4D space-time and 4D space

Now, suppose we can discover the significance of the difference between the positive and negative signs on the basis of the fourth dimension. If so, we may envision the four-dimensional world not in time but as a spatial picture. In other words, we will be able to perceive the 4th dimension as space.

  Usually, when we think of the direction of the 4th spatial dimension, we try to find it in the image of “an axis = a line segment“ orthogonal to the 3-dimensional space since the axes of the 3-dimensional direction are orthogonal to the 2-dimensional plane. However, no matter how hard we may search, we cannot find any other direction in the space except the x,y, and z axes as the vertical, horizontal, and height directions. As a result, we conclude that the 4th-dimensional direction is just beyond the reach of human perception. But is this really the right conclusion? We might just be missing the direction that exists in our sight. OCOT says the following about the 4D direction:

“The transition from the 3-D perception you currently have to the 4-D perception is quite different from the transition from the 2-D perception to the 3-D perception, even though they both have a gap by one dimension.” (2013: The Day God Sees God, Advanced Edition. Part1, p.86)

 The true meaning of OCOT’s phrase is that the direction of the 4th dimension is probably not following the way of dimensional ascension achieved by stacking dimensions from the 1st (line) to the 2nd (plane), to the 3rd (solid),  then to the 4th dimension (hypercube). However, this is the way how scholars explain dimensions in books generally. So what does OCOT mean by the difference? If we have no idea what 4-dimensional space is, we must look for clues from the time axis, which can be regarded as the negative 4-dimensional direction. So, let’s look at how time appears in relation to 3-dimensional space.

The mediator of time and space, in this case, is light. Since light propagates spherically through space, the equation for a two-dimensional sphere embedded in three-dimensional space can be expressed using time t as follows (see Figure 1 below).

Figure1 The equation for a two-dimensional sphere embedded in three-dimensional space.

Transposing (ct)^2 to the left-hand side yields

x^2+y^2+z^2－(ct)^2=0………（1）

Here, we get an equation that shows the 4-dimensional invariant distance in space-time.

Look at the relationship between x^2+y^2+z^2 and (ct)^2 in this equation (1). As you can see, x^2+y^2+z^2 is the space term, and (ct)^2 is the time term. Since the space term minus the time term is zero, space and time are, what Noosology calls, “Duality” relationship. It means one term is a positive factor and the other is a negative factor. To put it concretely, this means that the 3-dimensional space (x^2+y^2+z^2) extending from objects/matters and the 4-dimensional direction of time ((ct)^2) are in opposite directions in such a way that they cancel each other out. Because of this relationship between space and time mediated by the speed of light, we often conclude that a scene on the celestial plane seen one light year away is one year old.

 Now we dare to apply this spherical image to the spherical space of Ψ4 (Figure 2 below).

 I say “dare” here because the concept of 3-dimensional space that we deal with in mathematics derives from the identification of Ψ4 and Ψ*4 (i.e., the inversion relation between them is made invisible) in Noosology. To explain precisely how this identification occurs from a Noosological viewpoint, an explanation at the level of Ψ9-Ψ10 is necessary, but here we will “dare” to deal with Ψ4 to avoid complications.

By assuming the presence of “my face” as an observer in front of the object/matter (this was the meaning of the radius of the Ψ4 sphere space, the Inside of a human ), our consciousness separates the object/matter and “I.”  The distance created by the separation corresponds to the square root of (x^2+y^2+z^2) in space and ((ct)^2) in time, i.e., ct. The meaning of ct is that the object/matter seen by the observer is not the object/matter as it is now but is slightly in the past because it takes some time for the light emitted from the object/matter to reach the observer’s face.

Now let’s flip this Ψ4 sphere space to the Ψ3 side. The radius of this inverted spherical space cannot be expressed in terms of ct because this inversion operation means time turns inside out. In physics, the inversion of time is described by multiplying time t by imaginary number i, called imaginary time i t; t → i t. Then the equation of the sphere in the inverted spherical space Ψ3 becomes

x^2+y^2+z^2−(cit)^2=0

When we organize this equation, it becomes

x^2+y^2+z^2＋(ct)^2=0
(The identification of Ψ3 and Ψ*3 creates this sphere)

 If we consider ct as the fourth dimension, we can interpret this equation as a higher-dimensional sphere, in which a 3-dimensional sphere with its zero radius is embedded [1] in 4-dimentional space[2] . Well, if the radius is zero, it cannot be called a sphere, so perhaps it is better to say that it is a 3-dimensional sphere that has collapsed to zero radius. In any case, in the inverted spherical space Ψ3, time becomes the same entity as space, and “time ceases to exist.” In that sense, we can regard this as the light speed state (i.e., at light speed, the clock stops).

Recall the exercise we did earlier to imagine the spherical space of Ψ3. It was the spherical space created by the rotation of the observer’s gaze, penetrating [1] [2] the center of the object/matter and directed toward its background plane. I also mentioned that the direction is indistinguishable from the center point of the object/matter because the depth is identical to it. What do you think? Isn’t it quite similar to the image of this equation? — To be continued.