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
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