6. Distance  Time  Relativity
6.2. Time as a Dimension in SpaceTime Geometry
Time has more than one meaning, which are not clear enough and mostly confused. Let us discuss the concept of time as a dimension before examining why clocks tick in physical reality. 6.2.1. Time as a dimensionFirst, let us review the concept of time as a dimension in its largest meaning in unwrinkled epoch (the inflationary) of our abstract space geometry by ignoring matter and energy. Our geometry is closed (like spherical); hence, it has an intrinsic curvature as we discussed its logical cause in Section 4.3. If we visualize our geometry, circumference (onedimensional constantly curved line) represents space by leaving two dimensions of the three dimensions out. Time dimension is towards radial directions in our abstract geometry, and it is perpendicular to spatial dimensions. Figure 6.4 Coordinate system 6.2.2. Formation of time dimensionThis paper accepts time dimension as a basic direction in the geometry of physical reality, like other spatial dimensions. However, it is important that the time dimension differ fundamentally from spatial dimensions.
The geometry of physical reality is threedimensional in its essence. However, it is has an intrinsic curvature because of fundamental logical principles, and in our closed geometry, time dimension emerges on the direction of the radius of curvature, because of the curvature.
On the other hand, in abstract threedimensional Euclidean space, it is not possible for an additional (time) dimension to emerge, because Euclidean space does not have an intrinsic curvature that indicates a specific direction for an additional dimension. 6.2.3. Distinction of the time dimensionIn spatial dimensions, a point has a fixed location (on circumference). On the other hand, when time dimension emerges, a fixed point on circumference (space) has infinite possibilities of coordinates towards radial directions.
Interestingly, coordinates on time dimension (radial directions) are not equivalent like coordinates on spatial dimensions. In spatial dimensions, the next location does not have different intrinsic properties; all coordinates on the circumference (space) are equivalent at a time. However, in radial directions (time dimension) curvature decreases, when the distance from the center of curvature increases. 6.2.4. Flow of timeSince, curvature is not a natural state in geometry; space decreases its curvature by expanding in order to balance its intrinsic instability. Hence, in closed geometry, spatial coordinates constantly flow towards perpendicular time dimension. Geometric Generalization suggests that this is the basis of the expansion in the geometry of physical reality.
However, it is important to note that the flow of spatial coordinates towards time dimension is not a mechanic action that has energy content. More precisely, flow of expansion means that coordinates in radial (time) direction has sequential precedence, while all coordinates are equivalent in circumferential (spatial) directions.
Eventually, flow of the spatial coordinates towards time dimension (or the expansion) is the basis of clockticks and flow of time in physical reality. However, flow of expanding space does not present a direct definition of clockticks (or Einstein’s imaginary element clock, which is used to measure time). 6.2.5. Distance towards time dimensionIn fact, time as a dimension can be treated as a geometrical dimension like other spatial directions, although it has fundamental differences. Consequently, distance towards time dimension is a geometrical quantity like spatial distance.
In our unwrinkled abstract closed space, which does not have matter content, it is easy to handle time dimension theoretically. Distance towards radial (time) directions and distance on circumferential (spatial) directions are comparable quantities. Figure 6.5 Radial distance (A_{2 } A_{3}) and spatial distance (A_{3 } B_{3}) Simply, distance between two points towards time dimension is the difference of the radius of curvatures of these two points. In other words, it is the difference of how far apart two points are from the center of curvature. Figure 6.6 Distance towards time dimension (A_{2 } A_{3}) 6.2.6. Time dimension in the wrinkling epochAs we discussed in Section 5.3 on “Mass and Energy” with details, packages of matter and energy are strain formations that form against the universal strain on the expansion in closed and expanding geometry.
Additionally, during the universal strain on the expansion, there has happened a major change in the proportions of spacetime geometry as if spacetime is deformed under a pressure (like Poisson’s contraction). In the wrinkling epoch, original radial directions are not perpendicular to spatial directions, but radius has an oblique angle. Moreover, because of the universal strain on the expansion, a new shorter (contracted) radius of curvature emerges (Section 5.2). Figure 6.7 Oblique radial directions in the wrinkling epoch In fact, the degree of the wrinkling on radial direction or the degree of the obliquity (θradial obliquity) indicates the ratio of the universal strain on the expansion (Section 5.4). In physical reality, the path that light travels in spacetime represents the oblique radial direction in spacetime exactly.
Oblique radial direction of the expansion circulates in the confinement volume, and it forms strain packages of matter (knots and vortexes) in spacetime. However, since, the radial direction is not perpendicular to spatial directions in the wrinkling epoch; intersection point of radial direction and spatial direction moves (expands) in spatial directions with the expansion, and create the light cone. As a result, the complex dynamism of the intersection point is one of the main reasons of quantum mechanical behavior (nonlocality) in Nature. We will deeply discuss quantum mechanics in Chapter 8. 6.2.7. Distance towards time dimension in the wrinkling epochOf course, in physical reality, our basic definition of the distance towards time dimension remains the same: it is the difference of the radius of curvatures of two points in spacetime.
On the other hand, in physical reality, it is not possible to determine the distance between two points towards time dimension by making direct observations in Nature. Eventually, spatial coordinates flow towards the perpendicular time dimension as a whole, because of the expansion. It seems that there are no direct references (except logical reasoning) that can guide observers (who are also formed of knots and vortexes) to determine the geometric distance between two points towards time dimension.
Additionally, in the wrinkling epoch of physical reality, distance towards time dimension has a nature that is more complex. It is shorter than the original radial direction of the expansion, because of the universal strain on the expansion.
The graph above (Figure 6.7) demonstrates that the distance towards time dimension between two different time coordinates of the same spatial coordinate (P_{1} and P_{2}) is wrinkled, and the original radial direction has an oblique angle to the spatial plane in the wrinkling epoch.
Please note that angle of radial obliquity should not be confused with the constant of speed of light, which is related to the constant relation between the spatial distance metric and time as quantity of clockticks. Conversely, distance towards time dimension is a pure geometric property of spacetime, and it is not a relative property of matter like time as clockticks. 6.2.8. Formulation of distance towards time dimension (Dtd)As a result, distance towards time dimension between two points in spacetime can be calculated as a function of the spatial distance, only if the radial obliquity angle (θradial obliquity) is known. However, we will leave how to determine the numeric value of radial obliquity angle to further chapters, since it cannot be measured by direct observations, but it can be deduced by analyzing the nature of fundamental forces.
Definition: Dtd Dtd is the distance interval towards time dimension, or it is the difference of the radius of curvatures of two points in spacetime.
It is most easily calculated for two points that are on the surface of light cone, as a function of the spatial distance (Dspatial) between them: _{} E 6.4 Figure 6.8 Distance towards time dimension between two points, which are on the light cone Below graph demonstrates that all distance intervals towards time dimension between A_{0} and A_{8}, A_{0} and B_{8}, A_{0} and C_{8} are equivalent. Figure 6.9 Equivalent distance intervals towards time dimension On the other hand, please note that it is not necessary to calculate distance towards time dimension, unless the universe is analyzed at the large scale. According to the discussions above, it is clear that time as a dimension (flow of spatial coordinates  the expansion) is the basis of dynamism in Nature; however, it is should not be confused with time as (relative) quantity of clockticks.
In other words, in wrinkling epoch, distance towards time dimension is the projection of the original oblique radial distance (the path of light in spacetime) to time dimension, similarly to the spatial distance, which we defined as the spatial projection of the oblique radial distance.
Additionally, it is possible to calculate the distance interval towards time dimension between points that are not on the light cone, if the radial obliquity angle is known. However, we will skip its formulation in order to limit this paper. 6.2.9. NotesAs we overviewed in Section 5.4, obliquity angle also describes the ratio of the universal strain on the expansion. Please note that the strength of electromagnetic interaction indicates the ratio of the universal strain on the expansion or the degree of obliquity (it will be discussed in Section 7.4 on “Fine Structure Constant”).
As we will discuss in the next section, clockticks (or Einstein’s imaginary element clocks) is the property of matter (knots and vortexes), and it relatively varies with the tightness of the confinement volume. On the other hand, time as a dimension is the property of spacetime, since it is based on the radius of curvature at a coordinate in spacetime.
Now, we can examine why clocks tick in physical reality.

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