09-12-2016, 03:01 PM

(09-11-2016, 11:34 PM)secur Wrote: Ok, maybe I figured it out. First, consider a simply connected space, such as S^3. Suppose there's a path through it, point A to point B. Now traverse it in reverse, point B back to point A. That's a closed loop. Since the space is simply connected, it can be shrunk to a point - in particular, the identity of the space. So in this sense traversing the closed loop from A, to B, back to A, is equivalent to the identity. Therefore any path (A and B were of course arbitrarily chosen) is reversible.

But you can't do that with a connected space which is not simple, such as SO(3).

Is that what you're getting at?

It's absolutely what I'm getting at, secur. I rejoice that it is not so hard to understand!

Reversibility is the key property. As I wrote in a 2006 conference paper:

5.3.1 A move in time, i.e. a real continuous function – on the surface of a closed manifold (S^2 ) – makes no differentiation between a closed loop and a continuous line, but accommodates both. E.g. the Euler Network Formula for a flat plane, V – E + F = 1, becomesV – E + F = 2, for a closed (compact) manifold. If we were to speak of “hypertime,” we would find that what Brouwer took as a fundamental analytical fact of mathematical “twoity” [Brouwer, 1981] is in hyperspace a “fourity” of terms. (Indeed, the “hyper” classifications of numbers – quaternions, octonions and their extensions – due to W.R. Hamilton, Cayley et al – follow. We shall not need these.)

Bell loyalists can take refuge in formulating the shape of space as toroidal -- Euler characteristic 0, in which case no account is taken of time reversibility, and all events are independent. With an Euler characteristic of 2, spacetime is properly represented, and we have no worries about "attempting to breathe in empty space."