CHAPTER 1: INTRODUCTION

The aims of this study are:

1. To describe a pre-selected geometric structure of time.
2. To calculate the dynamic mathematical properties of this structure of time.
3. To relate these mathematical properties of the structure of time to known physical phenomenon, including the speed of light and gravitation.
4. To relate the mathematical properties of the structure of time to current mysteries of science, including the nature of the Planck constant, the accelerating expansion of the universe and the “pioneer anomaly”.

These aims were achieved by describing the pre-selected geometric structure of time and then performing mathematical computations on this structure to determine its properties. Logic and spatial visualization were employed to relate these abstract mathematical computations to known physical phenomenon of the speed of light, gravitation and the Planck constant. The mathematical computations were further considered to predict previously unexplained phenomenon, including the accelerating expansion of the universe and the “pioneer anomaly”.
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© John Evans 2011: Theory of Time Mechanics

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

CHAPTER 3: Relating Mathematical Formula to Known Physical Phenomenon

3.1 CONNECTION BETWEEN SPEED OF “NOW” TIME SHELL AND SPEED OF LIGHT

3.1.1 The Speed of “NOW” Time Shell and the Speed of Light

The mathematical formulas for the rate of speed, acceleration and the increase in the rate of acceleration of the “NOW” time shell have been determined in Chapter 2 for any future volume of time V(f). Each of these mathematical formula can be directly related to known physical phenomenon. The first to be considered is the speed of the “NOW” time shell.

Just like the exterior shell of the total time sphere representing the beginning of time and the center representing the end of time, the “NOW” time shell represents a specific time; NOW. Anything closer to the center of the total time sphere is in the future. Anything further from the center is in the past. Also, anything traveling at a speed that is faster than the “NOW” time shell is in the future, and anything slower is in the past. The speed of this “NOW” time shell corresponds to something that is familiar to us in everyday life. The key statement is that anything traveling slower than the “NOW” time shell is in the past. From the geometry of this study,  objects in the past must be traveling slower than the speed of the “NOW” time shell. Also, it is known that the speed of objects in the past, which is the same thing as three-dimensional space, can travel up to the speed of light, but not faster. Therefore, logically, the “NOW” time shell is traveling at the speed of light. This may be quite a leap of logic, so some visualization may be helpful. In section 3.1.4, a visual picture of how “energy” travels from one “NOW” time shell to another “NOW” time shell at the speed of light is described.

Logic and visualization, however, are not sufficient evidence that the speed of the “NOW” time shell is equal to the speed of light. The concept that the inward speed of the “NOW” time shell being equal to the speed of light is critically important to the overall study of time mechanics and, therefore, requires further analysis, as provided in the following three sections.

3.1.2 Time Dilation, Speed of “NOW” Time Shell and Speed of Light

To evaluate the speed of the “NOW” time shell;  first visualize a single point traveling through normal space (not the time dimension) on a straight line vector. The single point is in motion and travels a distance “c” over some period of time. Now in contrast, visualize a second single point (also traveling through normal space) in motion with the same starting position, but this point does not travel on the original straight line vector. Instead this second single point angles off in a different direction at an acute angle to the original straight line vector. This second single point travels the same total distance “c”, but at a distance “b” along the original straight line vector and at a distance “a” along a straight line vector at a 90° angle to the original straight line vector. This scenario creates the classic Pythagorean Theorem equality a²+b²=c². From this equality, the ratio b/c can be computed as follows:

The above formula represents the ratio of the distance the second single point traveled along the original line vector relative to the total distance traveled. This may not look terribly enlightening, but be patient.

It may be recalled that there are not only three spatial dimensions, but there is a fourth dimension representing the time dimension. In the above example, the two dimensions were both spatial dimensions and could easily be visualized. Now attempt to visualize that the original line vector in a spatial dimension, in the initial example,  is replaced by a line vector in the time dimension. From the geometry of time mechanics, this line vector is equivalent to the inward motion of the entire “NOW” time shell in the direction of the center of the total time sphere. Just like in the initial example, the single point is traveling, in the time dimension, the same distance “c”. Next, if this single point is redirected, by some force, it now has a velocity in a different vector direction. The angle of this different vector direction is always a 90° angle relative to the time dimension, just like evaluating vector angles in the three spatial dimension. Now substituting the values into the above  equations where “c” is the total distance traveled, “a” is the distance traveled in the vector represented by a spatial dimension and “b” is the distance traveled in the vector represented by the time dimension. The consequence of this substitution is that when “a” is zero, the single point is at rest in the three spatial dimensions and moving along the time dimension at its maximum distance (speed) which is represented by “c”. However, when the object is in motion with a positive (non-zero) distance traveled in a spatial dimension, the distance moved in the time dimension is represented by “b” distance (speed). The single point is consequently moving through time at a slower rate. The faster the single point is moving through the spatial dimension vector, the slower the object is moving through the time dimension vector. The single point is always, however, traveling the same total distance (speed), which is represented by “c”, regardless of the distance traveled (speed) in a spatial dimension. None of this information is new to science; this example is simply a description of time dilation that Albert Einstein explained many years ago. The effects of time dilation have been verified with numerous experiments. See section 4.5 for a further analysis of Lorenz transformations and time dilation.

The purpose of going through this evaluation of time dilation at this junction, however, is to point out that the value used for “c” in calculating time dilation is always the speed of light. Therefore, the total velocity of the single point in the above example is always the speed of light. When the single point is at rest in the three spatial dimensions, the motion of the single point through the time dimension is always at the speed of light. From the basic description of the geometry of time mechanics, the single point in our example is actually the same as a “NOW” time shell within the total time sphere.  The total time sphere is the fourth dimension, time. Therefore, the conclusion that the inward speed of the “NOW” time shell is always the speed of light is validated.

3.1.3 Conservation of Energy and Energy

One of the fundamental principles of physics is that energy must be conserved. In other words, energy is neither created nor destroyed. In section 2.4, concerning the acceleration of the “NOW” time shell, the natural conclusion is that some energy is being added to the “NOW” time shell to cause this acceleration. This conclusion is flawed, no energy is being added. The “NOW” time shell maintains a constant quantity of energy. It is correct that the speed of the “NOW” time shell is quantified by the formula

and this speed is also accelerating over the passage of time. However, this speed formula is only quantifying the speed of a single spot on the entire surface of the “NOW” time shell. Energy is actually distributed over the entire surface of the “NOW” time shell. Therefore, the total energy of the “NOW” time shell is the product of the speed of the “NOW” time shell multiplied by the surface area of the “NOW” time shell. From classical geometry the surface area of a circular shell is

Also, in an earlier section of this study, the radius of the “NOW” time shell was defined as

Substituting the “NOW” time shell radius formula into the formula for the shell surface area results in the formula

The product of the speed of the “NOW” time shell formula and the surface area of the “NOW” time shell formula is the total energy of the “NOW” time shell at any specific time, V(f). Therefore, the total energy of the “NOW” time shell is:

From the above computation, it is clear that the total energy of the “NOW” time shell is always constant. The accelerating speed of the “NOW” time shell acts to maintain this constant total energy and this total energy remains constant as the “NOW” time shell accelerates inward toward the end of time at the center of the total time sphere. Therefore, energy is conserved.

Now that the total energy of the inward moving “NOW” time shell has been calculated to be always constant with a value of 1, the energy E of the entire volume of future time V(f) can next be calculated. The energy of this volume of future time is calculated by taking the product of the total energy of the inward moving “NOW” time shell at any specific time V(f) (which is a constant 1) and the distance to the end of time (R-r) at the current time. Simplifying, the total energy of the volume of future time is just the distance to the end of time. It is very interesting that the computation of total energy of this volume of future time is determined solely by a distance measure, not a speed measure.

Next, change perspectives and view the volume of future time within the “NOW” time shell as a singularity within a time source point, instead of as the future part of the total time sphere. This view of future time as a singularity within a time source point is the perspective perceived in everyday experience. Recall that an individual time source point is an individual particle of matter. Therefore, the total energy E of a time source point (matter) is:

E – Distance to End of Time

The computation of the distance to the end of time is discussed in section 2.6 and calculated in section 4.2.

What would happen if the total energy within a time source point were released? What happens is the inward motion of the “NOW” time shell in the time dimension is redirected into the spatial dimensions. However, not just the energy of th current “NOW” time shell, but the energy of the future “NOW” time shells, to the end of time, are redirected and released all at once into the present. That would be a lot of energy to be released in one spot at one time. Quantifying the amount of this energy is discussed in the following paragraphs.

In section 2.3, the first derivative of the distance to the end of time (R-r) was calculated to be the speed of the “NOW” time shell. (The absolute value of the first derivative of (r) is the same as the absolute value of the first derivative of (R-r)). Thus, the first derivative of the total energy of the time source point would also be the speed of the “NOW” time shell. (Since the total energy of a time source point is equal to the distance to the end of time.) Further, from basic calculus, the first derivative of the formula, , is also mathematically equal to speed. Consequently, by equivalence,

Finally, by substitution, the total energy E of a single time source point is:

This is the total energy of a single time source point. If the total energy of a cluster of time source points is to be computed, just multiply the right side of the energy formula by the total quantity (m) of individual time source points to be measured. Therefore, the total energy of a cluster to time source points, according to time mechanics, is:

Obviously, this equation is very similar to th famous equation formulated by Albert Einstein;

The value in the time mechanics equation is explained by the way nuclear energy is traditionally computed with the splitting of particles into two parts. The two parts total the amount of energy released. A further discussion of the rationale for measuring energy as the sum of two individual time source points is presented at the end of this study, in the section covering the Planck constant. Measuring the energy from only one of the two pars, however, gives the in the above formula. The speed, c, in Albert Einstein’s formula is always the speed of light. Therefore, the speed of the “NOW” time shell toward the center of the total time sphere, to be consistent with Albert Einstein’s formulation, must also be the speed of light. This consistency of speed in the energy formulation, further validates the equality of the speed of light and the inward speed of the “NOW” time shell (which was the point of this long discussion about energy).

It is a common misperception that the equation for energy,, is dependent on the speed of light as the determining variable. In actuality, the time mechanics equation, , is a measure of the distance to the end of time. It just so happens that the derivative of the distance to the end of time formula is the speed of the “NOW” time shell. The derivative of this distance to the end of time is simply a calculation of the rate of change in the distance over time, which is speed. The computation of energy, in physics, should be focused on the distance to the end of time equation, not the speed of light squared equation. The distinction may be significant to the study of nuclear energy.

The equality of the speed of light and the speed of the inward motion of the “NOW” time shell validates the equation

When the inward speed of the “NOW” time shell is equal to the speed of light, c, by substitution the equation becomes:

The fraction     is eliminated if mass is computed for pairs of time source points. So mass m is the quantity of pairs of time source points and the equation is simply

 

From the above analysis, the conclusion that the inward speed of the “NOW” time shell being equal to the speed of light is further validated.

3.1.4 The Propagation of Light

In the previous sections, the equality of the speed of light and the inward speed of the “NOW” time shell was validated. This section describes the process of light (energy) traveling between two individual time source points, according to the principles of “time mechanics”. This process runs contrary to visual experience in everyday life, so careful attention must be paid to the distinction between an individual time source point and a “NOW” time shell in the explanation. Additionally, the frame of reference is important to visualization of the concept. Start with two individual time source points that are stationary relative to one another in three-dimensional space. The relative motion is not important, but it is simpler to visualize with no relative motion. In other words, the two time source points are just sitting there motionless in the spatial dimensions. Next, change perspectives and visualize these two individual time source points not as points, but as two individual “NOW” time shells. The first “NOW” time shell is traveling inward, toward its own center, at the speed of light, while the second “NOW” time shell is following and maintaining a constant distance from the first “NOW” time shell. The second “NOW” time shell must maintain a constant relative position with respect to the first “NOW” time shell, as required by our original relative positioning of the two individual time source points. The second “NOW” time shell is also traveling inward, toward its own center, at the speed of light, with the first “NOW” time shell also following and maintaining a constant distance from the second time shell. This may sound confusing, but each “NOW” time shell is maintaining a constant distance from each other. Relative to one another, the two “NOW” time shells are both stationary.

Next, consider that the first “NOW” time shell has excess energy to shed. It simply dumps the energy and continues on its way. The first “NOW” time shell is moving inward, at the speed of light, and the discarded packet of energy is stationary (in the shape of a shell). The second “NOW” time shell, which continues to follow the first “NOW” time shell, slams into the stationary packet of energy. The impact occurs at the speed of light, because the second “NOW” time shell is traveling at the speed of light and the energy packet is stationary.

Relative to these two “NOW” time shells, however, the packet of energy appears to be moving at the speed of light, away from the first time shell, toward the second time shell. From the original perspective of the two stationary time source points, the energy packet (light) appears to travel from the first stationary time source point to the second stationary time source point, at the speed of light. This is the “apparent motion” of energy perceived in everyday life, in that light appears to be traveling at the “speed of light” from source to target.

Understanding the propagation of light, according to time mechanics, is useful in the interpretation of the “two slit” experiment and “quantum entanglement” in quantum mechanics. The key to this interpretation is the concept that light (energy) is stationary and the target is in motion, relative to this stationary light (energy). The target is actually propagating toward that single point of light (energy). In the case of the “two slit” experiment, the “two slits” (and the background wall) are both moving toward the same point and make contact with the light (energy) together. In “quantum entanglement”, the stationary point of light (energy) can be viewed as a shell of light (energy). The entire surface of that shell of light (energy) remains entangled as if it were still a single point, so what affects one part of the shell, instantaneously also affects all parts of the shell.

3.2 ACCELERATION OF THE “NOW” TIME SHELL AND INCREASE IN RATE OF ACCELERATION OF “NOW” TIME SHELL.

Recall, as computed in section 2.4, the “NOW” time shell not only has a specific speed, but it is accelerating. Therefore, further expanding on the statement above, about the speed of the “NOW” time shell; if the “NOW” time shell is always moving at the speed of light and the “NOW” time shell is also accelerating, then, the speed of light is accelerating over the passage of time. The acceleration of the speed of light is very significant. This study shows, in a later section, that this acceleration of the “NOW” time shell, and the acceleration of the speed of light, is one of the two key ingredients in calculating the gravitational constant and, therefore, explaining the cause of gravitation. Also, it will be shown, at the end of this study, that the acceleration of the speed of light is a significant factor in the computation of the Planck constant.

One might ask the question, if the speed of light is increasing, why hasn’t this increase been measured? The most likely explanation is because the rate of acceleration of the speed of light is so small that it is simply difficult to measure.  The current accuracy of the measurement of the speed of light is 4 parts per billion. That indicates the accuracy of the speed of light is approximately plus or minus 37,844 Km/yr. This level of accuracy is quite impressive, but in a later section of this study the rate of acceleration of the speed of light is shown to be only 835 Km/yr/yr. Far too small a rate of acceleration to be measurable using current methodology. This possible answer does not  mean that the increase in the speed of light can’t be measured, just that it may be difficult. In section 4.4 of this study, the increase in the speed of light is used to explain some current mysteries in science, which may provide a methodology to measure the rate of acceleration of the speed of light with precision.

Next, recall that the rate of the acceleration of the “NOW” time shell is also increasing with the passage of time. This means the rate of acceleration of the speed of light is also increasing with the passage of time.

The implications of both, the acceleration of the speed of light and the increasing rate of acceleration of the speed of light, will be explored in later sections of this study. Specifically, the physical phenomenon existing as a consequence of this acceleration and increase in rate of acceleration are discussed.

3.3 CONNECTION BETWEEN THE ACCELERATION OF THE “NOW” TIME SHELL AND THE HUBBLE CONSTANT

In the year 1929, the famous astronomer Edwin Hubble, discovered that the spectral lines of light emanating from distant galaxies, all show a “red-shift.” One scientific explanation for a “red-shift” is that the source of the light is moving away from the observer of that light. Naturally, the conclusion of Edwin Hubble, and other astronomers of the time, based on this observational data, was that the distant galaxies are all moving away from the Earth in an expanding universe. Astronomers were able to calculate this rate of expansion and, working backward in time, concluded that the universe must have started approximately 13.7 billion years ago in a tremendous explosion. The force of this explosion caused the past and present expansion of the universe. This theory is called the “Big Bang” theory. Recently, it has been discovered that, not only is the universe expanding, it is expanding at an accelerating rate. The accelerating universe idea was a big surprise to most scientists, as they were expecting the force of gravitation to be decelerating the rate of expansion.

For this study, a different conclusion to the spectral line “red-shift” observations is explained. This alternative conclusion is that the “NOW” time shell is moving inward at the speed of light, and this rate of speed is accelerating with the passage of time. This increase in the speed of light over time is responsible for the “red-shift.” The speed of light was slower in the past, and between the time that the light originally emanated from the source, to the time that it reached the observer; the observer (and the source) accelerates to the current speed of light.

The spectral lines associated with events that occurred many years in the past are recorded when observing distant galaxies, and the increase in the speed of light over this period of time has stretched the light and red shifted the spectral lines within the light waves by the time they reach the observer. The red-shift that Edwin Hubble reported from his observations resulted from the “apparent motion” of the packet of energy. However, under time mechanics, as the packet of energy increases in apparent speed (it appears to be traveling at the speed of light which is accelerating) over time, the distance traveled per unit of time is increased, but the wave frequency is constant per unit of time. The result is a greater distance between wave peaks or troughs, therefore, the red-shift.
(See figure 4)

This “alternative conclusion,” that the spectral line “red-shift” is caused by the increase in the speed of light over time, implies that the relative positions of the galaxies to the Earth are actually fairly constant and the galaxies are, therefore, not speeding away from the Earth. The current speed of light is approximately 9,461 billion km/yr. and an estimate, based on time mechanics (see appendix), for the Hubble constant is 835.26 km/yr/ly (ly = light year). In a previous section, it was suggested that the speed of the “NOW” time shell is always the speed of light and, here, it is further suggest that the rate of acceleration of the “NOW” time shell (and the speed of light) is actually represented by this Hubble constant.

Since the Big Bang theory is predicated on the conclusion drawn from Edwin Hubble’s observations of the spectral line “red-shift”, the Big Bang theory may need to be re-examined based on a different perspective, as provided by time mechanics. According to the Big Bang theory, at the initial moment, the entire universe was compacted into a point. This point had no dimension or size. In other words, there was no space and no time. Due to some unknown event, (which for the moment can be attributed to the hand of God) this point began expanding and, over a period of at least 13.7 billion years, increased in size to encompass what we call the universe. The term “Big Bang” is somewhat misleading in that the event was not the result of a gigantic explosion in which all matter was thrown outward into th universe, like some giant supernova explosion. According to the Big Bang model, what happened was that space expanded to fill the universe and matter came along for the ride. Time mechanics provides a very simple picture of the events at the beginning of time and the smooth progression of events through to the present time. In the beginning, the universe consisted of an almost unlimited number of small points. These points had zero radius and, therefore, could be compressed into a dimensionless point. The individual points are actually individual time source points, as described in this study. At this point at the beginning of time, there is no past time and, therefore, the time source points are just dimensionless points, with no surrounding past time. (In the expanded image of the total time sphere, the “NOW” time shell is, at the beginning of time, in conjunction with the outer shell.) At the instant that time began, all the individual time source points began extruding past time in a steady outflow, filling a spherical volume, with each individual time source point at the center of its’ own volume of time. Relative to the individual time source points, the size of the universe expanded and increased in volume. Each individual time source point had its own volume of past time and all these volumes of past time melded together to form what is called space. Space is just the accumulated volumes of all individual past time volumes. The universe continued expanding from the beginning to the present time and will continue to expand. The rate of expansion is determined by the rate of acceleration of the “NOW” time shell within the total time sphere, as explained in the previous paragraphs. The rate of expansion was very slow at the beginning of time, but has steadily increased in rate of expansion to this present time. The rate of expansion will continue to increase until the end of time. The customary image of the Big Bang is that it all started with a tremendously hot, violent explosion. This is only partially correct; it was tremendously hot, because all the energy in the universe was compacted into a very small volume. That would meet the definition of hot. However, there was no explosion, according to the computations and formula in time mechanics, the volume of the universe expanded in a very slow and orderly rate at the beginning of time, with no violence. The Big Bang theory has been very useful in understanding the past history of the universe and is consistent (subject to some minor modifications) with the time mechanics model.

3.4 CONNECTION BETWEEN THE ACCELERATION OF THE “NOW” TIME SHELL AND GRAVITATION

3.4.1 INTRODUCTION TO GRAVITATION

Gravitation is commonly thought of as a force that causes objects to attract each other. On the Earth’s surface, dropped objects fall toward the center (down) and people feel the pull of gravity whenever they jump into the air and return to the ground. In space, the sun exerts a gravitational pull on planets to swing them around into elliptical orbits. This “force” of gravitation has been studied for hundreds of years and the effect of gravitation is well understood. The current mystery is what causes the effect of gravitation. That is the question that confounded both Sir Isaac Newton and Albert Einstein. Newton chose to ignore the question entirely and Einstein worked on the question for the last 30 years of his life, but could not find the answer. This section of the study will explain what causes the effect of gravitation and answer the question: Why is the gravitational constant a specific number? Time mechanics will show that gravitation is the result of the distorted geometry of the time “source point” universe, interacting with the acceleration of the “NOW” time shell.

During the discussion of a “time source point” and a “total time sphere,” the image of standing on the surface of the “NOW” time shell looking outward had been mentioned. One would see past time. What one would see if you turned around and looked inward toward the center of the time sphere is not the future, but more past time. The reason is because everything inward, toward the future, is moving inward at a speed that is faster than this “NOW” time shell. Therefore, no interaction is possible between this fast-moving future time and the “NOW” time shell and nothing from the future can be perceived, not even emptiness. Not only would this future time be invisible from the “NOW” time shell, but the whole volume shrinks down into a single point, a singularity. (See figure 5) The “NOW” time shell appears to occupy a single point with nothing inside. The future would no longer appear to exist. Future time actually does exist within this singularity and, by necessity, the geometry of past time (and the “NOW” time shell) is spatially distorted to fill this gap. The past time is stretched, very little at far distances, but in close, the stretching is significant. To make it clear; even though the time source point is the image of the universe known in daily life, future time is still present within the time source point and must be considered a part of our universe. The distortion of this space/time will be described in the next section.

3.4.2 THE TIME SOURCE POINT

To understand the concept of gravitation, the geometry of the “time source point” universe must first be described. The “time source point” universe is what remains when the entire volume of future time, within the total time sphere, is shrunken down into a singularity. The “NOW” time shell shrinks down to a single point (surrounding the future time singularity) and the surrounding universe of past time becomes spatially distorted around the “time source point.” The outermost shell representing the beginning of time does not shrink, but the entire volume of past time is stretched inward. The “time source point” is consequently a single point, without dimension or size, surrounded by all past time. The most significant characteristic of the time source point is that it continually extrudes a uniform volume of time outward in all directions, when the relative perspective of the time source point is taken. Recall, in the “total time sphere” universe the “NOW” time shell is moving along inward at the speed of light and a constant volume of time is flowing from the future into the past relative to this moving “NOW” time shell. From the perspective of the “time source point,” being stationary at the center, a constant time volume appears to flow outward from the center.

For the moment, the acceleration of the “NOW” time shell that was discussed in the first sections of this study is disregarded in order to concentrate on the geometry of the source point universe, but be assured it will be included in the final calculations.

As stated above, a time source point is a dimensionless point that extrudes time in an orderly manner. Time extends outward in all directions from the center of the point. (See figure 6) From the perspective of the source point, the source point is situated at “now” time and is surrounded by a sphere of past time. This extruded past time is what is commonly called three-dimensional Space. The farther away a particular past time shell is away from the center, the farther that event is into the past. The following geometry is the result of this extrusion of time: over one unit of time, a fixed volume of time is extruded from this dimensionless point to fill a sphere with a volume of one unit of time. The source point is at the exact center of this sphere of time. After two periods, this point is now surrounded by a sphere containing a volume of two units of time. After ten periods, the point is surrounded by a sphere containing a volume of ten units of time. These periods continue to be extruded from the source point until the end of time. These time spheres continually expand away from the original point like a balloon being filled with air.

3.4.3 THE TIME SOURCE POINT GEOMETRY

As was pointed out above, after one period of time, the shape of the source point universe is a sphere with a volume of one unit of time with the source point at the center. Given any volume of time involved, the radius of this time sphere can be numerically computed. From classical geometry, it is known that the volume of a sphere is determined by the formula

( See figure 7 )

This formula can also be written to solve for the value of the radius

( See figure 8 )

To determine this rate of change in the value of r for different values of v, evaluate the first derivative of the above formula as follows:

This formula quantifies the rate of change in the spatial distortion existing at any point within the “time source point” universe. In other words, this is the relative amount that the fabric of space/time is stretched at any particular location away from the center. The value for the rate of change can range from a value of almost infinity, close to the source point when r is small, to a value of almost zero, close to the outermost shell representing the beginning of time, when r is at its maximum. The fabric of space/time is stretched very little, far away from the center point, and is stretched a great amount near the center. This rate of change formula represents the first half of the equation that answers the question regarding the cause of gravitation.

3.4.4 THE ACCELERATION OF THE “NOW” TIME SHELL

There are actually two factors that cause gravitation. The first factor, which was discussed in the previous section, is the framework upon which the second factor does its work. The second factor is a force of acceleration. Acceleration was described in the first section of this study; the acceleration of the “NOW” time shell with the passage of time. To illustrate how the acceleration of the “NOW” time shell causes the “force” in the force of gravitation, two individual “time source points” are examined. Again, it is important to keep in mind the distinction between a time source point and a “NOW” time shell in this illustration. The first “time source point” is called the primary “time source point” and the other is called the secondary “time source point”. For this illustration, these two individual “time source points” start out stationary relative to each other. Next, if the primary source point is viewed instead as a “NOW” time shell, this “NOW” time shell is known to be moving inward at the speed of light, and the secondary “time source point” is also moving toward the center of the primary, at the speed of light. They are not only both moving at the speed of light, but they are both accelerating. As long as this illustration is confined to individual unconnected “time source points”, the primary and the secondary will continue to accelerate and maintain a constant relative position with respect to each other. However, when the primary individual “time source point” is replaced with a clustering of “time source points” (matter), the inward speed of this cluster of “time source points” is no longer capable of acceleration. The reason why a clustering of “time source points” can’t accelerate toward their individual centers is because the cluster would be geometrically unstable; therefore, the cluster must simply travel inward at the speed of light without acceleration.(How this speed slow down occurs is discussed in section 4.5.2) This non-acceleration of the primary cluster of “time source points” (matter) allows the accelerating secondary to catch up and actually impact the primary cluster. The secondary, in the form of an individual “time source point” or a cluster of “time source points” (matter), continues to accelerate toward the primary until impact. This acceleration toward the primary is the second of the two components of gravitation. The acceleration of the “NOW” time shell is the “force” behind the force of gravitation.

3.4.5 CONNECTION BETWEEN THE ACCELERATION OF THE “NOW” TIME SHELL AND GRAVITATION

When the rate of acceleration of the “NOW” time shell (acceleration of the speed of light) is multiplied by the spatial distortion framework of the time source point universe, the total effect of gravitation is determined. The product of these two factors (acceleration of the speed of light and the spatial distortion framework of the time source point universe) is the acceleration toward the center, called gravitation.

To demonstrate the mathematical calculations involved in the force of gravitation, “Time Mechanics” is used to calculate the value of the gravitational constant and the resulting force of gravitation.

In the year 1687 Isaac Newton formulated the theory of universal gravitation. This theory expressed the force of attraction between two bodies with mass by the equation:

F is the magnitude of the force between two point masses

G is the gravitational constant

is the mass of the first point mass

is the mass of the second point mass

r is the distance between the two point masses

Years after Isaac Newton formulated this theory the value for the gravitational constant was measured with some precision and the current value assigned to this “universal” constant is

To compute the predicted value G from “TIME MECHANICS”, the starting point is the classical formula for force expressed as:

In the simplest example of an individual source point, mass can be expressed as Mass=1 for each time source point. At this point it is just a mass = 1, Kg will later be added as the unit of measure. The formula for acceleration, however, is a little more complex. As stated in a previous section, the formula for acceleration of the “NOW” time shell was determined to be expressed by the formula

This value for acceleration must now be multiplied by the spatial distortion caused by the geometry of the “time source point” universe. Specifically, the distortion of past time that results from all future time shrinking down into one point must be factored into the computation of acceleration. The “time source point” spatial distortion is defined as the first derivative of the formula , which is . The value v in this spatial distortion formula represents the volume of time between the primary “time source point” and the secondary”time source point” and r represents the distance between the two points. Therefore, the “combined” acceleration for the gravitational force computation is expressed by the formula

The combination of the two formulas, above, is similar to the combination of two pulleys in lifting. Each individual pulley has a lift ratio, but the two pulleys used in combination are multiplied to determine the combined lift ratio.

At this point the gravitational force and the gravitational constant G can be calculated. In previous sections, the current “NOW” values for the following three formulas were computed:

And

And

Mass in this situation is defined as 1.

The gravitational force equation can now be computed based on th above formulas:

Since the value of Distance,r, in the above equation has no inherent units of measure, those units of measure need to be added. To add those units of measure we simply define the units of measure for r to be meters, so r squared would be so many meters squared. Therefore, in order to not change the total value of the equation, the original m (meters) in the numerator is now m cubed. Also, since the value of Mass of 1 has no inherent units of measure (as stated previously), those units need to be added. To add those units of measure, the units of measure of Mass is defined to be Kg. Mass is now 1 Kg and the denominator includes Kg to the minus 1 power, also, in order to not change the total value of the equation. Finally, this force equation is for one object. To make this a gravitational force between two objects, both sides of the equation need to be multiplied by another Kg unit of measure so that the numerator of the far right is now measured as the sum of the two individual objects measured in Kg’s. Therefore, the formula for the force of gravitation for a pair of objects, each with a mass of 1 Kg, at the current “NOW” time, is predicted to be

This predicted value agrees with the current experimental measurement of the gravitational constant. The significance of this calculation is that the “Time Mechanics”  predicted the value of the force of gravitation and the gravitational constant.

Continue to Chapter 4 >>

© John Evans 2011: Theory of Time Mechanics

CHAPTER 4: Relating Mathematical Formula to Current Mysteries of Science

4.1 CONNECTION BETWEEN SPEED OF LIGHT, HUBBLE CONSTANT AND THE END OF TIME

In a previous section, the ratio of the speed of the “NOW” time shell to the acceleration of the “NOW” time shell was computed as the equation

Additionally, the current speed of the “NOW” time shell is the current speed of light and the current acceleration of the “NOW” time shell is the current value for the Hubble constant. Substitute the current speed of light for the numerator (speed) and the current Hubble constant for the denominator (acceleration) to solve for the unknown value, V(f). This formula computes the total volume of time remaining in the future. In other words, the result of this computation tells in how many years will be the end of time! The computation is as follows:

Therefore, the end of time is approximately 7.5 billion years from now. No, this does not actually mean that time ends. Think of it like a boat floating down a river; the boat will eventually reach the mouth of the river and the journey will end for that boat, but the river remains. There may be other boats on the river of time that are further upstream.

It is important to understand that the measurement of time is in terms of a volume of time in three dimensions. In the next section, the measurement of linear years or years (linear) to measure a distance from one point to another is used. This is the same distinction between a meter as a distance measurement, and a cubic meter as a volume measurement. The next section will explore the measurement of the distance from the “NOW” time shell to the center of the total time sphere. Recall, the center of the total time sphere is the end of time.

4.2 CONNECTION BETWEEN THE SPEED OF THE “NOW” TIME SHELL, VOLUME OF FUTURE TIME AND DISTANCE TO THE END OF TIME

Now that the volume of time remaining until the end of time has been determined in the previous section, it is possible to compute the distance to the end of time. Two methods exist. The first will give results measured in years (linear) and the second method will give results in kilometers. The first method is fairly straightforward with use of the formula; Distance

Plugging in the value V(f) = 7,551,341,299 yrs into the Distance formula gets Distance = 1,217.060 years (linear). Not very enlightening without a familiar frame of reference showing how many kilometers are in a year (linear). Not very enlightening without a familiar frame of reference showing how many kilometers are in a year (linear). Therefore, the second method of calculating distance is considered. This second method uses the ratio of Distance divided by Speed as computed in Section 2.6.

Now inserting the speed of light for speed, 7,551,341,299 yrs for V(f), and solving for Distance, results in one year is equal to approximately Distance = 214,000,000,000,000,000,000,000 km. This means that the distance to the end of time is 2.14 times 10 to the twenty-third power kilometers away.

To compute the number of kilometers there are in a year (linear), divide the distance in kilometers by the distance in years (linear), which results in one linear year is equal to approximately 176,000,000,000,000,000,000 kilometers. That means a year, which is a volume, would fill a sphere measuring 1.76  times ten to the twentieth power kilometers in radius. This conversion ratio between years linear and kilometers is important when calculating the formula values based solely on the volume of future time V(f). (See table at end of this study).

The distance to the end of time of kilometers is very far. To give a scale frame of reference, the current estimated size of the known universe is approximately 93 billion light years in diameter and the speed of light is 9461 billion km / year. So the size of the entire known universe is kilometers. The known universe only includes the part of the universe that can be seen (according to time mechanics), which is the part of the universe that is in the past. So, the distance from the “NOW” time shell to the center of the total time sphere (and the end of time) is larger than the distance across the known universe. That is very far, even at the speed of light.

4.3 Connection between the Increase in the Acceleration of the “NOW” Time Shell and the Accelerating Expansion of the Universe

One necessary feature of any good scientific theory is the capacity to make testable predictions. Based on “Time Mechanics,” the following predictions can be made: The Universe should appear to be expanding at an accelerating rate over time and the actual rate of this acceleration can be calculated. Actually, the first part has already been shown by others. The second part, the predicted rate of this acceleration, can be calculated, based on time mechanics, by again using the ratio method. In this case, take the ratio of the formula for acceleration of the “NOW” time shell over the formula for the increase in acceleration of the “NOW” time shell, as follows:

Substitute the current value for the Acceleration of the “NOW” time shell for the numerator and the current volume of future time V(f) for the value on the right side and solve the equation for the denominator on the left.

Therefore, the rate of this acceleration should be approximately .000000184351 km/yr/yr/ly. This means that the further an object is from any point in the Universe, the faster it should appear to be traveling away from that point, and that the rate of increase should be increasing. The word “appears” is used because the expansion of the universe has been measured by the amount of spectral line “red-shift” and the “red-shift” was shown earlier in this study to be caused by the acceleration of the speed of light over time, not the actual expansion of the universe. Therefore, the “Hubble constant” would, based on the computations of “time mechanics”, not actually be a constant, but is currently very slowly increasing over time. The increase in the rate of apparent expansion in the Universe is extremely small and requires distances in the range of vast intergalactic scale, to be measurable. This measurement has recently been accomplished by two independent teams that have both concluded that the Universe is expanding at an accelerating rate. However, it is yet to be determined if the actual measured rate of acceleration agrees with the rate predicted by “time mechanics.” Confirmation of the rate of acceleration would be a good indication that this theory has some merit. If “time mechanics” is correct, it should be concluded that the cause of this apparent increase in rate of expansion of the Universe is not some mysterious dark matter or dark energy, acting as some form of negative gravity. The accelerating expansion of the Universe is the natural consequence of “time mechanics.”

4.4 Connection between the Acceleration of the “NOW” Time Shell and the Pioneer Anomaly

A few years ago, a team of scientists discovered that the actual trajectories of the two Pioneer spacecraft, launched in the early 1970’s, are both deviating from their expected trajectories. This deviation is equivalent to a sunward acceleration of , which was deduced to be a “blue-shift” in the spectral lines of the light originating from the spacecraft, relative to what was expected. Scientists believe they have accounted for all known causes of this acceleration, but the deviation still persists. This mystery created an excellent opportunity to test the theoretical principles of “Time Mechanics” against an actual real world problem. “Time Mechanics” can validate this mysterious sunward acceleration of the Pioneer spacecraft, and as a result, provide confirmation of the acceleration of the speed of light. As shown in previous sections, “Time Mechanics” predicts that the speed of light is very slowly accelerating over time by a specific rate. It has also been shown, in previous sections of this study, that the acceleration of the speed of light is the cause of the apparent expansion of the universe and is one of the two factors responsible for gravitation. The effect of the acceleration of the speed of light is apparent on the large-scale and the small-scale. Now it can be shown that the effect is in evidence on the intermediate scale as represented by the distances to the two Pioneer spacecraft. If this prediction is indeed correct, then the light coming from the Pioneer spacecraft should also be accelerating by that same rate as predicted under time mechanics. Note that I said the light was accelerating, not the spacecraft. An important point to keep in mind, at this point, is that whenever an object is observed, like the Pioneer spacecraft, that object is not seen directly, but instead what is seen is the light originated from that object. This is important because the “blue-shift” seen in the light is actually resulting from the acceleration of the speed of the light, not from the movement of the spacecraft. When light, originating from the Pioneer spacecraft travels to the observer, that light accelerates and, therefore, requires a slightly shorter period of time to reach the observer than anticipated. This shorter period of time between the spacecraft and observer corresponds to a “blue-shift” in the light from the Pioneer spacecraft, relative to the amount expected from Newtonian gravitation alone. So, does this mean that the actual Pioneer spacecraft does or does not have a sunward acceleration? The answer to that question depends on how the distance to the spacecraft is measured. If distance is measured by the amount of time required for light to traverse between the two points, then, yes, the spacecraft does possess a sunward acceleration. The rate of net acceleration towards the sun by the Pioneer spacecraft, predicted by “Time Mechanics”, is equal to the rate of acceleration of the speed of light. This rate of acceleration was shown in a previous section to be or when converted to meters per second squared, . The actual measured rate of acceleration toward the sun of the Pioneer spacecraft, unexplained by other means and referred to as the “Pioneer Anomaly”, is . Please note that the predicted amount is well within the margin of error of the actual data. This result is a good indication that “Time Mechanics” may have some validity.

This “Pioneer Anomaly” is not some form of additional gravitation. The acceleration is only affecting the light than is transmitted from one point to another as reflected in the “blue-shift”. The “Pioneer Anomaly” is not distance dependent, so it does not matter how far away the source of the light is from the target; the marginal rate of acceleration when the light reaches the target is always the same. Gravitation works very differently in that it has a direct effect on objects and its force diminishes with distance by the inverse squared law.

There may, at this point, be some explanation required for the apparent contradiction of the same theory predicting a “red-shift” and a “blue-shift”. The explanation is quite simple; it is true that, in both situations the cause is the result of the increase in the speed of light over time, however, two effects are at play. The first effect, which is responsible for the “red-shift”, is caused by the large increase in the speed of light over vast intergalactic distances and large periods of time, as was discussed in a previous section. This is reflected in the fact that the actual “Hubble Constant” is measured in terms of light years, which is a distance measure, not a time measurement. This means that the greater the distance from the source of the light to us, the larger the “red-shift”. The second effect, which is responsible for the “blue-shift” in the light from the Pioneer spacecraft, is caused by the small marginal acceleration in the speed of light from one moment to the next. Scientists have excellent trajectory measurements for the Pioneer spacecraft going back many years and can eliminate all other variables, so the “blue-shift” can be isolated. Consequently, for objects far away, at vast intergalactic scale, the first effect predominates and overwhelms the second effect, resulting in a net “red-shift”. For objects, like the Pioneer spacecraft, the first effect is too small to be measurable, but the second effect dominates and results in a net “blue-shift”. Therefore, the Pioneer spacecraft anomaly is another example in which the small, but measurable, acceleration of the speed of light is detectable.

4.5 Time Dilation, Lorenz Transformations and the Planck Constant

4.5.1 Introduction to Time Dilation and Lorenz Transformations

Imagine driving a car down a long highway at a constant rate of speed. Parallel to this highway is railway track with a very long train pulling freight cars. The train is traveling at the same rate of speed as the car. When the driver of the car looks out th side window at the train, he sees that he is even with the first freight car. He drives along for some period of time and again looks out his side window and sees that he is still even with the first freight car. He repeats this activity at various time intervals and continues to see that he is even with the first freight car. Eventually, the car must angle off to the left in order to go around a large mountain, but the train continues a straight path. There is a tunnel through the mountain for the train traffic. The car and the train continue to maintain the same constant speed. After some period of time, the car has made its way around the mountain and returns to a parallel path alongside the train. The driver looks directly out his side window and notices that he is even with the thirtieth freight car, with the engine and first twenty-nine cars in advance of the car. From then on the car and the train continue to progress at the same rate of speed with the car even with the thirtieth freight car.  It is quite clear what is happening in this scenario, but a similar (but not exactly) thing occurs when a spaceship leaves Earth at a high rate of speed and eventually returns. The clock on the spaceship appears to have progressed at a slower pace than clocks on Earth. This is called time dilation. To understand what is occurring, a change of perspective is required. In the above car and train example, instead of the train and car initially traveling in one of the three spatial directions, imagine they are both traveling in the time dimension. When the car angles off to go around the mountain, it is angling off into one of the spatial directions, but still also moving in the time direction. They both continue at the same rate of speed, but the train is progressing through the time dimension at a slightly quicker pace than the car. The result is that when the car and the train reunite after the car has angled off and returned, the train has progressed through the time dimension further than the car has progressed. It appears that the train has moved through time at a quicker rate. When we think of the fast-moving spacecraft scenario, however, it appears to create a paradox, because the spacecraft appears to be just moving through space, not the time dimension.

Time Mechanics can explain this paradox. The starting point, in the explanation of how the clock on a fast-moving spaceship appears to move slower than the clock on Earth, is the image of two individual time source points. The first time source point, called the primary, is stationary. The second time source point, called the secondary, is moving through space at a velocity v. The direction of motion of the secondary time source point is not important; it is always perpendicular to the time dimension. To illuminate how Time Mechanics works, the perspective of the primary time source points is switched from a time source point to a “NOW” time shell. The primary’s “NOW” time shell,  as mentioned in previous sections, is moving inward at the speed of light and the secondary time source point is following.  The problem is at what speed is the secondary time source point following the primary “NOW” time shell through the primary’s time dimension.  The secondary time source point can only travel through space and time at one total speed; the speed of light. The speed through space and time that this secondary time source point is moving is computed using simple geometry. As stated above, the secondary “NOW” time shell is moving through normal space at some velocity v. The direction of this motion through normal space is always perpendicular to the direction that the “NOW” time shell is moving through time. Motion in the time dimension is perpendicular to all spatial directions because it is a fourth dimension relative to the three spatial directions, just like the three spatial directions are all perpendicular to each other. This spatial vector speed, at right angles to the time vector, along with the total speed (the speed of light) provides the information necessary to compute the inward speed of the secondary time source point in the time dimension using the Pythagorean Theorem. The result of the formula (where c is the speed of light, v is the velocity through normal space and x is the speed through the time dimension) is used to make the computation. The secondary time source point’s total combined speed is always the speed of light, but its speed through the time dimension, when it has a positive velocity in a spatial dimension, would be slower than the primary’s “NOW” time shell through the time dimension. . This is similar to the analogy of the car and the train above. This is the effect that is called “time dilation”. To compute the amount of time dilation, the Pythagorean Theorem formula is used as follows:

Where x is the inward speed of the secondary time source point toward the center of the primary “NOW” time shell, v is the velocity of the secondary through normal space and c is the speed of light. (See Figure 9) This formula is algebraically modified as follows:

Plugging in the values of v and c into the right hand equation gives a fraction between zero and one. This is the ratio the secondary object’s clock is moving relative to the rate that the primary’s clock is moving. When the value of v is zero the ratio is one and the clocks are moving at the same rate. When the value of v is a large fraction of the speed of light, the clock on the moving object is slowed significantly. This formulation is named the “Lorentz Transformation” and was used extensively by Albert Einstein is his Theory of Relativity. Interestingly, this formulation shows that whenever a person moves through normal space, even walking across the street, that person is a “time traveler”, relative to stationary objects. Usually, however, the effect of this time travel is too small to perceive. Lorentz Transformations and Time Dilation is also critically important on the very small-scale as demonstrated in the following paragraph. Without their influence, matter and life would not exist.

4.5.2 Solitary Time Source Points vs. Cluster of Time Source Points

As mentioned in section 3.5.4, “time source points” sometimes cluster together to form matter and the inward speed of the “NOW” time shells making up this matter do not accelerate. This differential between the non-accelerating primary cluster of “time source points” and the accelerating secondary cluster of “time source points” is the current of acceleration in the speed of light. How this occurs is the subject of quantum mechanics, the details of which are beyond the scope of this study. However, (speculatively) there must be some form of resistance to inward motion caused by the geometry of the cluster of “time source points”and the energy from this resistance must manifest in some form. From the Lorentz Transformation computations above, it is necessary that this cluster of individual “time source points”, as they move inward at a reduced speed, progress through time at a slower speed. Also, this reduction in inward speed must be redirected into one of the three spatial directions in order to conserve energy. The physical manifestation of this movement may be a resonant vibration of the “time source point”. To visualize this resonant vibration of the “time source point”, consider the “time source point” not as a point but as a “NOW” time shell. Next, visualize a specific spot on the “NOW” time shell. This specific spot may trace small circles perpendicular to the inward motion of that specific spot. In essence, every location on the entire surface of the “NOW” time shell is tracing the same small circles in unison, creating a resonant vibration of the entire structure. The small circles may be either clockwise or counter-clockwise, but not both on the same “NOW” time shell. When tracked over the passage of time, any specific point on the “NOW” time shell would spiral inward toward the center of the total time sphere. Now returning to the image of the “time source point”; this resonating “time source point”, when tracked over the passage of time, would form a wave pattern. As a consequence, energy emitted from a solitary “time source point” would not vibrate and energy emitted from a cluster of “time source points” would vibrate. Therefore, a cluster of “time source points” would be considered normal matter and emit energy in a wave patter. A solitary “time source point” would be “dark” matter and emit “dark” energy without a wave pattern. “Dark” matter would still have gravity and, therefore, would be detectable by its gravitational field. “Dark” matter would not be detectable by its energy emissions because of the lack of a wave pattern. The amount of “dark” matter existing in a galaxy could be approximated by the amount of gravitational force measured that is unexplained by the existence of normal matter alone.

4.5.3 The Planck Constant

In the year 1899 the famous German physicist, Max Planck, formulated what is now considered one of the three primary “universal” constants in physics. This constant is called the “Planck Constant”. The other two “constants” are the speed of light and the gravitational constant. For this work he received the Nobel Prize in Physics. When this Planck Constant, “h”, is multiplied by the frequency of the vibrations of a particle of matter, the product is the rest energy, “E”, of the particle of matter. The formula is written as:

E = h * frequency

The numerical value of this “Universal” constant is(Joule Seconds). Note that this number is very small. The formulation of this constant was one of the basis for the development of quantum mechanics and is used extensively in the understanding of the interaction of matter on the very small-scale.

Time Mechanics is useful in understanding the nature of this Planck Constant. To begin this understanding of the Planck Constant: compare the difference in the inward speed of the “NOW” time shell for “normal” matter to the inward speed of the “NOW” time shell for “dark” matter. Recall, as discussed in section 4.5.2 above, the inward speed of the “NOW” time shell for “normal” matter is slower than the inward speed of the “NOW” time shell for “dark” matter. The differential between these two speeds is the current rate of acceleration of the “NOW” time shell:

 

In order to accomplish this slowdown in speed of the “NOW” time shell, the “NOW” time shell must vibrate in the spatial dimensions. This vibration was discussed in the previous section and was suggested, in that section, that this vibration is in the form of small circular motions at each location on the surface of the “NOW” time shell. The effect of this motion in the spatial dimensions is to take advantage of time dilation to cause a slowdown in the progression of the “NOW” time shell through the time dimension.

To calculate the rate of speed of this spatial vibration of the “NOW” time shell necessary to accomplish this very specific rate of slowdown, it is possible to use the Lorenz Transformation formula:

 

This formula can also be written as:

 

In this formula c is the speed of light, x is the inward speed of the “NOW” time shell toward the end of time, and v is the speed of the “NOW” time shell vibration in the spatial dimension. The current values for c and x are as follows:

 

Plugging in these two known variables into the above equation and solving for the unknown results in a value for v as follows:

 

This is the speed at which the “NOW” time shell is vibrating in the spatial dimension in the form of small circles perpendicular to the inward motion of the “NOW” time shell toward the end of time at the center of the total time sphere. Care should be taken to understand that this is the speed of the vibrating “NOW” time shell, not the frequency of the vibrations.

At this point, all the information required to compute the Planck Constant is available. However, before making the computation it is necessary to discuss what the Planck Constant is used for in physics. When the Planck Constant is multiplied by the frequency of vibration of regular matter, the product is the amount of energy contained in that matter when it is at rest. The frequency in the equation, E = h * frequency, is a measure of the quantity of matter being measured. When the quantity of matter is doubled, the frequency is doubled. When the quantity of matter is tripled, the frequency is tripled, and so on. The Planck Constant, h, is a measure of the amount of energy in a single unit of at rest matter. (Actually, it is the amount of energy in a pair of units of matter, since we are only dealing with normal matter. Dark matter is a single solitary unit of matter and has no spatial vibration.) A single unit of normal matter would be two “NOW” time shells bound together.

As was discussed in section 3.1.3, the total energy of a time source point is:

 

Also in the same section, the energy of a “NOW” time shell is:

E = Speed of “NOW” time shell formula * Surface Area of “NOW” time shell formula = 1

Therefore, the ratio of the energy of a “NOW” time shell to the total energy of a time source points is:

 

The total amount of energy of the time source point that is detectable in the spatial dimensions (not the time dimension) is proportional to the small part of the “NOW” time shell’s total speed detectable as motion in the spatial dimensions (not the time dimension) to th total speed of a “NOW” time shell:

 

Therefore, the Planck Constant is determined by the formula:

 

This is the portion of the total (at rest) energy of a pair of time source points (normal matter) that is detectable in the spatial dimensions. Recall, this is the formula for a pair of “NOW” time shells, hence the 2 in the equation. (This is necessary because if it were for only one “NOW” time shell, it would be “Dark” matter and there would be no spatial vibration). Albert Einstein and Max Planck were both consistent in using a factor of two in their famous formulations, even though neither understood the concept of Dark matter not vibrating in the spatial dimensions.

The mathematical computation is a follows:

The unit of measure conversion necessary to convert to Joule Seconds, J * s, is similar to the conversion used in the formulation of the Gravitational Constant. Therefore, the Planck Constant is:

 

There is another formula in physics referred to as the “Reduced Planck Constant,,” which is simply the Planck Constant divided by 2π. This reduced Planck Constant is useful when it is necessary to compute the cycles that the “NOW” time shell vibrations make as it traces a circular motion in the spatial dimensions. Recall, the circumference of a circle is simply 2πr. Therefore, to get the number of cycles, simply divide the speed by th circumference. (Radius, r, is assumed to have a value of 1 in this case). This reduced Planck constant is useful in a number of applications in physics.

The formula for the Planck constant can also be written in terms of Time Mechanics notation with the single variable being the volume of future time remaining at the present time, V(f). This formulation provides a unification of the Speed of light, Gravitation, Planck constant, acceleration of the expansion of the Universe and the rate of increase in acceleration of the expansion of the Universe. Each of the values can be derived from only one variable; the volume of future time, V(f). The value of the Planck constant in terms of Time Mechanics notation is:

 

The unification of the Speed of light, Gravitation and Planck’s constant is very significant in the general understanding of fundamental physics.

4.5.4 Unification of Gravitational Force, Speed of Light and Planck Constant

In section 3.5 the formula for gravitation was computed using the product of the rate of acceleration of the “NOW” time shell and the spatial distortion framework formula. The product of these two is the force of gravitation. In sections 3.1.1 through 3.1.4 the speed of light was determined to be equal to the inward speed of the “NOW” time shell. In section 4.5.1 through 4.5.3 the Planck constant was derived from the speed of the “NOW” time shell, the acceleration of the “NOW” time shell and the distance to the end of time. All three of these “fundamental constants” (the gravitational constant, the speed of light and the Planck constant) are ultimately derived as a function of the volume of future time V(f). The unification of these three fundamental constants has been the goal of scientists for years and is now unified under the theory of Time Mechanics.

Some scientists strive for the elusive goal of unifying the four  fundamental forces of nature under one unifying theory, referred to as a “Theory of Everything”. These four forces of nature are the Gravitational force, the Strong Nuclear force, the Weak Nuclear force and the Electromagnetic force. Whether the Theory of Time Mechanics is the foundation for a Theory of Everything, I will leave to others to decide. However, it is clear that the unification of the three fundamental constants, as was discussed above, is critical in achieving the goal of a Theory of Everything. Certainly, understanding the unifying basis for the speed of light, the gravitational constant and the Planck constant is critically important for any proposed Theory of Everything. 

© John Evans 2011: Theory of Time Mechanics

SUMMARY

By understanding the geometry and movement of the “NOW” time shell and the time source point, the fundamental physical properties of the universe become unified and can be defined as a simple function of the value of the total volume of future time remaining until the end of time. The speed of light, gravitation constant and Planck constant are unified under one theory. Even more importantly, the time mechanics model provides a fundamental basis for the existence of space, time, matter and energy. All of these four concepts are manifestations of one structure; the “total time sphere”.  The following table summarizes the mathematical relationships between the volume of future time, distance to the end of time, speed of light, acceleration of the speed of light, increasing rate of acceleration of the speed of light, the gravitational constant and the Planck constant. The value for each of these can be predicted for any time in the past and future. A surprising and satisfying observation about the details of “Time Mechanics” is that the mathematical solutions are elegantly simple.

This study has had a limited scope and there are plenty of unanswered questions that remain. I believe this study of “TIME MECHANICS” in conjunction with quantum mechanics is the key to understanding the formation of matter and the unification of physics.

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© John Evans 2011: Theory of Time Mechanics

APPENDIX

*Note: The observed value for the Hubble constant is actually a fairly large range of values. In my initial test of concept for using the value of the Hubble constant to predict the value of the Gravitational constant, 77 km/sec.mpc was used in the computation. The results were close enough to validate the concept, but not perfect. Knowing that the margin of error for the Hubble constant is large and the margin of error for the Gravitational constant is small, I reversed the computation and plugged in the value for the Gravitational constant working backward, computed a more exact value for the Hubble constant. The result of this computation gives a value of 86.40 km/sec/mpc for the Hubble constant. Based on the model of Time Mechanics, the value for the Hubble constant was smaller in the past. Additionally, the measurements of observational data used to determine this Hubble constant is from light emitted in the far distant past. Predictably, the observational value for the Hubble constant should be smaller than the current marginal value. Thus, the value of 86.40 km/sec/mpc is the current marginal rate of acceleration of the “NOW” time shell and the speed of light.

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© John Evans 2011: Theory of Time Mechanics