Proof that Rybczyk’s Energy Equations Are Different from Einstein’s
Copyright © 2017
Joseph A. Rybczyk
Abstract
When the author first derived his versions of the relativistic forms of the kinetic energy
and total energy equations1 through direct modification of the classical kinetic energy formula he
was told by officials of the SLAC National Accelerator Laboratory2 and the National Science
Foundation3 that they were simply different forms of Einstein’s equations4. In other words,
Einstein was given credit for the author’s findings. The official at SLAC attributed it to nothing
more than mathematical manipulation to derive the author’s form of the equations with respect to
Einstein’s form of the equations. Claiming therefore, that there was nothing original in the
author’s findings even though Einstein had never figured out how to do it himself. Only now,
after sixteen years of continued research, can the author finally show, as is documented in this
present work, that his equations are unique and different from Einstein’s, even though they give
virtually identical results.
1
Proof that Rybczyk’s Energy Equations Are Different from Einstein’s
Copyright © 2017 Joseph A. Rybczyk
1. Introduction
In a continuing effort to clarify his mathematical findings on the relativistic forms of
kinetic energy and total energy, the author has developed and presented three different versions
of his equations in nine previous papers5 during the past sixteen years. During that same time,
however, with regard to his many accompanying attempts to directly derive his form of the
equations from Einstein’s, or vice versa, he was unsuccessful. Yet, contrary to such failed
attempts to show equality, he was also unable to show that the two different versions of the
equations were not equal. The reasons for such outcome are many, including numerous
fundamental mathematical variables, and the false fundamental principles and rules of
mathematics by which we are guided. Only recently has the author finally come to understand
the underlying nature of all of this, and as a result succeeded in devising a way to show that his
equations for relativistic energy are different from Einstein’s. The method of proof is simple,
direct, and beyond dispute.
2. The Author’s First Version of Relativistic Kinetic Energy
Before showing that the author’s energy equations are different from Einstein’s it is first
necessary to show that the three different versions of the authors equations are equal to each
other. Starting then, with the first version of the author’s relativistic kinetic energy equation that
was derived via direct modification of the classical equation
1
2
1 we obtain
1
2 1
where the inserted gamma, γ symbol is defined by the following fraction
1
3 1
that itself is referred to as the gamma factor. Substituting this factor in place of gamma in
equation (2) then gives
1
1
1
1
1
4 1
2
for the complete version of this first form of the relativistic kinetic energy equation. This form is
then simplified by replacing the rest mass m0 and an associated gamma factor with the relativistic
inertial mass m as defined by
1
5 1
where inertial mass m varies with speed v, as given in equation (6) below. I.e., in reviewing
equation (5) it can be seen that as v increases from 0 to c, mass m increases from its rest mass m0
value to an infinitly high value.
1
1
1
1
6 1
1
This version can also be given in the simplified form of
1
1
7 where the gamma factor is replaced by the symbol γ as defined earlier by equation (3). Although
this version of the formula is correct, it is not the easiest to explain, so we will defer that aspect
of the analysis to the next version.
3. The Author’s Second Version of Relativistic Kinetic Energy
If we take the previously derived equation (6) and complete the implied multiplication
across the top and bottom of the fractions using the gamma factor to the right, we obtain
1
8
1
1
where the denominator to the left can be rearranged to give
1
1
9 1
for the second version of the author’s relativistic kinetic energy equation. This form of the
equation can then be expanded to its full form
3
1
1
1
1
10 1
by reverse use of the mass increase equation (5) given earlier.
Now we are in a position to more easily understand the nature of the author’s version of
the relativistic kinetic energy equation. In that regard, the factor to the right in the denominator
of the fraction of equation (9), and all of the other fractions in which it is used for that matter,
including the gamma factor of equations (3) and (5), is referred to as the Lorentz factor in honor
of its founder6. In analyzing this factor in equation (9) it is readily seen that as speed v increases
from 0 to c, the Lorentz factor decreases in value from 1 to 0. This then means that the value of
the entire fraction in equation (9) increases from the classical value of ½ to the total relativistic
value of 1, as speed v increases from 0 to c. In other words, the fraction has a value of virtually
½ for low Newtonian speeds and increases toward 1 as the speed of the mass increases toward
the speed of light c. At the same time, mass m is increasing in mass from its designated rest
mass value assigned to mass m0 toward infinity. This effect was discussed earlier in regard to the
mass increase equation (5). I.e., as v increases from 0 to c, the gamma factor increases from 1 to
∞ (infinity). When these two behaviors, (the increase in the ½ fraction, and the increase in mass)
are factored in with the squaring of the speed v given at the end of equations (9) and (10), we
have some idea of the difference between the classical form of kinetic energy, and the relativistic
form. In short, it would take an infinite amount of energy to exceed the speed of light. Now, let
us go to the final version of the author’s kinetic energy equation.
4. The Author’s Third Version of Relativistic Kinetic Energy
To derive the third version of the author’s relativistic kinetic energy equation, we simply
complete the multiplications contained in equation (10). This gives
11
1
1
1
and
12
1
1
for the first step of the process. Then we multiply the top and bottom of the resulting fraction by
c2 to obtain
13
1
1
4
14
1
15
1
and finally
16 √
as the third form of the author’s relativistic kinetic energy equation. With that accomplished, we
can now proceed to the next section in which the author’s total relativistic energy equation is
covered.
5. Rybczyk’s Total Relativistic Energy Equation
The author’s total relativistic energy equation is derived by simply adding the rest mass
energy to the various versions of his just derived relativistic kinetic energy equations. Since the
rest mass energy is defined by
17
this gives
1
1
1
1
1
1
√
1
1
1
18 1
19 1
20 and
5
1
1
1
1
21 1
1
1
1
22 1
using the previously derived relativistic kinetic energy equations (4) (10) (16) (6) and (9)
respectively. We are now ready to compare these equations for relativistic kinetic energy and
total energy to Einstein’s equations.
6. Comparing Rybczyk’s Equations to Einstein’s
As is very well known going all the way back to its introduction in 1905, Einstein’s
famous equation for total energy is given by
23 which is the simpler form of the more complete version given by
1
24 1
where the inertial mass m is replaced by the rest mass m0 times gamma as previously defined by
equation (5). The relativistic kinetic energy equation is then given by
1
25 1
were the rest mass energy is subtracted from the total Energy.
To compare Einstein’s total energy equation (24) and kinetic energy equation (25) to the
author’s corresponding equations, we simply need to evaluate both, Einstein’s and the author’s
equations, at the two extremes of v = 0 and v = c. Beginning with Einstein’s total energy
equation (24) we get
26 for v = 0, and
1
√0
6
27
0
giving
∞ 28 for v = c. Now, let us do the same for Einstein’s version of the relativistic kinetic energy
equation (25). In this case we get
29
giving
0 30 0
for v = 0, and
1
1 31
1
giving
1
√0
1 32
ending with
∞
1 33 for v = c.
Now, let us see what results when we do the same with the author’s total energy and
kinetic energy equations. Since the author has already proven that all of his total energy
equations are simply different versions of the same total energy equation, and likewise with his
relativistic kinetic energy equations, it is only necessary to evaluate one of each of those versions
of his equations for the purpose of comparison with Einstein’s corresponding equations. With
that understood we will now evaluate his total energy equation (19) repeated below.
1
1
1
1
19 1
For v = 0 we get
1
1
√1
1
√1
0
7
34
giving
35
0
or
36 0
which is the same as obtained for Einstein’s total energy equation at v = 0. Now let us see what
we get for v = c. Starting again with equation (19) we obtain
1
1
1
37
√0
√0
and
1
1
1
1 38
√0 √0
and
∞
1 39 for the author’s total energy equation at v = c. This result is not the same as that obtained for
Einstein’s E equation (28) at v = c. Before discussing the meaning of this in detail, let us first
complete the K analysis.
Beginning with the author’s kinetic energy equation (10) repeated below
1
1
1
1
10 1
we now evaluate it for the case where v = 0. This gives
1
1
1
√1
√1
0 40
and therefore
0 41 0
which is the same result as obtained for Einstein’s K equation (30) at v = 0. For the v = c result
we get
8
1
1
1
√0
√0
42
and therefore
∞ 43
or
∞ 44 which is different from Einstein’s K equation (33) for v = c.
7. Interpreting the Results of the Comparison
For ease of comparison, the final resulting equations involving Einstein’s and the
author’s total energy and kinetic energy equations are repeated below. Upon reviewing them, it
is seen that both researcher’s equations for total energy and kinetic energy give the same results
for v = 0. But for v = c, both researcher’s equations give different results. This is clear and
irrefutable proof that the two research’s equations are different from each other. In plain words,
the author’s equations are his own and not simply another version of Einstein’s equations.
Einstein’s role in this is similar to those7 responsible for the classical formulas for Energy. It
was their formulas that the author modified using Einstein’s principles of relativity to arrive at a
new and distinct version of formulas for total and kinetic energy. A relativistic version different
from those of all predecessors.
26 0
36 0
∞ 28 ∞
1 39 0 30 0
0 41 0
∞
1 33 ∞ 44 In reviewing the difference in the author’s equations in relation to Einstein’s it is readily
apparent that for v > 0, both, the author’s total energy, and kinetic energy equations, give a
9
slightly greater result than Einstein’s equations. This difference is so slight, however, that it is
not easily verified in a computer mathematical application. Even at a level of accuracy of 17
decimal places, the results are not entirely consistent with the findings presented here. I.e., as the
value for v approaches c, the results might fluctuate with regard to which equations give the
greater value. But that is due to the inadequacy of the computer application and not the findings
presented here. So, for all practical purposes, both, Rybczyk’s and Einstein’s equations give
identical results. (Except for very low values of v where, as shown in many of the author’s
previous papers, Einstein’s formulas give slightly erratic results.)
8. The Imperfect Laws of Science and Mathematics
It has long been apparent to the author during his many years of research that the, so
called, laws of science and mathematics are not entirely correct. Since we don’t know, and never
will know, all of the principles that govern life and the universe, we will never be able to
formulate a complete and perfect set of laws for science and mathematics. For example, in
mathematics, division by zero is illegal, yet the infinity that it leads to is an undeniable fact of the
universe. Another faulty principle of mathematics is that which calls for the simplification of
formulas and equations. It took the author years to realize that his first published equation for
relativistic kinetic energy, (equation (16)), was not the best version of his formula. In fact, the
first version he came up with, before the simplification process, (equation (4)), was superior to
the simplified version he published. I.e., with regard to any formulas, the most useful are the
ones that allow easy understanding of the underlying principles involved. In the example
mentioned, the gamma factor is no longer available to aid in the visual analysis of what the
formula predicts. So, contrary to the principles of mathematics, the more complex formula that
contains the gamma factor, is superior to the simplified version that doesn’t. At least when it
comes to a mental understanding the phenomena involved. Another example involving
mathematics is the principle of arranging the product of multiple variables in the order of the
highest exponent to the lowest. Even the world’s most famous equation, E = mc2, violates this
rule. In the author’s opinion, the laws of science and mathematics should be violated whenever
it leads to a better understanding of the phenomena involved.
9. Conclusion
From the very beginning of the author’s research into relativistic physics at the start of
the new millennium, the author was denied credit for his findings involving kinetic and total
energy. Not only did the science journals refuse to publish his findings, but the governmental
organizations that the author appealed to for support, also refused to help him. If they agreed
with his findings, as in the case with those involving kinetic and total energy, they simply gave
the credit to Einstein. This attitude toward independent researchers, who are not affiliated with
official higher educational, scientific, or governmental organizations, continues to this day.
What these organizations fail to realize is that they are impeding the progress of science. The
author does not fault them for their vigilance to ensure scientific validity in the work they
support, but for their unfair treatment of those who are not affiliated with associated official
organizations. Hopefully, this attitude will change in the future, but until then, the author is left
no option but to continue to fight back thorough the publication of his findings in papers such as
this on a website he designed back in 2002 and continues to operate to the present day for that
specific purpose.
10
Appendix – History of Rybczyk’s Energy Formulas Derivations
REFFERENCES
1
Joseph A. Rybczyk, Time and Energy - The Relationship between Time, Acceleration, and Velocity and its Affect
on Energy (2001) --- Available from the top of the home page of the Millennium Relativity website at
http://mrelativity.net
2
SLAC National Accelerator, Laboratory2575 Sand Hill Road, Menlo Park, California 94025-7015
3
National Science Foundation, 4201 Wilson Boulevard, Arlington, Virginia 22230
4
Albert Einstein, The energy equation is first mentioned in the paper, Does the inertia of a body depend upon its
energy-content? (1905), It is unclear when the equation was first given in the form E = mc2
5
Joseph A. Rybczyk, Time and Energy (2001), Time and Energy, Inertia and Gravity (2002), Relativistic Motion
Perspective (2003), Newtonian Mechanics Solution to E = mc2 (2015), The Ultimate Clarification of Relativistic
Kinetic Energy (2016), Relativistic Kinetic Energy Simplified (2017) --- All available from the home page of the
Millennium Relativity website at http://mrelativity.net
Using Special Relativity’s Gamma to Derive the Millennium Relativity Kinetic Energy Formula (2003), A
Mathematical Discrepancy in Special Relativity (2009), Most Direct Derivation of Relativistic Kinetic Energy
Formula (2010) --- All available from the Millennium Briefs Section at the bottom of the home page of the
Millennium Relativity website at http://mrelativity.net
6
Hendrik A. Lorentz, Dutch Physicist, discovered a new way to transform distance and time measurements between
two moving observers so that Maxwell’s equations would give the same results for both, (1899, 1904), Einstein
further refined the transformations based on the Lorentz factor for use in his special theory of relativity.
Wikipedia, Kinetic energy, The principle in classical mechanics that E ∝ mv2 was first developed by Gottfried
Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva. Willem 's Gravesande of
the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights
into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of
their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation.
7
The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early
understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled
Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord
Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51.
11
History of Rybczyk’s Energy Formulas Derivations – page 1
12
History of Rybczyk’s Energy Formulas Derivations – page 2
13
Proof that Rybczyk’s Energy Equations Are Different from Einstein’s
Copyright © 2017
Joseph A. Rybczyk
All rights reserved
including the right of reproduction
in whole or in part in any form
without permission.
Millennium Relativity home page
14