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Time and Energy, Inertia and Gravity
The Relationship between Time, Acceleration, and Velocity and its Affect on Energy, and the Relationship between
Inertia and Gravity
Copyright © 2002 Joseph A. Rybczyk
Abstract
Presented is a theory in fundamental theoretical physics that establishes the relationship
between time and energy, and also the relationship between inertia and gravity. This theory
abandons the concept that mass increases as a result of relativistic motion and shows instead that
the extra energy related to an object undergoing such motion is a direct result of the affect that
the slowing of time has on velocity. In support of this premise, new formulas are introduced for
acceleration and used in conjunction with a relativistic time transformation factor to develop new
equations for both kinetic and total energy that replace those of special relativity. These and
subsequently derived equations for momentum, distance, acceleration, inertia, gravity, and
others, establish a direct relationship between the presented theory and the principles of an
earlier theory, the millennium theory of relativity. A final consequence of this theoretical
analysis is the discovery of two new laws of physics involving acceleration, and the realization
that a unified theory of physics is now possible.
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Time and Energy, Inertia and Gravity
The Relationship between Time, Acceleration, and Velocity and its Affect on Energy, and the Relationship between
Inertia and Gravity
Copyright © 2001 Joseph A. Rybczyk
1. Introduction
This paper introduces a new physical science that bridges the gap between classical and
relativistic physics. It provides the final piece of the puzzle started by Newton with his
Principia1 in 1687, and expanded by Einstein with his Special Theory of Relativity2 in 1905.
What will be shown is the true relationship between acceleration, time, and velocity, and
the affect this relationship has on energy and thus our perception of other physical phenomena.
We will see that time expansion is real and its affects on velocity and energy are real, but the
affects on mass (see appendix A) and distance are only perceptual. We will accomplish this by
evaluating and correcting the classical formula for acceleration, and employ the new formula
together with a relativistic transformation factor to directly derive a correct equation for kinetic
energy. This new equation replaces both the Newtonian and Einsteinian equations for kinetic
energy and is subsequently used in the formulation of a new total energy equation that replaces
Einstein’s famous E = MC2 equation. In the process, the limitations of the former formulas and
equations will be clearly demonstrated and their replacements validated. The final results of this
work are new equations for acceleration, momentum, kinetic and total energy, and
transformation factors not directly dependent on velocity. With these new equations it is no
longer necessary to distinguish between classical physics and relativistic physics. In their place
is the beginning of a new physics from which future discoveries may be anticipated.
2. The Millennium Transformation Factor and Time Transformation
In relativistic physics it is widely accepted and supported by evidence that time slows
down in a moving frame of reference. The relationship of moving frame time relative to
stationary frame time can be expressed by the time transformation formulas,
t ` t
c2  v2
c
(1)
and,
t  t`
c
c v
2
2
(2)
Time
Transformation
where v is the relative velocity, c is the speed of light, t` is moving frame time relative to
stationary frame time, and t is time in the stationary frame. In the given equations t and t` can be
used to represent a unit of time, or an interval of time. Also of note is the fact that the more
familiar Lorentz transformation factors,
2
v2
1 2
c
1
(3)
and,
v2
1 2
c
(4)
Lorentz
Transformation
Factors
have been replaced by their respective millennium theory of relativity equivalents3 (see appendix
B),
c2  v2
c
(5)
and,
c
c2  v2
.
(6)
Millennium
Transformation
Factors
In the past, these factors were normally associated with time and distance transformation.
In the present work their use will be expanded to include modification of velocity and a
numerical constant associated with acceleration.
3. Constant Time and Relative Time Defined
There is often confusion about time transformation and the terms used to define
stationary frame time vs. moving frame time. It should be understood that time in the moving
frame is identical to time in the stationary frame. That is, time in both frames may be
represented by the variable t. Thus time t is actually constant time, or time that increases at a
constant rate. Time t` then is actually relative time, or time whose rate of increase is affected by
relative motion. It is used to show that time t in the moving frame is slower than time t in the
stationary frame. Or stated another way, that constant time in either frame has a reduced rate
relative to constant time in the other frame as defined by the transformation formulas. From this
point forward, it should be understood that when we say stationary frame time, we actually mean
constant time t, and when we say moving frame time, we actually mean relative time t`.
4. The New Kinetic Energy Equation
Although the classical equation for kinetic energy seems to be supported by the evidence
for low values of velocity, it is refuted by the evidence for high values of velocity in the area that
represents a significant fraction of the speed of light. At the other end of the spectrum is the
relativistic equation that seems to be supported by the evidence for velocities that are a
significant fraction of light speed, but as will be shown later, is not representative of very low
velocities in the area of classical physics. In fact, in both the real and theoretical sense, neither
equation is truly correct at any velocity. The causes by which this happened are as follows:
1. Relativistic effects were unknown during the development period of classical
physics. The problem is then compounded by the manner in which the classical
kinetic energy equation is normally derived that in turn obscures its true meaning.
2. The relativistic effects, when discovered, were then used in an indirect derivation
for the new kinetic energy equation that in turn obscured the true meaning that
equation.
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To avoid such problems we must derive the classical equation in a manner that makes its
true meaning unambiguous. Then with our knowledge of relativity it becomes possible to
correct the equation in such a manner as to make it properly represent the physical laws of nature
along the entire range of valid velocities.
In classical physics the following formulas are given for momentum, and constant
acceleration:
Where p is momentum, m is mass, and v is velocity, we are given,
p  mv
(7)
Where a is the rate of constant acceleration, and t is the time interval, we are given,
v  at
(8)
v
t
(9)
and therefore,
a
And lastly, where d is the distance traveled by an object under constant acceleration we
are given,
d
1 2
at
2
(10)
Although these equations provide satisfactory results for Newtonian levels of velocity,
none of them are truly correct at any velocity. For the moment we will defer treatment of the
momentum equation and proceed with the equations for constant acceleration. That is, if
relativistic physics is a correct model for physical phenomena, then all of these equations and
others to follow, including the classical equation for kinetic energy will have to be modified.
Until now, however, no one has been successful in figuring out how it could be accomplished.
The problems are many, and involve a very difficult analysis due to circular relationships that
involve many interdependent variables. To proceed we must first decide the order in which the
variables will be used. Additionally, we will need to make certain assumptions that appear to be
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reasonable and in agreement with the evidence. And finally, we must assure that the resulting
circular arguments are self-enforcing throughout the analysis and lead to correct final results.
The first assumptions to be made involve the classical formulas for constant acceleration.
The evidence shows that relative velocity increases asymptotically and therefore equations, 8 and
9, cannot be correct for constant acceleration. If the rate of acceleration, a is constant, and time t
increases at a constant rate, then the instantaneous velocity v, given by equation 8, will also
increase at a constant rate. Such a result is of course inconsistent with the evidence and therefore
in disagreement with the principles of relativistic physics. On the other hand, where v is the
instantaneous velocity, ac is the rate of acceleration, and t` is the time interval, our analysis will
show that,
v  act `
(11)
and,
ac 
v
t`
(12)
Constant
Acceleration
are the correct formulas for constant acceleration. That is, since t` increases asymptotically at
the same rate as v, the rate of acceleration will remain constant as the velocity increases toward
c. To verify this, in a computer math program for example, we will first need to derive different
forms of equations 11 and 12, those that do not depend on the variable t`. This is because, as
seen in equation 1, time interval t` is itself dependent on the value of v. Such treatment,
however, must be deferred to a later point in the analysis. The forgoing is an example of the
circular dependency commented on earlier.
The subscript c in the preceding equations is needed to distinguish between constant
acceleration and relative acceleration, which occurs simultaneously and is defined by,
v  ar t
(13)
and,
ar 
v
t
(14)
Relative
Acceleration
where ar is the rate of relative acceleration. These are nothing more than correct versions of
equations 8 and 9 respectively, rewritten to show their true meanings. Thus, if t increases at a
constant rate, and if v is asymptotic, ar the rate of acceleration relative to the stationary frame
will decrease as the velocity increases toward c.
For consistency with the modifications used in equations 11 and 12, we must also modify
the associated distance formula given in equation 10. This gives us,
d
1
act `2
2
(15)
where the travel distance is denoted by the variable d. This change, however, is insufficient for
arriving at a correct formula for the distance traveled by an object or particle under constant
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acceleration. One of the problems involves the constant, ½. This constant is also affected by the
asymptotical nature of velocity. That is, as c is approached, further increases in velocity
approach 0. Thus, the distance traveled approaches vt`, and not ½ vt`. If this were the only
consideration, we would modify the constant ½ so that its value approaches 1 at a rate that is
consistent with the rate that v approaches c. But there is another factor to consider. One
involving the energy required for the acceleration, and this too affects the required modification.
In the interest of clarity, this modification is best deferred to a later part of our analysis where
such modification can be more easily understood.
Now, by using the right sides of equation 7 (p = mv) and equation 15 we can formulate a
new formula where the temporary variable ka = momentum over distance traveled by an object
under constant acceleration. Thus we have,
1

ka  (mv) act `2 

2
(16)
By substituting v/t` from equation 12 (ac = v/t`) for ac in the above equation, we obtain,
ka  mv
1v 2
t`
2 t`
(17)
which simplifies to,
ka 
1 2
mv t `
2
(18)
Now, since the classical formula for kinetic energy is,
k
1 2
mv
2
(19)
we can state,
6
k
ka
t`
(20)
and by substitution get,
1 2
mv t `
k 2
t`
(21)
The reason for deriving the kinetic energy equation in this manner is to make it clear what the
various variables and constant represent. This last version of the equation can now be evaluated
from a relativistic perspective and corrected to properly represent all values of velocity, v. At
this point it is necessary to call upon the millennium transformation factor, previously introduced
as expression 6.
Referring now to figure 1, we can compare the relativistic motion of a particle to the
Newtonian motion when a constant force is applied. Whereas in Newtonian motion the velocity
increases without limit, in relativistic motion the velocity increases asymptotically as the object
approaches the speed of light c, and of course the speed of light is never exceeded. From what
we understand about relativistic effects, the transformation factors 3 through 6, are mathematical
definitions of the behavior. If time slows down in the moving frame of reference, it is not
unreasonable to assume that this slowing of time directly affects the velocity of the moving
object. Thus, as velocity increases, time slows down causing further increases in velocity to
require increasingly greater amounts of energy. With respect to kinetic energy this is a
paradoxical contradiction of Newton’s second and third Laws of motion. Since kinetic energy is
a direct function of velocity it would increase at a slower rate along with the velocity increases
when at the same time it must increase at an increasingly greater rate along with the energy
causing the acceleration. Obviously it cannot do both. Of the two choices, reason and
experience support the view that the kinetic energy must always equal the input energy.
If we now refer to the classical kinetic energy equation 19, we can see there are only two
variables to choose from should we wish to modify the formula to bring it into conformance with
the evidence. The choices are, mass and, velocity. Einstein made what appeared to be a
reasonable choice at the time and selected mass. If mass increases with velocity, it would
explain the observed behavior. This, it will be shown, appears to have been the wrong choice
and at very least results in an anomaly that is well hidden in the E = Mc2 equation, but is very
apparent in the resulting relativistic kinetic energy equation, K = Mc2 - Moc2. But even more
than this, as the analysis continues the evidence builds in favor of the new theory thus
challenging the very premises upon which Einstein’s equations are founded. It will be shown
now, that neither variable should be modified. This is not to say that the kinetic energy equation
itself should not be modified.
7
v
Newtonian motion
v=c
relativistic
motion
t
FIGURE 1 The velocity of a particle starting at rest
when a constant force is applied.
Referring back to figure 1, and also to equations 12, 15, and 21, (repeated below) it can
be seen that equation 21 must be modified in such a manner as to offset the relativistic effect that
the motion has on velocity.
v
ac 
t`
(12)
1
d  act `2
2
(15)
1 2
mv t `
2
k
t`
(21)
That is, if equation 21 is to produce the correct result for kinetic energy, the experienced
relativistic effect on velocity must in the mathematical sense, be reversed. The obvious
conclusion is that we must factor the velocity by the reciprocal millennium factor, expression 6.
If we are right, however, we must also factor the fractional constant, ½, in equations 15 and 21.
This conclusion is supported as follows: We can assume from equation 15 that the distance, d,
traveled by an object under constant acceleration should = ½ act`2. Referring to equation 12, we
can see that this is the same as saying distance ½ vt`. In other words, the distance traveled by an
object under constant acceleration should = ½ times the velocity, v, achieved for the interval, t`,
during which the acceleration takes place. However, when we study the relativistic motion curve
in figure 1, we can see that as velocity increases toward c, there is less and less change in
velocity over an interval of time. Stated another way, as velocity increases toward c, the distance
traveled by the object approaches vt`, and not ½ vt`. This implies that the constant ½ should
increase toward a value of 1. But since at this point, v, is not only near the value of c but, with
our first change, its value is being dramatically increased by the millennium factor, the constant
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½ must actually be modified to decrease rather than increase in order to compensate and bring
the final result into agreement with the evidence that supports relativity. When these changes are
properly implemented, the distance will progress toward vt` as correctly assumed while at the
same time the resulting kinetic energy equation is a correct equation for all values of v.
Proceeding now with the necessary modifications to equation 21, we derive,
1
c
1
c  v2

c
m v
2
2
 c v
2

 t `

2
k
,
t`
(22)
which reduces to,
k
mc 2v 2
c c2  v2  c2  v2
.
(23)
Kinetic Energy
This equation replaces both, the Newtonian and Einstein’s equations for kinetic energy. (see
appendix C) With it, and other equations presented in this work, there is no longer a need to
differentiate between classical and relativistic physics. For convenience purposes, this new
physics will be referred to as millennium physics from this point forward.
5. The New Total Energy Equation
If we now add the internal energy term, mc2 to equation 23, we arrive at the millennium
equation for total energy that replaces Einstein’s E = Mc2 equation. Thus, where E is the total
energy for an object or particle in motion,
E
mc 2v 2
c c v c v
2
2
2
2
 mc 2 ,
(24)
Total Energy
produces a correct result for all values of v. In view of these findings, it is no longer appropriate
to refer to mass, m, as a rest mass, or for that matter the term, mc2 as the rest energy of an object.
As can be seen, mass appears to be unaffected by velocity. (This assumption will be explored
further in section 11.) Nonetheless, the present theory is in agreement with the relativistic
concept that matter does contain internal energy as defined by the expression, mc2.
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6. Comparison of the Millennium Equations with the Relativity Equations
Under close evaluation it will be seen that the relativity equation for kinetic energy
becomes erratic at low values of v. This behavior is apparent at velocities as high as 100,000
km/hr and becomes very noticeable as the velocity drops below 1000 km/hr. It continues to
intensify as the velocity drops below 100 km/hr, and at approximately 16-20 km/hr, the equation
produces a result of zero. Obviously then, this equation is not reliable at those velocities where
most of our experience resides. There we have to rely on the Newtonian equation for accurate
results. The Newtonian equation, however, is non-relativistic and therefore losses accuracy as
the velocity increases. Subsequently, neither equation provides a good program for analyzing
the entire range of velocities. To make matters worse, it is unclear where reliance on the
Newtonian equation should end and reliance on the relativity equation should begin.
Comparison of Energy Equations
1400
1200
Kinetic Energy
1000
800
Rybczyk
Einstein
600
400
200
0
0
5
10
15
20
25
30
Km per hour
35
40
45
50
Figure 2 Comparison of the millennium kinetic energy equation to the special relativity
equation.
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Figure 2 is an actual graph comparing the millennium kinetic energy equation to the
relativity equation for velocities below 50 km/hr. At these velocities, and with the 15 decimal
place precision used, the millennium curve is indistinguishable from, and therefore synonymous
with, the Newtonian curve. Very evident in the graph is the erratic behavior of the relativity
equation as previously discussed.
Interestingly, analysis of the two total energy equations shows the millennium results to
be identical to the relativity results throughout their entire range. The reason for this is simple.
At the lower end of the scale, the kinetic energy is insignificant in comparison to the internal
energy, mc2 accounted for in the equations. Thus the relativity equation for total energy obscures
the fact that the kinetic energy result is erratic at low velocities. Since the internal energy is also
a component of the millennium equation, and since it is very significant at low velocities, both,
equations will produce the same values for the level of precision used. On the other hand, at the
upper end of the scale, the kinetic energy component is the most significant component of both
equations. And, as was alluded to earlier, at the higher levels of velocity, the relativity kinetic
energy equation becomes increasingly less erratic. Therefore, it is not surprising that both total
energy equations produce identical results.
7. The New Momentum and Distance Equations
Referring back to equations 16, 20 and 22, (repeated below) it should be remembered that
the top of the right side of equation 22 represents momentum over distance traveled.
1
1

ka  (mv) act `2  (16)
2

k
c
1
ka
t`
c  v2
2

 t `

2
k
(20)

c
m v
2
2
 c v
t`
(22)
Therefore we can place the top of right side of equation 22 = to pd, where p is momentum and d
is the distance traveled by an object or particle under constant acceleration.
1
pd 
1
c
c2  v2

c
m v
2
2
 c v
2

 t `

(25)
Now we can rearrange the right side of the resulting equation and separate the momentum
component from the distance component as follows:
11




c
c
1

pd   mv
v
c
c 2  v 2  1 
c2  v2


2
2
c v



t `



(26)
By removing the left component on the right side of the equation and placing it equal to p, we
obtain the equation for momentum.
mvc
p
c2  v2
Momentum
(27)
Here Einstein was only partially correct. Although the most apparent difference between the
new millennium equation and the older relativity equation is the use of the millennium factor in
place of the Lorenz factor, there is a more important difference. In the millennium equation it is
the velocity that is operated on by the factor. In Einstein’s equation it is the rest mass that is
operated on. This of course is a profound distinction between the two theories and will be
discussed further in section 11.
If we now take the remaining component from the right side of equation 26 and place it =
to d, we have one form of the distance equation for an object under constant acceleration,
1
d
1
v
c
c
c2  v2
t`
(28)
c2  v2
which simplifies to,
d
cvt `
c c v
2
2
.
(29)
Distance based on
v and t`
Since, v = act`, for an object under constant acceleration, by way of substitution in equation 29,
we obtain,
12
d
cact `2
c  c  (act `)
2
2
.
(30)
Distance based on
ac and t`
This gives us an alternate form of the equation for distance traveled by an object under constant
acceleration. To arrive at an equation that uses constant time t instead of relative time t` we need
only to substitute the right side of equation 1 for t` in equation 29. After simplification we arrive
at,
d
vt c 2  v 2
c c v
2
(31)
2
Distance based on
v and t
This gives us our final form of the millennium equation for the distance traveled by an object
under constant acceleration.
8. The New Constant Acceleration Equations
Now that we understand how the velocity of an object under constant acceleration is
affected by the millennium factor, we can formulate new equations for constant acceleration that
do not rely on moving frame time. As stated earlier in section 4, equations such as equations 11
and 12 have limited practical application because of the interdependency between velocity and
moving frame time. What we need are equations that use stationary frame time, such as will be
developed now. Given the relationships of equations 1 and 11, (repeated below)
t ` t
c2  v2
c
(1)
and,
v  act `
(11)
we can substitute the right side of equation 1 for t` in equation 11 to obtain,
v  act
c2  v2
.
c
If we now solve for the variable v, we obtain,
13
(32)
v
actc
c 2  act 
2
(33)
Instantaneous
Velocity based on
ac and t
where, v, is the instantaneous velocity of an object or particle under constant acceleration. This
is an important equation. By breaking the direct link between v and t` it allows for the type of
analysis given in the addendum. Such analysis not only allows us to verify the many
assumptions of the presented theory, but also permits a direct comparison of the new kinetic and
total energy equations to those of special relativity over an unlimited range of masses, velocities,
and acceleration rates. Many such analyses were used to verify the validity of all of the
assumptions presented in this work.
If we now solve equation 33 for the variable ac we arrive at its variation for finding
constant acceleration that also does not rely on moving frame time.
ac 
vc
t c v
2
(34)
2
Constant Acceleration
based on v and t
It should be noted that equations 33 and 34 do not invalidate equations 11 and 12. They are
simply the respective constant time equivalents of those two equations. It should also be noted
that in regard to the millennium transformation factor it should normally not matter how velocity
v is arrived at. That is, it is not normally necessary to make a distinction between the velocity of
uniform motion and the instantaneous velocity arrived at through constant or relative
acceleration. Uniform motion is after all, the final result of acceleration.
9. Consistency with Newton’s Second Law of Motion
According to Newton’s second law of motion, where F is the applied force, the
relationship between force and the acceleration of an object or particle is given by,
F  ma, (35)
while according to the present theory, ac taken from equation 12, and ar taken from equation 14,
are expressed as,
ac 
v
t`
ar 
(12)
14
v
t
(14)
The question is, which form of acceleration, ac or ar is intended by Newton’s force equation?
Most, if not all, textbooks, even those of a higher level intended for scientists and engineers do
not make a distinction here. The practice has been to use the classical version of equation 14
even when constant acceleration is explicitly stated. This is because at Newtonian levels of
velocity, t and t` have essentially the same value. On the other hand in disciplines such as
particle physics or theoretical physics where such distinction is necessary, an appropriate switch
is made to relativistic physics. In millennium physics it is recommended that the appropriate
form be used at all times. This permits the investigation of physical phenomena throughout the
entire spectrum of velocities and acceleration rates without concern about which branch of
physics is appropriate.
Continuing now, for constant acceleration, by way of substitution we can then state that,
F  mac
(36)
and,
v
F m .
t`
(37)
Constant Force
Since it is found through mathematical modeling on a computer that v/t` is indeed a constant, this
relationship between force, mass, and acceleration is consistent with the present theory. In
accordance with the present theory, an object under constant acceleration does not experience an
increase in mass. Thus, if the acceleration is constant, and the mass is constant, the applied force
is constant. Because of time expansion, however, it does require greater amounts of energy than
can be attributed to velocity alone to maintain a constant force on an object undergoing
significant acceleration. In other words, a small amount of energy, in a frame of reference that is
moving at a significant fraction of the speed of light, is equal to a large amount of energy, as
defined by the energy equations, in the stationary frame of reference. This is a phenomenon of
which there are countless analogies in everyday life, involving only Newtonian physics.
Consider for example a bullet fired from a gun. If you are in the frame of reference in which the
gun was fired and tried to stop the bullet with your hand, you couldn’t because of its large
amount of kinetic energy. Yet, if you were moving ahead of the bullet at, say, one kilometer per
hour faster than the bullet, you would have no trouble at all of stopping it. That is, in your frame
of reference the bullet has very little kinetic energy. Relativistic effects on energy are nothing
more than an extension of this principle. The only thing that makes it appear strange is that we
are approaching light speed, the standard of measure by which time itself is gauged. This causes
time to slow down, which in turn causes required energy increases, to maintain a constant rate of
acceleration, to be greater than that predicted by the classical principles of Newtonian physics.
In view of these findings, it is not unreasonable to suggest that the new acceleration
equations 11 through 14 are worthy of elevation to the status, “laws of acceleration.” This
recommendation was the subject of a paper subsequent to the original version of this revised
work.
10. The Connection Between Inertia and Gravity
In accordance with Newton’s first law of motion: An object at rest tends to remain at
rest, and an object in motion tends to continue in motion in a straight line unless acted upon by
an outside force. This property of matter that causes it to resist any change of its motion in either
15
direction or speed is referred to as inertia, which is in turn used to define mass. It will now be
shown that such property can be expressed mathematically. Thus, where Ir is the rest inertia of
an object or particle and m is its mass we can state,
I r  m.
(38)
Rest Inertia
This simple acknowledgement that Newton’s first law actual refers to the rest inertia of the
object or particle in question allows us to establish the relationship between inertia and gravity,
or inertial force and gravitational force. If we now combine this formula for rest inertia with one
that defines motion inertia, we will have a complete formula for inertia. As it is, formulas for
motion inertia already exist. They are the momentum formulas 7 and 27. Such formulas express
Newton’s first law with respect only to the contribution that motion has on inertia. Whereas the
rest inertia is always a factor regarding total inertia, momentum is a factor only when motion is
involved, but it is by far the most prominent factor even at relatively low velocities. Thus where
I is the total inertia, the complete formula for inertia is,
I  I r  p.
(39)
Total Inertia
By substituting the right sides of equations 38 and 7 for Ir and p respectively, we obtain,
I  m  mv
(40)
giving,
I  m(v  1)
(41)
for what could be called the classical version of the formula. We can derive the relativistic form
of the formula by using equation 27 instead of equation 7 for p in the above substitution. Thus
we obtain,
I m
mvc
c v
2
2
(42)
giving,
 vc

I  m
 1
2
2
 c v

(43) Total Inertia
for the relativistic form of the formula. Now we are ready to establish the relationship between
inertia and gravity. Where g is the acceleration of gravity, G is the gravitational constant
6.6720·10-11, and r is the distance to the center of gravity, the Newtonian formula for gravity is,
16
g G
m
.
r2
(44)
If we now solve the two inertia equations, 41 and 43, and the above gravity equation for the
variable m, we get respectively,
m
I
v 1
m
(45)
I
vc
1
c2  v2
(46)
m
gr 2
G
(47)
If we then place the right side of the inertia equation, 45 equal to the right side of the gravity
equation, 47 and solve the resulting equation, first for g and then for I, we obtain,
g G
I
2
r v  1
Ig
(48)
r 2 v  1
.
G
(49)
This gives us the classical formulas that relate inertia to gravity. To gain confidence in the
correctness of these two formulas we need simply to substitute the right side of equation 41, in
place of I in equation 48. Since equation 41 is, I = m(v + 1) it should be obvious that the terms,
(v + 1) will cancel and we will end up with the equation for gravity (44) originally given.
Doing similar, using the right side of the relativistic inertia equation, 46 with the right side of the
gravity equation 47 and solving first for g and then for I, we obtain,
g G
I
vc


 1
r 2 
2
2
 c v

(50)

 vc
r 2 
 1
c2  v2
.
Ig 
G
(51)
Gravity
and
Inertia
for the relativistic form of the equations that relate inertia to gravity. Since we know that
classical formulas are not completely valid it is the relativistic formulas, 43, 50, and 51 that we
are most concerned with. As before, if we substitute the right side of equation 43 in place of I in
equation 50, the terms enclosed in brackets will cancel and we will again end up with the
equation (44) originally given for gravity. This proves that the values we initially assigned to
inertia are not only consistent with Newton’s laws of motion, but are also consistent with his
universal law of gravitation.
17
The analysis just completed will be used in the following section as one of the proofs that
mass does not increase as a result of motion.
11. The Mathematical Proofs
Proof 1.
Where k is kinetic energy, p is momentum, d is the distance through which ac acts, and t`
is the time interval, the formula for kinetic energy can be expressed as,
k
pd
.
t`
(52)
Kinetic Energy
For analysis purposes this equation can be expressed in its expanded form as,




1
c
c
 mv

v
c
c2  v2
c 2  v 2  1 


2
2
c v

k
t`


t `



(53)
Since this equation produces the same results as Einstein’s equation for kinetic energy it
is not unreasonable to claim that it is supported by the same evidence that supports the latter
equation. For similar reasons, it can be further claimed that the momentum component of this
equation is also supported by the evidence. With this in mind, if we now examine the
composition of this equation we can draw certain conclusions from it.
1. If the momentum component is valid and the overall equation is valid, it is then
reasonable to assume that the distance component is also valid.
2. Since there is no mass element m in the distance component, and since the interval t`
element disappears when the equation is simplified, there can be no question as to the fact
that the millennium factor operates on the velocity element v in this component of the
equation.
3. In view of the above observations one is then left to explain by what rationale do we wish
to claim that the millennium factor in the momentum component is operating on the mass
m element and not the velocity v element in that component? In plain words, why should
velocity be treated differently in one component as opposed to the other?
18
4. It could be argued that since t` remains an element of the distance formula when it is used
separately, it is actual t` that is operated on in this component of the kinetic energy
equation. That is, it is only an illusion that it is the velocity operated on when t` cancels
out of this component in the kinetic energy equation. However, if that is the case then the
distance formula is converted to a constant time formula as shown in equation 54 below:
1
d
1
vt
c
(54)
c2  v2
That is, t` acted upon in such manner is converted to t as shown by equation 2 (repeated
below):
t  t`
c
c  v2
2
(2)
However, although equation 54 is correct in the mathematical sense, it is exactly that, a
conversion. Equation 54 does not truly represent the physical laws of Nature any more
than Newton’s laws do when constant time is used in place of relative time. It is simply a
conversion for purposes of mathematical convenience like all of the other conversions
given here. Although, unlike Newton’s laws, the conversion given here is correct for all
velocities, like Newton’s laws it misleads us, and thus does not accurately portray the
physical laws of Nature. It has been shown throughout this work that only relative time t`
is truly correct whenever relative motion and its associated relationships are defined.
Proof 2.
In the analysis on inertia and gravity in section 10, it was shown that the results
fundamentally agree with Newton’s laws of motion and his law of universal gravitation.
Yet the analysis clearly shows that gravity is unaffected by relative motion. It is
reasonable to conclude therefore that Einstein’s theory involving the effect velocity has
on mass is invalid. In Einstein’s relativity, mass and energy are seen to be
interchangeable in recognition of the apparent effect velocity has on mass. But if mass
represents a given quantity of matter then such assumption means that matter increases as
a result of relative motion. And if matter increases, so must its gravity. The validity of
such assumption was found to be false in the inertia, gravity analysis.
Perhaps one of the reasons the effect motion has on velocity was overlooked in
favor of an effect on mass is because of the apparent interdependency velocity has on
itself when the transformation factor is applied to velocity instead of mass. That is,
velocity is represented within the transformation factor. This is another of the
problematic circular dependencies discussed earlier in this paper. In this case the
19
problem is not so much one involving the functionality of the math, but rather one
involving comprehension. As it turns out, this problem is easily remedied. For that
matter, the transformation factors used to this point appear to be nothing more than a
misleading form of the true transformation factors involved. This problem is further
discussed and resolved in the next section.
12. Acceleration Based Time Transformation Formulas
Once the fundamentals of relativistic behavior are sufficiently understood, the following
order of simultaneously occurring phenomena seems to be self-evident:
1.
2.
3.
4.
5.
6.
Energy is required in application of force.
Force causes acceleration.
Acceleration causes time to slow down.
The slowing of time reduces the rise in velocity, and thus the rate of acceleration.
A reduction in the rate of acceleration implies a reduction in the magnitude of the force.
An increase in energy is therefore needed to maintain the force and thus the acceleration.
Apparent in the above analysis is the fact that acceleration and not velocity causes time to
slow down. It is also apparent then, and rather obvious, that the slowing of time requires the use
of energy whereas, in the absence of an outside force, uniform motion does not. Since any
velocity, however, is actually the instantaneous velocity resulting from acceleration, its use in the
transformation factor though possibly misleading is nonetheless correct. Nevertheless, it could
be of benefit to develop transformation factors that use the rate of acceleration over an interval of
time instead of the velocity achieved during the interval. Such development may be
accomplished as follows:
By taking the right sides of the velocity equations 11 and 32 and placing them equal to
each other we obtain,
act ` act
c2  v2
.
c
(55)
Since v = act`, (equation 11) we can substitute the right side of this equation for v in the above
equation to arrive at,
act ` act
c 2  (act `) 2
.
c
(56)
If we solve equation 56, first for t` and then for t we obtain the following two equations for time
transformation:
20
c
t ` t
c 2  (act ) 2
(57)
and,
t  t`
c
c 2  (act `)2
(58)
Acceleration
Based Time
Transformation
Thus, for finding relative time t` we have equation 57, and for finding constant time t we have
equation 58. (Note the change of signs.) These of course could also have been arrived at by
substituting act` for v in equation 2 at the beginning of this paper.
From equations 57 and 58 above we can deduce the following new transformation factors
that do not directly rely on velocity: (I.e., the two factors used above and their reciprocals)
c
c  (act )
2
2
(59)
and,
c 2  (act `) 2
c
(60)
Stationary Frame
to Moving Frame
Transformations
(61)
and,
c 2  (act ) 2
c
(62)
Moving Frame to
Stationary Frame
Transformations
c
c  (act `)
2
2
It should be noted that because the time variables t and t` are included in these four factors, only
two, 59 and 60, are useful for time transformation.
13. The Contradictions of Special Relativity
It is well known that the controversy over special relativity has never ended. Although
many of the arguments against it may be flawed in the scientific sense they are often correct in
the logical sense. The problem with special relativity is that contrary to scientific claims there
are no true cornerstones in it. Everything is dependent on everything else. From the purely
scientific line of reasoning this may appear to make sense because after all, all things are relative.
But this line of reasoning is actually in itself scientifically unreasonable. With it we are led
down a road of endless contradictions.
It appears that there are absolutes in the universe after all. It is just that these absolutes
are different than we originally assumed. The existence of matter, for example is absolute, and if
mass is a true measure of a quantity of matter, then the true mass is also absolute. Whereas all
things do appear to have a relative value they also have an absolute, or constant value. For
example, if a second of constant time in one frame of reference is the measure of a certain
amount of atomic activity in that frame, then under identical conditions, that same amount of
identical atomic activity represents a second of constant time in any other frame of reference.
This is the absolute value of time. Relative time, on the other hand, is simply the relative value
of absolute time in one reference frame to the absolute time in another reference frame. The
same is true of mass. If mass represents a given quantity of matter in one reference frame, then
under identical conditions it represents the exact same quantity of matter in any other reference
frame. This then is the absolute value of mass.
21
In the presented theory the contradictions of special relativity do not arise because there
is an acceptance of absolutes. Matter and energy are absolutes and are not interchangeable with
each other. Such are the cases with the momentum and kinetic energy associated with matter.
Matter has physical existence and energy does not. Since energy has no physical existence, it
can only be measured indirectly by its effects on matter. If energy had physical existence it
should be measurable directly. For example, we should be able to measure potential and kinetic
energy without regard for the matter that is causing it. The opposite is actually true. We can
determine energy only by measuring the effects it has on matter.
Force as treated in general relativity is yet another enigma we have to deal with. In the
case of inertia, it is regarded as a fictitious force, or fictitious representation of gravitational
force. But then at the same time, gravitational force is shown not to be a real force at all. Space
curves around massive objects or bodies of matter. So we are left with a fictitious representation
of something that does not truly exist.
While it is true that force does not exist in the physical sense, it does indeed exist. Like
energy it is actually a measured of an effect on mass and is not directly measurable independent
of those effects. Gravitational force, for example, is a measure of the effect gravity has on
matter. And gravity, although we little understand it, does in fact exist. The evidence to that
effect is overwhelming, yet for some reason we feel compelled to disprove it. Objects enter into
orbit, naturally only by chance, or when accomplished by man, only with a great deal of effort.
It also takes effort to remove an object from orbit. But it takes no intervention on our part
whatsoever for an object to fall to earth. Does this sound like curve space influencing the
outcome? No, it is gravity doing exactly what it does, and whether we understand it or not, it
does in one manner or another exist.
14. Conclusion
The evidence presented here strongly supports the view that mass is not affected by
acceleration. The evidence shows convincingly that only those things that are a function of time
are affected by the slowing of time. This includes velocity and acceleration. Mass is a function
of the quantity of matter contained in an object, and this quantity is not a function of time. These
conclusions are not only sensible and logical, but they appear to be self-evident. Time and
distance (space), however, have no physical presence and it appears equally self-evident that
their quantitative values are interdependent on the activity (motion) of the matter upon which
they are referenced to. Even here, it appears that only the variance of time is a real fact of
Nature. It was pointed out in the millennium theory of relativity that the shrinking of distance is
not a real effect but rather a perspective effect related to the slowing of time. Thus it appears that
even distances have constant, or absolute values in addition to relative values.
Whereas gravity appears to be unaffected by inertia it is itself a force that causes
acceleration. And since acceleration causes time to slow down it is equally apparent that gravity
indirectly affects the relative nature of time. This relationship between gravity, acceleration and
relative time are worthy of further exploration in a future work.
Of equal importance here, is the direct link established between the present theory and
the millennium theory of relativity. This connection was established throughout the analysis and
provides strong supporting evidence as to the validity of both theories. Also of importance here
is unifying role that time variance plays in both theories. This suggests that time may be the
unifying element of all physical science.
22
Appendix
A. For the purpose of this paper mass is defined as an absolute quantitative measure of matter. Mass is
detectable through either its gravitational, or inertial force which have been conveniently set equal to each
other at the earth’s surface by use of the gravitational constant G. To be useful as a quantitative measure,
however, it is recognized that apparent changes of mass due to changes in velocity or gravitational field
must be taken into account. Thus it is recognized that even on earth, variances of the gravitational field or
the experience of g forces imposed by acceleration cause only an apparent change of mass, and not a real
change. True mass in the sense meant here is that which is experienced in an isolated system outside of any
gravitational field. In such a system it is the inertial resistance such mass offers to acceleration.
B. In the millennium theory of relativity it is shown that the millennium factor is directly derived from the
evidence. By use of the same evidence the Lorentz factor can also be arrive at but requires more
mathematical steps. Thus, the Lorentz factor and its reciprocal are simply another form of the millennium
factor and its respective reciprocal. If one wishes to go through the exercise, the Lorentz factors can be
used in place of the millennium factors throughout the analysis including in equation 22. After
simplification, the resulting equation will be unchanged from that given as equation 23.
C. Although equation 23 may appear a bit more complicated then the special relativity counterpart, one is
reminded that K = Mc2 - Moc2 is really an abbreviated form of,
K
M oc 2
1
2
v
c2
 M oc 2 .
Similar is true for the special relativity total energy equation, E = Mc2 which in the unabbreviated form is,
E
M oc 2
v2
1 2
c
.
Note: This is an MSWord Office 2000 document. MSWord must be in the print layout view to
view equations, figures 1 and 2, and the addendum. The addendum was copied over from a
Mathcad file to show the method used in verifying mathematical results.
23
Addendum
Addendum - Time and Energy, Inertia and Gravity - J. Rybczyk - February 14, 2002
c
299792458
c . .00000000001732
ac
a c. t. c
v
c
2
v
100000000000
= 0.866019052628739
v . 60
c
t.
t1
c
2
a c.t
t
2
mo
v
t 1 = 5 10
c
kn
1.
2
n = Newton, r = Rybczyk, e = Einstein
2
m o.v
2 2
m. c . v
kr
2
c. c
ke
k n = 3.370282486553152
v
m o.c
c
2
v
10
kr
19
k r = 8.98715633073946
10
k e = 8.98715633073946
10
c
kr
19
2
ke
2
ke
=1
= 2.666588443727394
kn
Total energy
2 2
m. c . v
Er
c. c
= 2.666588443727394
kn
2
2
m o.c
10
19
2
v
1
2
mo
Note: t 1 used in place of t` due to
s/w program unreliability in
handling variables with prime
designators.
km/hr
1000
kinetic energy
m
2
2
= 934653529.6646437
1000
2
v
2
m. c
c
2
v
2
2
m o.c
Ee
2
1
v
c
E r = 1.797470811810764
10
20
Er
2
E e = 1.797470811810764
2
10
=1
Ee
20
Momentum
m. v . c
p r
c
2
p r = 5.192 10
v
p e
11
m o.v
2
1
p e = 5.192 10
v
2
c
2
11
p r
=1
p e
Distance from acceleration
c.v .t 1
d 1
c
c
2
2
c.a c.t 1
d 2
v
2
c
Distance from uniform motion.
c
d u
2
a c.t 1
v.t
d 3
2
d u = 2.596 10
v . t. c
2
c
2
c
19
Constant acceleration
v 1
a c.t 1
c
2
a c.t
2
2
v
2
d 1 = 8.654 10
v 1 = 2.596 10
8
v 2 = 2.596 10
8
a c1
18
d 3 = 8.654 10
18
a c2
v 1
t1
2
t. c
24
a c1 = 0.005
v.c
a c2 = 0.005
v
2
18
d 2 = 8.654 10
a c = 0.005
a c. t. c
v 2
v
Relative acceleration
v
ar
v r a r. t
t
v r = 2.596 10
m = 1 10
8
a r = 0.003
3
Constant force
m. a c
F1
F 1 = 5.192
F2
v
m.
t1
F1
F 2 = 5.192
=1
F2
Inertia and Gravity
Ir
m
I1
Ir
pr
v.c
m.
I2
c
G
g1
6.6720 . 10
11
r
m
G.
2
r
2
1
v
2
14
I3
c
g 2.
2
I 2 = 5.192 10
11
I2
G.
g2
I1
=1
I2
2
r .
2
14
g 2 = 6.672 10
v.c
c
2
r .
11
1000
g 1 = 6.672 10
v.c
I 1 = 5.192 10
1
v
2
1
v
2
I 3 = 5.192 10
G
I2
11
g2
=1
I3
=1
g1
Transformation factors
e1
1
v
c
2
2
1
c
r3
c
2
1
e2
2
c
2
c
r1
2
v
c
c
c
2
a c.t 1
c
2
r5
2
a c. t 1
2
c
r6
c
e 1 = 0.5
r 1 = 0.5
r 3 = 0.5
r 5 = 0.5
e2=2
r2 =2
r4 =2
r6 =2
e1
=1
r1
e2
r2
e1
e1
=1
r3
=1
c
r2
2
2
v
2
c
r4
a c.t
v
2
e2
r4
e2
r6
25
a c.t
c
= 1.000000000000001
r5
= 0.999999999999999
2
=1
2
REFERENCES
1
Isaac Newton, The Principia, (1687) as presented in Physics for Scientists and Engineers, second addition, (Ginn
Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975)
2
Albert Einstein, Special Theory of Relativity, (1905) as presented in Physics for Scientists and Engineers, second
addition, (Ginn Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY,
1975)
3
Joseph Rybczyk, Millennium Theory of Relativity, Unpublished Work, (2001)
Time and Energy, Inertia and Gravity
Copyright © 2002 Joseph A. Rybczyk
All rights reserved
including the right of reproduction
in whole or in part in any form
without permission.
Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link
below:
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