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a4112

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A parenthesis : Space-time orientation.

** ...**In 2d's world we assimilated geometrical objects to letters. In 3d's world they were assimilated to "right hand" and "left hand".

Four-dimensional structures were assimilated to animated holograms.

**...**What the hell could be a five-dimensional structure, 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 the right from the 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 sens. 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 figures the arrow of time.

If this space-time can be oriented in time ( we suppose it is closed ) at given time, space is a S2 sphere :
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**...**But suppose the surface representing space is single-sided. Take a Boy surface, for an example ( which is a closed single-sided surface. See the section "Mathematics" of the site ).
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You can build a one just gluying Mbius strips together. I show a one :
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You know that you cannot define an oriented normal vector :
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**...**The two folds cover of a Boy's surface is a sphere S2. If we assimilate our three dimensional space-time to a sets of spheres S2, arranged like russian dolls, each corresponding to a given value of a cosmic time t, we can think (hardly ) to 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-folds cover of a Mbius strip :
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We see how this conjugates space-antipodal regions with opposite time's arrows.
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**...**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 a S3 sphere, which can be shaped as the two folds 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.

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