Incandescence

"On the garmside, the natural path must be more sharply curved than a circle of the same size, because the weight there is directed toward the center of the orbit. On the sardside, the natural path must be less curved than the circle that's actually followed, because the weight is directed outward, away from the center of the orbit."

«That makes sense,» Roi said, «but where does the three come from? Or the two and a quarter?»

«That depends on the precise rule that describes how the natural orbital period varies with the size of the orbit,» Zak replied. «If the period multiplied by itself grows in proportion to the size multiplied by itself, then by itself again, weight should grow three times faster when you move garm or sard than when you move shomal or junub.»

Roi said, «Why choose that rule, though, and not another one? Is it really the simplest one imaginable?»

«I thought there was a way of seeing it that made it perfect,» Zak replied. «So simple it couldn't be otherwise. But now I don't know what to think. The measurements aren't lying, so I'm wrong about something.»

Roi said, «What made it seem so simple? 'The period times the period grows in proportion to the size times the size times the size.' Why is the size taken three times? Why not two? Why not four? Why not five?»

Zak held up the skin and scratched a circle. «This is the orbit of the Splinter. Focus on a small part of the curve, so small it looks like a straight line. That's the Null Line. Now tell me what happens to the direction of that line, as you follow it around one orbit.»

Roi stared at the picture. «It always lies perpendicular to the direction from the Splinter to the center of the circle. Rarb is always perpendicular to garm.»

Zak said, «Yes. But if rarb stays perpendicular to garm, what happens to garm? If you draw a line from the Splinter to the center of the orbit, what happens to that line as the Splinter moves?»

«It moves with the Splinter, it follows it around the circle.»

«Its direction rotates?»

«Yes.»

«So what happens to the Null Line, which is always perpendicular to it?»

Roi tapped her carapace in self-reproach. «The Null Line rotates! With each orbit the Splinter makes, the Null Line itself rotates.»

Zak said, «Yes. So the Splinter doesn't simply travel around this circle, it's also turning as it moves. It rotates around the shomal-junub axis, in exactly the same time it takes to make an orbit. If it wasn't doing that, there would be no Null Line: we would not be weightless anywhere, except at a single point, the center of the Splinter.»

Roi was growing dizzy. First the Splinter was flying through the Incandescence in a giant circle, now it was spinning as it went. «What about the first experiment, though? The one where you spun the tube and the stone gained weight?»

«Yes?» Zak sounded pleased that she'd raised this. «Tell me what that means for the Splinter.»

«We're on the Null Line, but I'm sure we're not precisely at the center of the Splinter. So we're spinning around the center, just like that stone. So why don't we have weight from that spin?»

«I believe we do. But it's balanced by something else.»

«What?»

Zak said, «Suppose we're thirty-six spans rarb along the Null Line from the center of the Splinter. If we're spinning around the center, which way should our weight point?»

«Rarb. Away from the center.»

«But we feel no weight at all. So if you took away the spin, then which way would our weight point?»

«In the opposite direction,» Roi supposed. «Back toward the center of the Splinter.»

«Right. Now, if we were thirty-six spans shomal of the center, which way would our weight point?»

Roi was confused. «Are we spinning now, or not?»

«It makes no difference. The spinning stone had no weight in the direction of its axis. Spinning the Splinter along the shomal-junub axis has no effect on weights in that direction.»

«All right,» Roi said, «then nothing has changed. Our weight would point back toward the center of the Splinter.»

Zak said, «So for any direction other than garm or sard — the directions that take us closer to, or further from, the center of the orbit — our weight would point toward the center of the Splinter itself. And what's more, if you look closely at the calculations, the weight at any given distance from the center of the Splinter would be equally strong, whether you traveled rarb, sharq, shomal or junub. Shomal-junub weight depends on the time for the Splinter to make one orbit, and your distance from the Null Line, in exactly the same way as the weight from spinning depends on time and distance. So if you took away the spin, shomal-junub weight and rarb-sharq weight would be exactly the same.»

Roi said, «Where does that leave the garm-sard weight?»

«You tell me. If the spin was absent, would the garm-sard weight be less or more?»