In figure 2, I place point P at (x₁,0), point Q at (0,y₁), and point O at (0,0), which is the origin. In this figure, an eventon travels straight from P to Q, crossing

PQ is the distance-time given in distance units. In order for this eventon to travel from P to Q, it must cross

PO is the distance-time parallel to the X axis and

Consequently, PO and QO are dilated along the path that the eventon takes between P and Q when PO and QO are smaller than PQ.

PO and QO dilated between P and Q occur slower than speed c. Dividing PQ by c, I change PQ from distance units to time units, T. Since PO distance-time and QO distance-time occur in a period of time T, I divide Eqs. (7) and (8) by T, which gives

Both Eqs. (9) and (10) show that the distance in PO and QO, dilated between P and Q, have a speed smaller than or equal to c. If, in figure 2, I send a different eventon traveling straight between P and O or Q and O, it travels at speed c; therefore, actual distance occurs at speed c between P and O as well as between Q and O. However, the PO and QO distances in Eqs. (5) and (6) are dilated between P and Q, and are only parallel to the X and Y axes in figure 2. They do not necessarily lie on the X and Y axes. According to Eq. (9), PO has a speed c if PO=PQ. According to Eqs. (4) through (6), this only occurs when QO equals zero distance making PO the actual distance occurring between P and O. Also, QO has speed c and is the actual distance-time between Q and O when PO equals zero distance.

Visible space is the space an observer sees. Since light particles crossing distance-time give an observer a special view, I shall derive a three-dimensional view of space by placing an observer in a distance-time manifold.

In this subsection on visible space, I assume that light is traveling in a vacuum. In figure 3, an observer stands a distance from the broadside of a wall. Also, in figure 3, I place points P, Q, and O from figure 2 such that the observer is at point Q, and on the wall’s surface lay points P and O. Light beams travel a straight path in figure 3 from P to Q and from O to Q. The Observer at Q sees that the light that travels from P to Q, crosses PO distance parallel to the wall’s surface. Because the path from Q to O is perpendicular to the wall, the observer sees that the light that travels from O to Q does not cross any distance parallel to the wall’s surface. Therefore, the observer sees P extended out from O a distance of PO. Since P can be any point on the wall’s surface, the observer sees the entire wall’s surface extended out, a distance in any direction on the wall’s surface from O, when the light rays reflected off of the wall’s surface reach the observer.

The fraction of distance in PQ parallel to the straight path between Q and O is QO distance. Since P can be any point on the wall’s surface, every light ray from the wall’s surface reaching the observer must traverse QO distance parallel to the straight path between Q and O. Consequently, the observer sees the wall’s surface a depth of QO distance away and in the past.

I make the plane represented by the wall’s surface infinitely large, and I make the distance QO any possible distance in any direction from the observer and perpendicular to the wall’s surface. The point P can now be any point within the three-dimensional distance-time manifold, and the observer sees a three-dimensional space in the past. However, the distances within this visible space do not always travel as fast as actual space at speed c. Visible space is only the space that an observer sees. It is not necessarily the actual space, which has a speed c. According to Eq. (9), the distance of PO that the observer sees on the wall’s surface has a speed smaller than or equal to c. The observer only sees the actual distance PO occur at speed c if the observer is located at point O, making QO equal to zero distance and PQ=PO. However, If the observer is at a distance away from O, the distance that the observer sees between P and O is dilated along the path between P and Q, and is slower than speed c. Hence, visible space has a speed smaller than or equal to the speed of actual space.

Objects given a location in a traditional space are stagnant if no provision for the motion of these objects across space is defined. Therefore, the traditional view of a space requires a fourth dimension of time. In this view of space and time, or space-time, space moves across the fourth axis of time. Each distinct point of time is a distinct point of existence for space. At a given point of time, an object can exist at a different location in space compared to previous or later locations in space. This ability of relocation at different points of existence allows an object to move in space while space moves across the time axis. In distance-time theory, however, there is no fourth axis of time with a space extended out separate from the time axis. Instead, in distance-time theory, motion is defined by eventon motion occurring at speed c relative to a reference frame at rest. A reference frame is a coordinate system that is a reference for a perspective. Any motion of another reference frame at velocity v, relative to the reference frame at rest, is a part of this motion of eventons at speed c. Therefore, relative to the reference frame at rest, a fraction of the distance-time (in the speed c) of these eventons goes into the velocity v of the other reference frame. This relationship of c to v is in contrast to traditional theories of space and time, which do not view all velocities v as only fractions of a whole number c. Since both reference frames share the same eventons, the same eventons define distance and time for both of these frames. As a result, the distance-time in v (the speed of the moving frame) is subtracted out of the distance-time of the moving frame relative to the reference frame at rest. However, the distance-time relative to the reference frame at rest does not have distance-time subtracted out. Using the S and S’ reference frame, I illustrate in figure 4 the relative motion of one reference frame to another.

In figure 4, S’ has a velocity v in the positive X axis direction. The X and X’ axes, as well as the Y and Y’ axes, lie parallel. The S clock is at the S origin, and the S’ clock is at the S’ origin. At the S and S’ clock measurements of t=0 and t’=0, the S and S’ origins coincide with an eventon A. Relative to S, S’ moves from (0,0) at t=0 to (x,0) at t=t, and eventon A moves straight from (0,0) at t=0 to (x,y) at t=t. However, relative to the S’ frame this eventon A moves from (0',0') at t’=0 to (0',y’) at t’=t’, which is straight along the Y’ axis. Also, rod measurements within the S frame give

In figure 4, the distance-time that eventon A crosses between (0,0) and (x,y), in time period of (t- 0), is given by the distance-time Euclidean metric function, which is

However, the amount of ct parallel to the X and Y axes are given by Eqs. (11) and (12), and they are dilated across ct. Thus, eventon A crosses D" distance in t time parallel to the X axis. Also, parallel to the Y axis, eventon A crosses D’ distance in t time. The speed of eventon A parallel to the X axis is

The speed of eventon A parallel to the Y axis is

The ratio of distance to time in which an eventon crosses is always c. Thus, Eqs. (14) and (15) are not distance-time metric functions. Instead, relative to S they are only rod measurements taken along the X and Y axes, and divided by the S clock measurement, which gives partial velocities v and r of the total velocity c. Dividing Eq. (13) by t and combining the result with Eqs. (14) and (15), produces,

the relationship between the total and partial speeds of eventon A. Combining Eqs. (11) and (14), I arrive at

Combining Eqs. (12), (15), and (16), yields

Both Eqs. (17) and (18) give D" and D’ as fractions of ct. Since D’ and D" are crossed by eventon A while eventon A crosses ct, D’ and D" are dilated according to Eqs. (17) and (18).

Relative to the S frame, the S’ frame moves with velocity v along the X axis. The distance-time crossed by this velocity v along the X axis is defined in Eq. (14) to be the distance-time D"=cT", which is the part of ct that occurs parallel to the X axis in figure 4. This distance-time in v does not occur relative to S’. Only in the S frame does it occur as a fraction of ct distance-time. Relative to S, eventon A travels across D’=cT’ parallel to the Y and Y’ axes and inside S’. D"=cT" is the distance-time in the rest speed, r, of S’ relative to S. Combining Eqs. (11), (12), and (13), I derive

Similar to Eqs. (19) and (18), the distance-time in v, which is D"=cT", is subtracted out of the total distance-time ct, which eventon A crosses in S. This leaves D’=cT’ distance-time within S’ relative to S. Although other eventons have different paths from eventon A’s event line, relative to S, all the paths of eventons within S’ still have the distance-time in v subtracted out according to Eqs. (19) and (18), leaving the distance-time in rest speed r, which is D’=cT’. I used eventon A’s event line in figure 4 because with this event line the relationships are apparent between the distance-times in speeds c, v, and r. The distance-time in v is the difference between S and S’ reference frames relative to S. Consequently, D"=cT" is the difference between the here-nows of the S and S’ reference frames. The events in D"=cT", therefore, are located here-now relative to the S’ reference frames. However, these events are still dilated across D=cT relative to the S reference frame.

It is now appropriate to examine a second event line in figure 4 defined by an eventon B. I place eventon B in figure 4 traveling in a positive direction along the X and X’ axes. At t=0 and t’=0, B coincides with the origins of S and S’. According to the metric function, within the S frame, B crosses a

quantity of distance-time, and within the S’ frame, B crosses a

quantity of distance-time. Using Eq. (20) for substitution into Eq. (17), I arrive at

Dividing Eq. (22) by c, I change Eq. (22) into time units, getting

In Eqs. (22) and (23), D"=cT" is dilated along the X axis. Since D"=cT" is the difference between distance-time occurring within S and S= relative to S, Eqs. (22) and (23) give this difference of D"=cT" along the X axis in terms of x. In order to put Eqs. (22) and (23) in terms of x’, I substitute x’ for x and - v for v since S is moving in a - X’ axis direction relative to S’. These substitutions give

                                                    and

                                                                 Equation 25

Both Eqs. (24) and (25) give the difference of D"=cT" in velocity - v between S and S’ along the X’ axis in terms of x’. I shall use Eqs. (22) through (25) at a later point in the paper.

Since D"=cT" is the difference in distance-time between the S and S’ frames, it is also the difference between the various sets of events occurring here-now in the S and S’ frames. Consequently, for every distinct reference frame, there is a distinct set of events occurring here-now in that frame. Any of the Eqs. (22) through (25) can be used to describe the difference between the here-nows of the S and S’ frames. I shall give a clearer description of this after I have derived a few more equations.

I next substitute the distance-time of x in Eq. (20) into ct of Eq. (18), deriving

                                                                Equation 26

In Eq. (26) D’=cT’ is the actual distance along the X’ axis in the S’ frame which is dilated relative to S.

Distance-time theory offers two explanations not found in relativity of how speeds faster than speed c cannot be attained by particles crossing distance and time. The first reason is that space travels at speed c. Therefore, at speeds faster than c, space has not happened yet; thus there is no space to travel across. The second explanation proposes that c is the only speed intrinsic to the structure of distance and time in the universe. All other speeds possess only a part of the speed c according to Eq. (16). Consequently, the maximum speed that v can be is c, according to Eq. (16).

A three-dimensional distance-time manifold has an indefinite number of eventons moving along all possible paths and directions within matter and along the same path with light. Relative to any observer, a reference frame within this manifold possesses distance and time according to how distance and time is traversed by eventons. Whatever periods of time the eventons move across, relative to the observer, the reference frame is defined across those periods of time too. The distance that the eventons cross along their paths relative to an observer is the distance occurring along that path in that reference. The motion of the eventons is intrinsic to the distance-time manifold, and the speeds of matter that are slower than c are part of the motion of these eventons in matter. Hence, these speeds of matter, which are smaller than c in and relative to a reference frame, are fractions of the speed that the eventons experience in and relative to that reference frame. In essence, all experiences of time, distance, and motion in and relative to an observer’s reference frame are given by the eventons. Nonetheless, eventons are defined only as integral parts of matter and light. They are never located outside of light and matter. Therefore, a reference frame is essentially all the time and space that can be measured by particles of light or matter relative to an observer. This perception of a reference frame is a proper perception of a distance-time manifold. Moreover, such a perception is in reverse to the relativistic and classical theories of time and space. In these traditional theories, first time and space are defined, and then particles are placed in these prefabricated structures of time and space. In contrast, distance-time theory uses an observer who measures with particles to define time and space. This approach corresponds to how an observer actually experiences time and space via particles.

In a distance-time manifold, the position of a particle in space is defined by the relative measurement of its location by an observer who measures with another particle. In other words, light traveling from an object to my eye not only allows me to see the object, but it also literally defines the object’s location in space relative to me. This is not the case with relativity theory, in which space-time is defined first, and then objects are placed in the space-time structure, independent of the measurement of their locations by an observer who measures with particles. Also, space is infinitely fast and defines both position and velocity of a particle in special relativity. This means that at any given point of time the exact location and speed of a particle are defined in relativity theory, contradicting Heisenberg’s uncertainty principle but agreeing with classical space and time theory. [4, 5] Therefore, special relativity is a classical theory. In distance-time theory, on the other hand, the location of an object in space is dependent on the measurement of its location by an observer who measures with particles. This is the argument that Heisenberg used to defend his uncertainty principle. He stated that knowing a particle’s exact position and speed is not possible because an observer would always measure with a particle that would always influence the measurement. [4] Therefore, the defining of an object’s location and speed in a distance-time manifold agrees with Heisenberg’s uncertainty principle.

In classical and relativity theories, time is defined along one dimension on a fourth axis. Relative to a body at rest in its own reference frame, all points of space, therefore, exist at the same point of time. Thus, there is no difference in time among different vector coordinates, and time must be treated only as a scalar in these traditional theories. Also, in these theories, distance may be treated as a scalar and a vector because distance is defined as a magnitude of difference between distinct points in a three-dimensional coordinate system. Although these vectors may be arranged in negative directions relative to each other, the magnitudes of these vectors are always a positive distance.

In distance-time theory, time is defined as a magnitude of difference between points in a three-dimensional coordinate system. See Eqs. (1) and (3). Consequently, periods of time can be treated as vectors. Although these time vectors may be defined in a negative direction with respect to each other, their magnitudes always represent positive periods of time. Thus, these time vectors cannot define a negative period of time relative to an observer. For example, let’s suppose that a photon is traveling straight from point P to Q, and simultaneously another photon is going straight from Q to P. I will assume that the photons pass through each other. Both photons will define a time vector in opposite directions to each other, but these photons still define positive quantities of time relative to a reference frame for matter. Therefore, negative magnitudes of time would not be defined. Since distance and time are defined as the difference between vector coordinates in a distance-time manifold, distance and time can be treated either as a scalar or a vector in distance-time theory. The same thing is true for momentum and energy. Although time can be defined as a vector, only the scalar quantity of a period of time can be divided into a distance.

Traditionally, momentum was treated as a vector, and energy was treated as a scalar. Yet, in distance-time theory, both distance and time are treated as a vector or a scalar. Therefore, both energy and momentum can also be treated as a vector or a scalar in distance-time theory. The traditional types of energy potential, kinetic, and rest energy can be redefined as potential, kinetic, and rest scalar momentum respectively. This scalar momentum, as I shall later show, is equivalent to the energy of an object. Also, the traditional concept of a momentum can be redefined as a vector energy, analogous to a vector period of time. Although this vector energy is equivalent to the momentum of an object, its magnitude is not equivalent to the kinetic energy of an object. Vector energy can have a negative direction just as momentum can. However, the magnitude of a vector energy is always positive. Since energy and momentum are both conserved, scalar momentum and vector energy are also conserved in similar fashions. I shall derive the vector energy and scalar momentum in section 5 of my paper when I derive the special theory of relativity.