum, the attraction of the nucleus can’t squeeze the electron’s wave too tightly, because confining its position just increases its momentum.”
“Yes. But increases it how much? Confining a wave spatially has an inverse effect on the spread of its momentum. Kinetic energy is proportional to the square of momentum, making that inverse-square. So the effective ‘force,’ which is the rate of change of kinetic energy with distance, is inverse-cube.”
Orlando’s face lit up for a moment with the sheer pleasure of understanding. “So in three dimensions, a proton can’t ever make an electron crash, because the uncertainty principle is just as good as centrifugal force. But in five dimensions, that’s not good enough.” He nodded slowly, as if coming to terms with the inevitability of it. “So the lepton’s wave shrinks down to the size of the nucleon. Then what?”
“Once the lepton’s inside the nucleon, it’s only pulled inward by the portion of the charge that’s closer to the center than it is itself, which is roughly proportional to the fifth power of the distance from the center. That means the electrostatic force stops being inverse-fourth-power, and becomes linear. So the energy well isn’t bottomless; outside the nucleon it’s too steep for the lepton to ‘brace itself’ against the sides, the way an electron does in three dimensions, but inside the nucleon the sides curve together and
meet in a paraboloid.”
They moved on to the first chemistry exhibit, which showed the paraboloid bowl at the bottom of the well, with a translucent electric-blue bell-shape superimposed over it: the lepton wave in its lowest-energy, ground state. Orlando reached in and touched it; it flickered into an excited state, breaking apart and deserting the center to form two distinct lobes, one of them color-coded red to indicate an inverted phase. After a few tau the whole wave flashed green, spontaneously emitting a photon, and fell back to its lowest energy level.
“So this is the macrosphere’s equivalent of a hydrogen atom?”
Paolo prodded the wave himself, trying to get it to the next highest level. “More like a cross between a hydrogen atom and a neutron. There are no neutrons in the macrosphere, but a positive nucleon with a negative lepton buried in it to cancel its charge makes a rough imitation of one. Blanca called it a ‘hydron.’ If you try to join two of them together to make a ‘hydron molecule’ you end up with something more like deuterium.” The exhibit, overhearing him, obligingly provided an animated demonstration.
Orlando exhaled heavily. “I don’t know how you can take this so calmly. Do you really trust anyone in C-Z to build an entire working polis according to these rules?”
“Maybe not, but if they get it wrong we won’t even know about it. I can’t see us shipwrecked in the macrosphere with the hardware disintegrating slowly beneath us. It’ll be all or nothing: a working polis, or a cloud of random molecules.”
“You hope. How are they even going to make molecules, if every chemical bond triggers nuclear fusion?”
“Not every bond does. If you throw enough hydrons together, the leptons fill up all the energy levels where they’re confined tightly within the nucleus, so the outermost ones end up protruding sufficiently to be able to bind two atoms together with a respectable separation between the nuclei. You have to fill up the first two levels completely, which takes twelve leptons — so every stable molecule needs to contain a few judiciously placed atoms of number 13 or higher. Atom 27 can form fifteen covalent bonds; it’s the closest thing in the macrosphere to carbon.” The exhibit showed them a three-dimensional shadow of a five-dimensional, sixteen-atom molecule: one atom of 27, joined to fifteen hydrons. Paolo said, “Think of this as a souped-up version of methane. If you knock off any of these hydrons and substitute a side branch, you can build all kinds of elaborate structures.”
Orlando was beginning to look besieged. As he glanced down the hall toward distant speculation on biochemistry and body plans, something caught his eye. “‘U-star polymers.’ What does ‘U-star’ mean?”
Paolo followed his gaze. “That’s just another name for the macrosphere. U is the ordinary universe, and the star is mathematical notation for its ‘dual space’ — that’s a term used for all kinds of role-reversals. The universe and the macrosphere are both ten-dimensional ... but one has six small dimensions and four large, the other has six large and four small. So they’re inside-out versions of each other.” He shrugged. “Maybe it’s a better name. ‘Macrosphere’ captures the difference in size, but that hardly matters; once we’re there, we’ll be operating on roughly the same scale as any comparable lifeform. It’s the fact that the physics has been turned inside-out that will make all the difference.”
Orlando was smiling faintly. Paolo asked, “What?”
“Inside-out. It’s nice to know that’s the official verdict. It’s how I’ve felt about it all along.” He turned to Paolo, his expression suddenly, painfully naked. “I know I’m not flesh and blood. I know I’m software like everyone else. But I still half believe that if anything happened to the polis, I’d be able to walk out of the wreckage into the real world. Because I’ve kept faith with it. Because I still live by its rules.” He glanced down and examined an upturned palm. “In the macrosphere, that will all be gone. Outside will be a world beyond understanding. And inside, I’ll just be one more solipsist, cocooned in delusions.” He looked up and said plainly, “I’m afraid.” He searched Paolo’s face defiantly, as if daring him to claim that a journey through the macrosphere would be no different from a walk through an exotic scape. “But I can’t stay behind. I have to be a part of this.”
Paolo nodded. “Okay.” After a moment he added, “But you’re wrong about one thing.”
“What?”
“A world beyond understanding?” He grimaced. “Where do you get that shit? Nothing is beyond understanding. A hundred more exhibits, and I promise you: you’ll be dreaming in five dimensions.”
* * *
16
–
Duality
« ^ »
Carter-Zimmerman polis, U*
Orlando stood outside the cabin and watched the last visible trace of his universe recede into the distance. The dome of sky above the Floating Island offered a pinhole view of the macrosphere, revealing only two faint stars; the station they’d built beside the singularity appeared just above the western horizon as a tiny, flashing white light, fading rapidly. The singularity itself was invisible at this distance, but the station’s beacon echoed the regular stream of photons emanating from it to mark its position.
If the team on Swift ever stopped creating those photons, the singularity would vanish from sight. A massless anomaly in the vacuum, small as a subatomic particle, it would be almost impossible to find. But then, if no one was sending, no one would be listening either, so there’d be no point scouring the vacuum for the home universe; any data blasted back at the singularity would trigger beta decays in Swift’s neutrons to no avail. Some people had expected the singularity to be surrounded by Transmuter artifacts, but Orlando hadn’t been surprised to find the region abandoned, given the absence of machinery on the other side of the link.
The beacon seemed to dim with unnatural speed, as if the polis was accelerating away wildly. Yet another manifestation of the inverse-fourth law: anything that spread out in all directions thinned more rapidly here. Orlando watched the reassuring pulse of light fade from view, then managed to laugh at his visceral sense of abandonment. It was possible to be stranded anywhere. On Earth, he’d once almost died of exposure less than twenty kilometers from home. Scale meant nothing. Distance meant nothing. They’d either make it back, or they wouldn’t — and nothing this world could do to them could begin to compare to a slow death from cold and dehydration.
He addressed the scape. “Sweep the sky.” At any one moment, the ordinary view from the Island — a mere two-dimensional dome — could only encompass a narrow portion of the macrosphere’s four-dimensional sky. But the hemisphere could be swept across the sky, scanning it like a Flatlander scanning ordinary space by rotating the plane of vis slit-like view. Orlando watched the sparse stars come and go, far fewer than he’d have seen from Atlanta beneath a full moon. Still, it was remarkable that he could make out so many, when they were scattered so widely and their light was spread so thin.
A brilliant rust-red point of light appeared in the east, then faded rapidly as the view swept over it: Poincaré, the nearest star to the singularity, their first target for exploration. It would take forty megatau to reach Poincaré, but no one was tempted to freeze themselves for the journey; there was too much to think about, too much to do.
Orlando braced himself. “Now show me U-star.” His exoself responded to the command, spinning his eyeballs into hyperspheres, rebuilding his retinas as four-dimensional arrays, rewiring his visual cortex, boosting his neural model of the space around him to encompass five dimensions. As the world inside his head expanded, he cried out and closed his eyes, panic-stricken and vertiginous. He’d done this in sixteen dimensions to view the Orphean squid, but that had been a game, a dizzying novelty, like riding a comet or swimming with blood cells, adrenaline-pumping but inconsequential. The macrosphere was no game; it was more real than the Floating Island, more real than his simulated flesh, more real, here and now, than the ruins of Atlanta buried in a distant speck of vacuum. It was the space thro
ugh which the polis sped, the arena in which everything he thought and felt was truly happening.
He opened his eyes.
He could see many more stars at once now, but they seemed more sparsely distributed; there was so much more emptiness to fill. Almost without thinking, he began joining up the dots, sketching simple constellations in his head. There were no striking figures here, no Scorpios or Orions, but a single line between two stars was a thing to be marveled at. His vision now stretched beyond its ordinary field in two orthogonal directions; Paolo’s friend Karpal had suggested calling them quadral and quintal, but with no obvious basis for distinguishing between them Orlando seized on the collective term: the hyperal plane.
Networks in his new visual cortex and spatial map attached a raw perceptual distinction to the hyperal directions, but it still required a conscious effort to make cognitive sense of them. They were definitely not vertical; that realization carried the most immediate force. The direction of gravity, of his body’s major axis, had nothing to do with them; if he was like a Flatlander seeing the world beyond his plane, that plane had always been vertical, and his slit-vision had now spread sideways. But the new directions weren’t lateral, either; unlike a vertical Flatlander, his “sideways” was already occupied. When he consciously divided his visual field into left and right halves, all the purely hyperal pairs of stars lay solely in one half or the other, just like all the purely vertical pairs. And whatever common sense dictated as the only remaining possibility, there was no sense of the sky having gained depth, of the stars looming toward him like a holographic image leaping out of a screen.
Orlando held these three negations in his mind at once. The hyperal plane was clearly defined by his anatomy, so long as he remembered that it was perpendicular to all three of his body’s axes.
