The volume of an n-dimensional sphere increases with r^n, r = the radius of the sphere.
When space has a high negative curvature, homogeneous and isotropic (everywhere the same and in all directions the same) then volume increases faster with r than its volume-formula describes for flat space.
Suppose we have a 5-dim sphere in flat space. Its volume increases with r^5. When not mentioned otherwise in this page all considered spaces are static, that is: not changing in time, constant in time. Suppose we too have a 4-dim sphere in highly negative curved space. The curvature is such the volume increases with the r^5.
I assume such a negative curved space can exist. I mean, the volume increase to be proportional to r^5 in every point of space and in all directions, while the sphere is 4-dim. I donít know whether this assumption is right but in this page I assume curvature can be adjusted like that.
Suppose you are a 3-dimensional observer “O”. You observe a matter distribution of C-60 molecules around you, just ordinary 3-dimensional C-60 molecules. All molecules are standing still with respect to you. You can measure the apparent diameter of a molecule and thus infer the distance r to you - if not, you should use a bigger type of molecule. The distribution has a quite low average density, the nearest molecules are about 1 kilometer away. The observed distribution is isotropic but the observed number-of-molecules m at a certain distance r increases with r^5; m is proportional to r^5.
We state that the following possibilities have identical observations of matter distribution. Provided those spaces can exist physically, they are indistinguishable for the observer O and thus for O, as “experienced” by O, will superpose.
1) A flat 3-dim space in which C-60 molecule density increases in space with r^5, being r the distance to the observer;
2) A flat 5-dim space with homogeneous and isotropic distribution of C-60 molecules;
3) A highly negative curved 4-dim space of the described kind with homogeneous and isotropic distribution of C-60 molecules;
4) A 3-dim flat expanding space with homogeneous and isotropic distribution of C-60 molecules, in which the expansion simulates the effects of the highly negative curved 4-dim space of number 3. I tried to discuss its existence in page 4 of the storyline EXPANSION OF THE UNIVERSE, the column at the right.
The observer observes its local environment to which he, she or it belongs, as being 3-dim. The transition between your local 3-dim space and the space further away around you is, as to speak, “not your business”. It consists of vacuum that you cannot observe.
Consider the molecules within a sphere of r = 10 km. In the second possibility (flat 5-dim space) no molecules can be found with a mutual distance larger than 20 km. They can see that from each other. In possibility 3 the molecules with largest mutual distance, this distance certainly will exceed 20 km. But that is not of your business. The mutual distance consists of vacuum that you cannot observe. The molecules have no mind or memory, they will not give a sign to you what they see. Initially we assume such signs to be absent and the superposition will be there. Later, if there arises some sort of memory or any indication of mutual distance, this would be decisive and forces you into a choice.
Then you are in flat 5-dim space. Or in highly curved 4-dim space. Or in flat 3-dim space with a peculiar C-60 molecule distribution - peculiar but not forbidden. Or in flat 3-dim expanding space. You will have to find out.
If space turns out to be 4-dim or 5-dim, how then the transition is made between your local environment, obviously being 3-dim, and the 4-dim or 5-dim space further away? I donít know, but IF nature provides an existable way THEN those possibilities will superpose.
Well, I guess under normal circumstances this will not be the case. But circumstances have been not normal in the far past. Nor are they everywhere in space.
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