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General Features of the Space

Suppose the global feature of the space is like this: Subscripts 0..3 (t,x,y,z) represent the usual spatial dimensions. 4..7 (s,u,v,w) represent momentum space. Global boundary conditions on txyz are unknown, and all action takes place without impinging on the perimeter. But suvw are periodic in the manner of a Klein bottle. The s direction is not flipped; the other three (all together as a unit) are flipped 180 degrees going around the bottle. In order that the space be isotropic, uvw form a hypersphere (rather than a torus).

A geodesic in the s direction is closed and passes through two points in a slice perpendicular to s, except that for at least one such geodesic (exactly 2, in the 2D case) the two loops coincide.


    {ut,x} = -a    (large but turns out not to be constant)
    {xt,x} = a

then for a geodesic whose tangent is V (/t = covariant derivative by t, !t = ordinary)

    0 = V[x][/t]V[t] = V[x][!t]V[t] - {ut,x}V[u]V[t] - {xt,x}V[x]V[t]

So the ordinary derivative of V[x] is 0 if V[u] == V[x], and if {xt,x} is positive then the topography is stable and V[x] will approach V[u] rather than run away from it (V[t] being positive).

Assuming similar relations on the other dimensions, this binds together the spatial and momentum coordinates in pairs, so the geodesic has the same velocity in the spatial and momentum coordinates. But since momentum space is periodic, phase rotation in the various directions is created.

Look at the geodesic around the Klein bottle whose two loops coincide. This corresponds to "absolute rest", which of course is bogus. A connection component (force) that increases the x and u components will force the trajectory around the bottle, but the force will be less effective there because it has to be continuous through the flipped periodicity, and hence (probably) has a cos(th) dependence on the u "angle". Thus is produced the increasing mass with relative velocity, and a specific upper bound on velocity.

More specifically on the wrapping: Let s range in +-pi with 0 at the "top" of the bottle and a discontinuity at the bottom of the loop. Let u range in +-pi with 0 at the "front" and a discontinuity at the back. This is something like a torus, except flipped. The loop of the bottle is so oriented (toward u=pi/2) that a line with u==0 starting at (0,0) returns to (0,0) without crossing s==0 elsewhere.

The connection is singular (by components, not intrinsically) across (some of) the coordinate singularities. A continuous vector field will have these singularities:
Crossing: s = +-pi u = +-pi
V[s] none none
V[u] sign reverses none

Thus, the function a( ) above (assumed continuous) cannot be constant. (a sin(s/2)) or (a cos(s/2)) (and all derivatives) are continuous across the joint due to the sign change. Since for stability the Christoffel symbol has to have a particular sign, (a cos(s/2)) is the preferred functional form. (a cos((n+1/2)s)) is also continuous (but not stable) (integer n), but (a cos(ns)) is discontinuous.

What is the source term associated with the connection assumed here? We shall assume geodesic normal coordinates, bypassing details of the metric tensor. Then:

    C[J][t][I] = C[t][J][I] = -a cos(s/2)    (J = uvw, I = xyz)
    C[I][t][I] = C[t][I][I] = +a cos(s/2)    (all others zero

    C[...][!s] is nonzero, only for s and for the above subscripts.
    C[j][k][m] g'[m][n] C[i][L][n] is nonzero for:
    C[J+I][t][i] C[J+I][t][i]        J+I = uvw xyz (two independent),
                                     summed i over xyz
    R[u][t][x][t] = C[t][t][u][!x] (0) - C[t][x][u][!t] (0)
        - C[u][t][x] C[x][t][x] + C[u][t][u] C[x][t][u] 

(Still working on this)

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