Blanca had originally planned to spend no more than twenty or thirty megatau on the problem, then sleep for the rest of the voyage, satisfied that ve'd struggled long and hard enough to understand exactly how difficult it would be to find a solution. Ve'd guarded against investing too much hope in the prospect of helping Gabriel out of his post-Forge depression, despite fanciful visions of greeting him when he woke with the news that his soul- destroying 'failure' had been transformed into the key to the physics of the next two thousand years. But the fact remained that Renata Kozuch had invented a universe of unsurpassed elegance, ruled by a set of economical and harmonious laws—and the bulletins from Earth were beginning to portray this marvelous creation as some kind of hideous mistake, as disastrous as the Ptolemaic epicycles, as wrong-headed as phlogiston and the aether. Blanca felt that ve owed Kozuch herself a spirited defense.
Ve ran vis Kozuch avatar; an image of the long-dead flesher appeared in the scape beside ver. Kozuch had been a dark-haired woman, shorter than most, sixty-two years old when she'd published her masterpiece—an anomalous age for spectacular achievement in the sciences, in that era. The avatar wasn't sentient, let alone a faithful re-creation of Kozuch's mind; she'd died in the early years of the Introdus, and no one really knew why she'd declined to be scanned. But the software had access to her published views on a wide range of topics, and it could read between the lines to some degree and extract a limited amount of implicit information. Blanca asked, for the thirty-seventh time, 'How long can a wormhole he?'
'Half the circumference of the standard fiber.' The avatar, not unreasonably, injected a hint of impatience into Kozuch's voice. And though it paraphrased inventively, the answer was always the same: about five time ten- to-the-minus-thirty-five meters.
'The standard fiber?' The avatar gave ver something approaching a look of exasperation, but Blanca pleaded stubbornly, 'Remind me.' Ve had to go back to the foundations; ve had to re-examine the model's basic assumptions and find a way to modify them that made sense of the Distance Problem, but left the fundamental symmetries of the wormhole mouths intact.
The avatar relented; in the end it always cooperated, whether Kozuch herself would have or not. 'Let's start with a two-dimensional spacelike slice through a Minkowski universe—flat and static, the simplest possible toy to play with.' It created a translucent rectangle, about a delta long and half a delta wide, then bent it around so that the two halves were parallel, a hand's width apart, one above the other. 'The curvature here means nothing, of course; it's necessary in order to construct the diagram, but physically it has no significance at all.' Blanca nodded, feeling slightly embarrassed; this was like asking Carl Friedrich Gauss to recite multiplication tables.
The avatar cut two small disks out of the diagram. one in the top plane and the other directly beneath it. 'If we want to connect these circles with a wormhole, there are two ways of doing it.' It pasted a thin rectangular strip into the diagram, joining a small part of the top hole's rim to the matching segment of the bottom rim. Then it extended this tentative bridge all the way around both holes, spinning it out into a complete tunnel. The tunnel assumed an hourglass shape, tapering to a waist but never pinching closed. 'According to General Relativity, this solution would appear to have negative energy in some reference frames, especially if it was traversable. The two mouths could still have positive mass, though, so I pursued some tentative quantum-gravity versions of this for a while, but in the end I could never make it work as a model for stable particles.'
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 Schwarzschild 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 worm hole 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
