In this section, we will discuss the exact definition of the concept of spatial distance in physical reality. Although, this section may sound unchallenging at first, it is one of the key sections in this paper.

First, please imagine an empty three-dimensional Euclidean space (that does not contain any kind of matter or energy). By Euclidean space, we mean a metric space of three independent dimensions that does not have any kind of curvature. Now, let us ask a simple question: How can distance be defined between two points on this geometry; simply, how far apart are these two points?

Interestingly, in Euclidean geometry, there is no concrete answer to this question, because, Euclidean distance does not have an intrinsic metric standard or a built-in coordinate system as a function of a property of the Euclidean space.

In a Euclidean space, distance (between A-B) can be measured by comparing it to any other arbitrary distance (C-D) that is chosen as a standard for the measurement. Hence, the choice of the standard metric (the length of the standard ruler) is completely arbitrary.

Additionally, coordinate system in Euclidean space, which is used as an imaginary reference to map the space, can be oriented arbitrarily too, since Euclidean space does not have concrete metrics or special locations that appear as intrinsic properties of space itself. Simply, Euclidean geometry does not define a concrete coordinate system intrinsically.

Additionally, the indefiniteness of the metric standard in Euclidean space might be accepted as an additional argument supporting that Euclidean geometry is not self-consistent enough to be the basis of physical reality.

In Euclidean space, the distance between two points can be calculated most easily by an arbitrarily chosen Cartesian coordinate system: distance= sqrt [(x₁-x₂)²+ (y₁-y₂)²] (in two dimensions).

Now, we can go one-step further. In Section 4.3, we discussed that the geometry of physical reality cannot be Euclidean; and we assumed that it expands and it is closed (having constant positive curvature on any location, spherical). Let us now discuss the concept of distance on our closed geometry by ignoring matter and energy content and the expansion property.

In order to make visualization easy, we will again reduce spatial dimensions, and we will consider space as a one-dimensional line by leaving two dimensions of the three dimensions out. Simply, one-dimensional curved line (circumference of circle) represents our closed space.

Now, let us ask the same question on our abstract closed geometry, which does not contain any matter or energy. How can distance be defined between two points (A-B) on closed (e.g. spherical) geometry?

Of course, the easiest way is to choose an arbitrary standard metric, and to measure the distance as a ratio of this arbitrary metric. However, such an arbitrary choice of metric in our closed space (e.g. spherical) is a sign of ignorance as to the properties of the closed space.

Our closed geometry has significant differences with an abstract Euclidean space, because any distance can be described by the intrinsic property of the closed space. This property is the size of the circumference (space) or the magnitude of the radius of the curvature of the space. Hence, distance between two points can be described as a ratio of the circumference (or the radius of curvature) very concretely without making arbitrary choices.

Figure 6.1 Distance on circumference and radius

However, we do not suggest measuring the distance in physical reality as a ratio to the size of the universe, because of practical reasons, of course.

Additionally, our space geometry fundamentally expands. As we discussed in Section 4.2, the expansion means that any distance between two points constantly extends. However, since the circumference also expands, the ratio of the distance between any two points and the circumference stays constant with expansion. Therefore, the expanding size of the whole circumference (space) as a metric standard has confusing consequences.

Moreover, the concept of distance has an additional meaning in curved (like spherical) geometry. Now, distance is not only a magnitude describing how far apart two points are on the circumference (spatial direction), but also it describes how far apart a point is from the center of curvature (time direction). Although, circumferential (space) and radial (time) directions fundamentally differ, we will see that they can be equivalently measured in physical reality.

The coordinate system can still be oriented arbitrarily in our closed space geometry, because our space does not imply any special locations in space. However, since our closed geometry is elliptic (with constant positive curvature on any location, like spherical), it is not possible to map the whole space consistently by a coordinate system like Cartesian.

A closed space (like spherical) can be mapped by increasing the dimensional order of the space geometry to a higher order flat geometry, but in reality, such an approach should be considered incorrect, since higher order flat geometry also maps regions beyond physical space imaginarily, and ignores the intrinsic properties (curvature) of the closed space. Our space can be mapped completely by a coordinate system like polar (spherical coordinates), where the distance from the origin is the radius of curvature (time dimension), as we discussed in Section 4.1 on “Coordinate System”. In polar coordinates, the distance between two points is calculated as a function of the radial distances and angles. Distance= sqrt [r₁²+r₂²-2r₁r₂cos (theta₂-theta₁)] (two-dimensional).

In Section 5.3, we discussed the formation of matter and energy; we described energy as spatial wrinkles, and matter as the knots - vortexes that emerge in the wrinkling epoch of the expanding space.

We have simplified our closed space to a circle where the one-dimensional circumference represents space. Now, we may visualize the wrinkling epoch of our geometry that includes matter by a circle with knots tied on it.

Figure 6.2 Circumference with knots tied on it

(We will discuss exact definitions, interactions, and formation principles of the knots and the vortexes in further chapters. Readers’ tolerance is appreciated for the simplified visualization, which serves to discuss the concept of distance.)

Now, our wrinkled space with knots comprises three intrinsic properties that can be used to ratio the distance between any two points on it, without choosing metric standard arbitrarily. First, it is the wrinkled size or the (radius of) curvature of the circumference. Second, it is the non-wrinkled (original) size of the circumference; and third, it is the length, which is confined in a single knot or vortex (the tightness of the confinement volume). (Please note that all of these intrinsic properties are related to each other.)

The first two properties need exact information about the whole circumference (space); hence, they cannot be practically used as standard distance metric. Additionally, these properties have arguable consequences in expanding space.

On the other hand, the length that is confined in a knot seems to be the most practical standard distance metric, which emerges as an intrinsic property of the space-time continuum. The distance between two points on physical space-time can be described as a ratio of the length that is confined in a knot or vortex.

Standard distance metric as length that is confined in knots or vortexes has significant advantages to any other distance metrics, including any arbitrary choices. First, any observer (conscious physical body formed of knots and vortexes) can universally determine the standard distance metric by observing constituents of its body (length confined in knots or vortexes in its own inertial frame of reference). Second, the confinement volume contracts and tightens in relativistic cases (including gravitational effects); hence, the metric based on the tightness of the confinement volume intrinsically derives relativistic transformations (we will present its details in this chapter).

Let us leave discussing the properties of “The Universal (Natural) Unit System” to Chapter 11, and discussing “The Exact Meaning of Relativity” to Chapter 9. This chapter aims to examine the basic concepts of distance and time and their relativistic correlations.

Now, we can suggest the exact formulation of the spatial distance in physical reality.

According to the discussions above, the ratio between the length of circumference (space) and the distance between two points (on space) is kept constant in our expanding closed space. Careful readers may claim that space does not expand, but knots are tied to smaller volumes and they contract constantly. In fact, such a claim seems to be right too, since these two alternatives describe the same relation between the whole and its constituents. There are no references or referees guiding us to choose between spatial expansion and the knot (matter) contraction (except the logical cause of space geometry). Eventually, it seems to be easier to assume that we (observers formed of chain of the knots) have a constant size.

Please note that, basically, the concept of distance is related to the curvature at a location (size of the universe) in physical reality, even though the constant size of vortexes is suggested as the standard metric in this paper. It may not be practical to measure the size of the universe, but curvature at a location can be determined roughly by indirect measurements. In fact, metric contraction in gravitational fields is a result of the variation of the curvature at a location. In this papers Universal (Natural) Units System, gravitational constant is reduced to 2.787 x 10^-46 automatically, which indicates the degree of spatial curvature (relatively to the Compton wavelength of electron). We will deeply examine gravitational effect in a complete chapter on “Fundamental Forces and Gravity”.

Definition: Nrotation

Nrotation is the amount (number) of circulations in the confinement volume (with constant speed of light). It shall not be presumed as conventional frequency (1/s). It is not time dependent, but it generates clock-ticks.

Definition: Unit distance

Unit distance is a standard spatial distance metric (a measurement rod), which is universally invariant for any observer.

E 6.1

According to the equation above, unit distance is the spatial distance light travels while a single full circulation is completed in the confinement volume of a certain type of inert elementary “particle”.

Practically, we may also say that unit distance is the wavelength of the energy, which is equal to the rest mass of a certain type of inert elementary “particle”. In other words, it is the length of the circumference of the vortex. In Chapter 11, we will discuss the basic properties of “The Universal (Natural) Unit System”, and suggest the wavelength of the energy that is equal to the rest mass of electron (Compton wavelength of electron) as the standard distance metric.

The 4-dimensional (three spatial + time dimension) helical path of the intrinsic circulation (like vortex in space-time) in the confinement volume of electron can be visualized as seen in figure below (ignoring the exact definitions).

Figure 6.3 Four-dimensional helical formation of an inert electron (vortex) can be opened as in the figure; unit distance is the spatial projection of a single circulation of the helix.

E 6.2 Unit distance can also be defined as a function of the radius of the confinement volume or the wavelength of the energy that is equal to the rest mass of electron.

Eventually, our standard spatial distance metric (unit distance) is a physical definition of the Einstein’s imaginary element “measuring rod”. However, our standard metric derives relativistic transformations intrinsically, since our unit distance contracts with acceleration relatively. However, relativistic transformation will be discussed after crucial concept of time is examined.

Now, according to Geometric Generalization, it is possible to make a concrete definition of the spatial distance in physical reality:

E 6.3

Practically, this equation gives the spatial Distance (Distancespatial, Dspatial) between two points in space-time as multiples of the (Compton) wavelength of inert electron. (Compton wavelength electron = circumference of the confinement volume of electron). More properly, spatial distance between two points is described as the amount of circulations in the confinement volume (Nrotation) while light travels that distance.

In Einsteinian relativity, space and time are joined together to make space-time. Consensually, our definition of distance is not about spatial directions only. Similarly, we will soon suggest a joined four-dimensional distance in our expanding space-time geometry. However, the two different meanings of the concept of time (time as a dimension and time as the quantity of clock-ticks) should be clearly identified first.

Our standard distance metric (unit distance) is an intrinsic property of the physical reality, and it is based on degree of the expansion that is confined in formations like knots or vortexes. In other words, in closed and expanding space, distance between two points is described by comparing it to the expansion itself. Practically, the distance that light travels represents the deformed oblique radial direction of the expansion in the wrinkling epoch of our closed space geometry. Our concept of distance integrally describes the distance between two points in both spatial and time directions, since radius (time dimension) and circumference (space) are directly dependent to each other. Now, let us examine time…