Diaspora

"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.