CHAPTER 2: THE STRUCTURE OF TIME

2.1 STRUCTURE OF TIME

The concept of “time” in science is usually linked with the three dimensions of normal space. For this reason, time is usually referred to as the fourth-dimension. The customary image is for normal space to occupy three dimensions composed of an x-axis, a y-axis and a z-axis. An item is located within this space by reference to these three spacial coordinates. This information is sufficient for location when referring to something that is stationary, like your home. However, when referring to something in motion, like an airplane, a fourth dimensional reference is required to specify the location at a specific time. Therefore, the location of an airplane in flight may, for example, be 29.89° latitude, -81.31° longitude, 1,000 feet of elevation above sea level at 3:00 pm eastern time. Within this customary image of four-dimensional space/time, the time dimension is viewed as one single dimension. Therefore, a stationary point in space, traced through the time dimension, result in a straight line. This time dimension is difficult to visualize, but it can be represented  mathematically. This customary image of space/time, however, is incomplete because the stationary point in space is only indicating location at the current “NOW” time. That straight line through time only shows the path of the current “NOW” time as time progresses. The concept of past and future are ignored completely in this image of time. “Time mechanics”, however, expands upon this customary image of time to include, not only the current “NOW” time, but the entire geometry of time, which includes past, present and future time.

The study of time mechanics commences with a geometrical model of the structure of the time dimension as the initial starting condition and , from there, proceeds to the exploration of some interesting dynamic characteristics of time. It is recommended, to the reader, that the presented geometrical model should initially be viewed as an abstract structure. Later on, the model will be correlated to the real world. This initial abstraction may help avoid confusion at the beginning.

The initial description of the structure of time starts with a total emptiness; no space, no time, no matter and no energy. The first object that is introduced into the model is something familiar; an individual point. This individual point is called a “time source point“. A time source point represents the present; “NOW” (See right side of figure 1). Next, this individual time source point is surrounded by a spherical volume containing all past time, relative to that individual time source point. For the moment, please ignore the concept of space and just think in terms of a volume of past time. The “distance” outward from a time source point determines how far objects are into the past; close to the time source point is the recent past and far away is the distant past. The maximum distance, “r”, from the time source point is the distance to the outer shell, which represents the beginning of time, relative to that time source point.

Now that a superficial description of a time source point and its surrounding universe of past time is described, future time needs consideration. If one is to think of the beginning of time as the outermost shell of the time source point universe and time progresses inward until reaching “NOW” at the center, it may be deduced that time continues inward and future time is inside the time source point. Figure 1 shows how the time source point “universe” (the right side of the Figure 1) is perceptually expanded to include a representation of future time (the left side of figure 1). This expanded structure, called a “total time sphere”, includes all past, “NOW” and future time. This total time sphere is the structure of time that is the primary subject of this study. (Please view the representations of the time source point universe and the total time sphere in Figure 1 as three-dimensional spherical objects, even though the diagram appears in two dimensions).

An individual “total time sphere” (see figure 2) consists of spherical volume of time. To help visualize this spherical volume, imagine an inward moving three-dimensional spherical shell within the total time sphere called a “NOW” time shell. This “NOW” time shell starts out large in conjunction with the outermost perimeter of the total time sphere at the beginning of time, and as time progresses, shrinks down to a single point in the center of the total time sphere, at the end of time. At each instant, the image of the “NOW” time shell is a three-dimensional shell, but over the whole of time, from beginning to end, the shells add up cumulatively to a total time sphere. Each individual total time sphere is located in its own separate fourth dimension, the time dimension. There is no universal fourth dimension of time. It is important to keep in mind the concept that the entire “NOW” time shell is conceptually equivalent to an individual point; the time source point, as described at the beginning of this illustration. The only difference is that the “NOW” time shell contains all future time within its shell in an expanded form and the time source points contains all future time within its point as a singularity. Otherwise, they are the same phenomenon.

In accordance with the total time sphere geometry of Time Mechanics, the outermost exterior shell of an individual total time sphere is the beginning of time. The center of the total time sphere is the end of time. Located between the exterior shell and the center, is an inward moving spherical shell that is the “NOW” time shell.  The volume of time outside the “NOW” time shell, out to the exterior shell, contains all past time, and the volume of time inside the “NOW” time shell, in toward the center, contains all future time. The “NOW” time shell originally started at the beginning of time at the exterior shell, and with the passage of time, travels inward toward the center at some speed.

This “NOW” time shell contains a volume and has a radius. Both the volume and radius get smaller as the “NOW” time shell travels inward. If one visualizes standing on this inward moving “NOW” time shell looking outward, only past time is visible. The further any object is from this position on the “NOW” time shell, the further the object is in the past.

What is seen upon turning around, while standing on the “NOW” time shell, and looking inward toward the center of the total time sphere? Some naturally guess the future. That guess is not correct, because of a relationship explained in a later section of this study. Future time is not visible while standing on the “NOW” time shell, and not only is future time invisible, but the whole interior volume shrinks down into a single point, a singularity. The “NOW” time shell appears to occupy a single point with nothing inside. The future no longer appears to exist, but it does. Further, and very importantly, the past time volume is distorted or stretched to fill this missing future time volume. So, when turned around, all that is seen is more past time. The implications of this apparent geometry are very important and will be explored in later sections of this study.

Before proceeding, however, the definition of a “NOW” time shell and a “time source point” needs re-emphasis. The “NOW” time shell and “time source point” are, in actuality, the same phenomenon, with one major exception:  the “NOW” time shell is a spherical shell, within the total time sphere, which includes the volume of all future time within its shell in an expanded form.  The “time source point”, however, includes all future time, within this time source point, as a singularity. The time source point universe is the geometry observed in everyday experience, however, a clear vision of the geometry of the total time sphere is essential to understanding fundamental physics. Next, the geometrical model of the time dimension is correlated to the real world of three-dimensional space.

Where does “space” fit into this geometry of time mechanics? The conventional image of space incorporates a single universal three-dimensional volume existing independent of matter and energy. The Theory of Relativity maintains that, even though this spatial volume is warped by the presence of matter, only one universal space exists and all matter and energy is located within this volume. The existence of this single universal space created a problem for scientists; what are the properties of this space and what is its origin? This conventional image of space also implies that time is a separate dimension. Time Mechanics, however, eliminates the requirement of a single universal three-dimensional space: For this theory, space does not exist as a distinct structure independent of time. To explain; each individual total time sphere occupies its’ own individual universe of time. Within each individual total time sphere is a “NOW” time shell which is the boundary between past and future. When this “NOW” time shell is perceived as an individual “time source point” singularity, it occupies a distinct individual position, relative to other time source points. For this reason, individual time source points are matter. Additionally, each individual time source point is surrounded by its’ own spherical volume of past time extending out to the beginning of time at its’ own exterior shell, relative to that time source point. These individual spherical volumes of past time all meld together to form a structure called “three-dimensional space.” Three-dimensional space is simply the overlapping of all individual past time volumes. The concept of space (and our image of the Universe) is only a simple human convenience for quantifying the relative relationships between all the individual time source points. Empty three-dimensional space and past time are synonymous.

To give a visual picture of how space fits into the time mechanics model, visualize two individual “NOW” time shells (“NOW” time shell “A” and “NOW” time shell “B”) located adjacent to one another, with some separation between the two structures. Future time for “NOW” time shell “A” is located inside “NOW” time shell “A” and future time for “NOW” time shell “B” is located inside “NOW” time shell “B”, respectively. (See Figure 3)

Keep in mind; each “NOW” time shell is spherical even though the diagram only represents them visually as circular. When these two individual “NOW” time shells are perceptually converted into their representation as two individual time source points, the entire volume of future time within each “NOW” time shell is shrunken down into two individual singularities. (Why this occurs is explained in later sections) The two “NOW” time shells are now two individual points. These two points are represented in Figure 3 as points “A” and “B” within the three-dimensional spatial universe. There is still a distance between the two points, just like there was a separation between the two original “NOW” time shells in their expanded form. Past time relative to point “A” and relative to point “B” overlap each other in the spatial universe and make up what is usually referred to as empty space. Point “B” is in Point “A”‘s past and Point “A” is in Point “B”‘s past. Because of the shrinking of the individual “NOW” time shells into individual points, the spatial universe is warped in the presence of these points. Close to the individual points exhibits extensive warping, but far away the warping is minimal. (Albert Einstein extensively studied the effect of this warping of space/time in the presence of matter). This image of two “NOW” time shells and two time source points will be used in chapters three and four to illustrate the propagation of light, gravitation and other physical phenomenon. The above illustration has shown the “NOW” time shells frozen in time, without any forward progression into the future. In the next few sections, this study expand upon the geometric structure of an individual total time sphere and explains how its’ “NOW” time shell progresses through time into the future.

2.2 THE GEOMETRY OF TIME

Now that the general shape of the total time sphere has been described, specifics about its geometry and how it changes over the passage of time can be expanded upon.

Using the basic description of the structure of an individual total time sphere as the starting point, classical geometry is next used to calculate the relative relationship between radius and volume. First, visualize the total time sphere as a three-dimensional volume containing  time within its exterior shell. From classical geometry, the total Volume (V) within this total time shell is defined by the formula:

Radius (R) is the radius of the entire time sphere. Next, if Radius (R) is defined as the radius of the entire time sphere and Maximum Distance (r) is defined as the distance between the exterior shell and the “NOW” time shell, then the volume of future time V(f) is defined as:

The expression (R — r) is equivalent to the distance from the “NOW” time shell to the end of time at the center of the total time sphere. Also, the volume of past time V(p) is defined as:

From the basic parameters of this study (see Figure 2), the “NOW” time shell starts out at the beginning of time, in conjunction with the exterior shell of the total time sphere, and moves inward at some speed toward the center. Therefore, the distance (r) is increasing over the passage of time. As the volume of past time V(p) increases, the volume of future time V(f) decreases. The exchange of future time into past time is a one-for-one exchange, in that, for every minute of future time that is used up, an equal minute of past time is created. Of course, this is all relative to the inward moving “NOW” time shell. The total volume of all time remains the same within the exterior shell of the total time sphere. This is similar to the image of an old-time hourglass, with future time represented by the sand in the top of the hourglass and past time represented by the sand in the bottom of the hourglass. A volume of time (sand) passes from future to past as it passes the “NOW” time at the narrow point at the center of the hourglass. Sand, in this example, is only a representation of time, not time. The volume within the total time sphere is also only a representation of time, but it is being used in this study as a means to measure the passage of time.

2.3 SPEED OF THE “NOW” TIME SHELL

As stated earlier, the “NOW” time shell is moving away from the exterior shell of the total time sphere toward the center at some speed. Next, the rate of speed that the shell is moving is calculated. Speed is defined as the amount of distance traveled for a given period of time. Distance is measured by the amount of net change in the distance (r) between the exterior shell of the total time sphere and the inward moving “NOW” time shell. The passage of time is measured by the volume of time passing from the future into the past relative to the “NOW” time shell. The formula, is used to determine speed at any point in time. First, it is necessary to algebraically convert this volume formula into a formula to solve for (r), which is a distance measurement. The result of this conversion is:

To calculate the rate of change in distance (speed) for any particular future time volume, V(f), simple calculus is performed. The calculus required is taught in the first few weeks of a high school or college calculus class. So, don’t be too alarmed by the introduction of calculus into the computation. To determine the speed of the “NOW” time shell; take the first derivative of the above formula. The result of the first derivative is:

(Absolute value is used in all the results to avoid confusion when the direction of time is inward, but the volume of future time is getting smaller).

Recall that R is a constant. This formula represents the speed of the “NOW” time shell at any specific volume of future time V(f).

2.4 RATE OF ACCELERATION AND INCREASE IN RATE OF ACCELERATION OF “NOW” TIME SHELL

From the above speed formula, one can see that as the volume of future time decreases, the speed of the “NOW” time shell accelerates, assuming the progression of time from future to past is at a constant rate. The rate of this acceleration of the “NOW” time shell is computed by again using simple calculus. The second derivative of the above formula is the rate of acceleration of the “NOW” time shell. The second derivative of the radius formula is:

The rate of this acceleration is also increasing over time, so the rate of the increase in acceleration of the “NOW” time shell would be the third derivative of the radius formula which is

To summarize, the inward motion of the “NOW” time shell, toward the center of th total time sphere,  has a speed, an acceleration and an increase in the rate of acceleration as reflected in the first, second and third derivatives of the radius formula, respectively.

2.5 THE END OF TIME

In this section, one of the two endpoints of time is considered; the end of time. In section 2.3 the formula for speed was computed and in section 2.4 the formula for acceleration of the “NOW” time shell was computed. Next, divide the formula for speed by the formula for acceleration.

Simplifying the numerator and the denominator results in the new equation

Determining the current rate of speed of the “NOW” time shell and the current rate of acceleration of the “NOW” time shell provides the numerator (speed) and the denominator (acceleration). Solving for the remaining unknown, the value of V(f), determines the total volume of time remaining in the future at the current time. In other words, the result of this computation indicates in how many years will be the end of time. The determination of these two unknowns and solving for the total volume of time in the future is covered in section 4.1.

2.6 DISTANCE TO THE END OF TIME

The distance to the end of time is computed in a similar manner as the computation for the amount of time remaining until the end of time, as covered in the previous section. The formula for the current distance to the center of the total time sphere is divided by the formula for the current speed of the “NOW” time shell.

Simplifying the numerator and the denominator results in the new equation

Determining the current speed of the “NOW” time shell and the current volume of time remaining in the future provides solutions for two of the above variables. Solving for the remaining variable (distance) provides the distance to the center of the total time sphere and, therefore, the distance to the end of time. The determination of these two variables (speed of the “NOW” time shell and the volume of future time) and solving for the distance to the end of time is covered in section 4.2.

2.7 THE BEGINNING OF TIME

So far, the time volume and distance measurements for the section of the total time sphere represented by the future have been calculated. This section covers the beginning of time and the distance to the outer edge of the total time sphere universe as defined by this study. Ideally, the definition of the beginning of time, under this study, is the specific location and time when the “NOW” time shell was in conjunction with the outermost limit of the total time sphere and have a velocity of exactly zero. This seems like a straightforward calculation. Simply use the formula for speed and keep inserting values for V(f) until the speed is zero. Unfortunately, no matter how large the value for V(f), there is always an amount of V(f) that is larger and closer to zero speed. It is possible to insert a value for V(f) that results in a speed extremely close to zero, but not quite zero. The implication is that the past stretches out into the past for an infinite volume of time and distance. This result produces an interesting thought, a structure that has no beginning point, but has an ending point. That thought creates an interesting paradox, the discussion of which is beyond the scope of this paper. Fortunately, the determination of the exact location of the beginning of time is not critically important to the study of time mechanics. Actually, the volume of future time V(f) is the variable that is most important and is included in every formulation.

One possible solution to this paradox, however, is that at some point in the distant past there may have been something or someone who gave the “NOW” time shell its initial push to break the “inertia” of zero speed. If this initial push occurred a very long time ago, the initial speed required would have been minimal. This initial push may have been some natural physical force or action, the cause of which, I do not understand.

2.8 MATHEMATICAL FORMULAS AND RATIOS

Now that the mathematical formulas for the various properties of time mechanics have been calculated, the interesting part of this study may begin. The third chapter of this study relates these mathematical formulas to known physical phenomenon and the fourth chapter relates these mathematical formulas to some mysteries of science. Each of the formulas computed in Chapter two can be directly correlated with a physical process. Not only can the mathematical formulas be directly correlated with physical processes, but the mathematical ratios between the formulas can be determined. Within each ratio equation, when two of the three variables are known, the third variable can be calculated. Therefore, all the formulas are related to each other. The mathematical formulas and ratios are as follows:

Continue to Chapter 3 >>

© John Evans 2011: Theory of Time Mechanics

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