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(151)
A parenthesis: Space-time orientation.
**...**In the 2d world we assimilated geometrical objects to letters. In the 3d world they were assimilated to "right hand" and "left hand".
Four-dimensional structures were assimilated to animated holograms.
**...**What the hell could a five-dimensional structure be, or a ten-dimensional one? Sometimes I envy God, don't you?
**...**He must laugh, looking at our miserable four-dimensional structures.
**...**But a theoretical physicist, and even a mathematician, are nothing but oriented four-dimensional structures. If they were not, they could not distinguish past from future, and right from left.
**...**The universe, as a whole, is a four-dimensional structure. Let us think about it as a closed object, with locally spherical topology. Call t the time. At a given time we can make a cut, which is a 3d hypersurface. If this last is a hypersphere S3, time makes sense. The vector time crosses this hypersurface and we get no paradoxical situation.
**...**Let us reduce the number of dimensions. Imagine a closed two-dimensional world, some sort of space-time (x,y,t).
**...**We can cut it at t = constant, then we get a geometrical object whose dimension is 3 - 1 = 2: a 2d surface. At any point the oriented normal vector represents the arrow of time.
If this space-time can be oriented in time (we suppose it is closed) at a given time, space is an S2 sphere:
(152)
**...**But suppose the surface representing space is single-sided. Take a Boy surface, for example (which is a closed single-sided surface. See the "Mathematics" section of the site).
(153)
You can build one just by gluing Möbius strips together. I show one:
(154)
You know that you cannot define an oriented normal vector:
(155)
**...**The two-fold cover of a Boy's surface is a sphere S2. If we assimilate our three-dimensional space-time to a set of spheres S2, arranged like Russian dolls, each corresponding to a given value of a cosmic time t, we can think (hardly) of some sort of space-time where antipodal points could be put together. That was the topological structure suggested in the paper:
Jean-Pierre Petit: "The missing mass problem". Il nuovo Cimento B, vol. 109, July 1994, pp. 697-710.
**...**Then we know that antipodal points, located on an "equator" of a sphere can be arranged as the two-fold cover of a Möbius strip:
(156)
We see how this conjugates space-antipodal regions with opposite time's arrows.
(157)
**...**By the way, we see how it would conjugate enantiomorphic objects.
**...**Space is a four-dimensional hypersurface. If we can define a cosmic time t, we can make cuts at t = constant and these cuts are 3d spaces. If space is closed, we could assimilate it to an S3 sphere, which can be shaped as the two-fold cover of a projective space P3 (the equivalent of a Boy's surface, in 3d). This operation would put opposite time's arrow regions into interaction.
