<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Buffoil wrote:
But how? Consider the reflective spherical segments facing one another along the center line between the two spheres as infinitesimal planar reflecting surfaces. Better yet, to invoke the crude classic model, consider them as points on directly opposite sides of a pool table. If a pool ball ("photon") follows the nominal rule of incident deflection it would seem that there is no way for it to "get inside" the line connecting those two points unless a player ("light source") intentionally places a ball there and then shoots it so as to bounce back and forth along that line. What am I missing?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The pool analogy works for me---bearing in mind that CG's are perfectly elastic balls. The way to get your CG (cue) ball on the line between two MI balls is to use both balls in multiple combinations. Given adequate separation between the MI balls, the CG ball can bounce almost directly along the center line on the second bounce. On subsequent bounces, the angle between the CG ball's path and the center line is a small fraction of what it was on the previous bounce. Each time the cue ball bounces it gets closer to the center line until it eventually crosses over and starts bouncing farther and farther on the opposite sice of the center line. If the shot is aimed perfectly, the CG ball will end up permanently bouncing back and forth along the center line.
Of course there are pie slice forbidden regions (or cones in the 3D case), on opposite sides of the two balls, from which the shot cannot originate. The closer together the balls, the wider that region. If the angular size of the forbidden region is nonzero, then a net force of attraction between the MI balls will result.
If the MI balls are separated by less than the diameter of the CG ball, the forbidden region is 360 degrees, and the shot is impossible. However, in the LeSage model, the CG ball's diameter is presumed to be a small fraction of the diameter of the MI ball diameters. I think Slabinski's math assumes that the effective diameter of the CG is zero while the MI diameter is nonzero. Therefore, there are no impossible shots.
Regardless of whether there are impossible shots, we must consider whether the net result of all allowed shots is zero or not. I think Slabinski has shown that it is, in fact, zero. He convinced me when I studied his proof several years ago. Unfortunately, I am not enough of a mathematician to add any clarity to his proof.
Fractal Foam Model of Universes: Creator
