"Hyperspace" is the necessarily curved space on the "surface" of the hypersphere defined by Equation 3; the lower-dimensional circle/spherical surface are embedded in this curved space and it has a predictable antipode. Although the antipode has been referred to consistently over the years (because it occurs in more than one model), the formal description of precisely how an antipode affects our perception of objects in its vicinity is rarely discussed.
So what happens to our observations if we do look toward an antipode?
Since we have deduced that a spherical surface is embedded in this space, we can simplify the question: relative to any position on the spherical surface, what happens when we observe objects that are on the opposite side of the surface? (Don't forget: whatever happens on the spherical surface also happens at right angles to the spherical surface.)
Suppose we're at the north pole of a spherical surface and that light is confined to the surface. There is a cloud nearby that we can move along any line of longitude as far as, and even "beyond", the south pole. As we move the cloud away, at first it seems to become smaller, as expected, but only until it reaches the equator. Once we move the cloud into the southern hemisphere, it starts to appear larger from our vantage point at the north pole. And if the cloud would just happen to lie over the south pole, it would stretch all the way around our horizon, since we always look toward the south pole in every direction from the north pole.
Therefore objects actually pass through a maximum in size as they approach the opposite side of a spherical surface, which corresponds to the antipode of the hypersphere.
Next what happens to our perception of luminous objects near the antipode?
We'll simply repeat the previous thought experiment, using a flashlight instead of a cloud. Here we'll assume: (1) the flashlight is nearby, aimed right at us, and moves along the same line of longitude the cloud does, (2) each side of the flashlight's beam follows a great circle (geodesic) around the surface, and (3) the relative brightness of the flashlight varies inversely as the width of the beam when it reaches our position at the north pole.
At first the flashlight appears fainter with increasing distance because the beam is getting wider when it reaches us. However, the beam ceases to become wider with increasing distance once the flashlight crosses the equator. Instead the beam gradually becomes narrower when it arrives at our north pole position, which means the flashlight is starting to appear brighter. And it will continue to get brighter with increasing distance until we move it "beyond" the south pole.
Again we have deduced a very unusual phenomenon in hyperspace: objects actually pass through a maximum in luminosity as we look toward the opposite side of a spherical surface (or the antipode of a hypersphere).
A similar thought experiment reveals that any lateral movement - called "proper motion" in astronomy - will appear exaggerated if it occurs near the opposite side of a spherical surface (or the antipode of a hypersphere).
Therefore if we're looking past an antipode, discrete luminous objects actually pass through a maximum in size, luminosity, and (lateral) velocity. We could say that the curvature of space near the antipode is behaving essentially like a gigantic lens, causing objects in its vicinity to appear larger, brighter, and to move across our field of view faster than they would at the equivalent distance in the conventional spherical universe (without an antipode).
If we did observe discrete objects very near an antipode, we would probably attach prefixes like "ultra-", "super-", and "hyper-" to the adjectives describing them. (Hint: some of these prefixes are used to describe certain objects in deep space, aren't they.)
Obviously in just a few minutes we've derived some very unusual phenomena that never happen in ordinary Euclidean space.
If we actually do look past an antipode, you might think we'll detect fewer objects in its vicinity. After all, we're looking into and out of a smaller volume of space than would occur at the same distance in ordinary Euclidean geometry.
But don't forget that objects actually appear brighter as they get nearer the antipode. Since most astronomical distributions have many more fainter objects in them than brighter ones, it's entirely possible that we'll detect the more numerous fainter objects if, and only if, they are near the antipode. Depending on whether the number of fainter objects we detect offsets the smaller volume of space we're looking into and out of, we may notice either a "peak" or a "valley" in the observed distribution.
But we ought to notice something if we're looking past an antipode. It would take a very unlikely set of circumstances for us not to notice anything: two functions varying exponentially in opposite directions with the radius, r, from the antipode (volume changing as r3 and luminosity changing as r-2) would have to nullify each other in a given distribution for the antipode to escape our attention.
[Probably the best and most obvious example of an astronomical distribution with an increasing number of fainter objects is the stars in the night sky. There is almost exactly a threefold increase in the number of stars at each fainter magnitude: 22 stars at the first magnitude (or brighter), 68 stars at the second magnitude, 197 stars at the third magnitude, 599 at the fourth magnitude, and so on. While the apparent brightness we observe is affected by distance, in our own galaxy the intrinsically fainter K- and M-type stars (orange and red dwarfs, respectively) are at least a hundred times more numerous than the much brighter O- and B-type stars.]
In fact, since the perception of objects changes so dramatically as we look past an antipode, its presence will probably affect the normal isotropy of the universe. Isotropy means "uniformity": the tendency of objects to appear identical in every direction and at all distances, although we should make some allowance for evolution when we look out billions of light years into space (and therefore billions of years back into time).
Our best chance to find an antipode, if one is close enough to affect our observations, is to look for unusual isotropic behavior in whatever class(es) of objects lie in its vicinity. A less precise technique is to look for a variety of observations that seem unusual in a spherical universe: larger, brighter, faster. If these anomalies occur consistently at the same red shift, they could very easily be more clues suggesting the antipode.
But is it possible to define the universe in a way that antipode is expected and where should we expect the antipode to occur?
These are the topics in Essay #3.
Essay 1 & Essay 2 were created and written by: Michael R. Feltz. Theses and MORE essays dealing with the fourth dimension can be found on his homepage. Click here to go to his homepage!
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