The Ultimate Clarification of Relativistic Kinetic Energy
Copyright © 2016
Joseph A. Rybczyk
Abstract
Up until now, all of the author’s attempts to show the direct relationship between the
Newtonian principles of kinetic energy and the relativistic principles of kinetic energy have been
somewhat short of that goal. Although the author has come close to achieving that goal in
previous papers, he never found the perfect translation between Newton’s kinetic energy
principles and the relativistic principles of Einstein’s theory. In Einstein’s treatment, the
translation is basically nonexistent. Having finally resolved this issue, it is now clear as to what
the various problems were. All of this is covered in as clear a manner as possible in what is
intended to be this author’s final paper on the subject.
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The Ultimate Clarification of Relativistic Kinetic Energy
Copyright © 2016 Joseph A. Rybczyk
1. Introduction
The author’s first attempt to derive the relativistic form of kinetic energy1, although
mathematically correct, and superior to Einstein’s formula2, looks nothing like either Newton’s
formula3 or Einstein’s. And, although his most recent attempt4 came very close to showing the
direct connection to Newton’s formula, it still fell short of that goal. Moreover, whereas
Newton’s treatment shows the direct connection between the principles of force and distance that
kinetic energy is based upon; that form of the formula is lost in the process of Einstein’s indirect
derivation of the relativistic form of the kinetic energy formula. This present paper deals with all
of this and clarifies the connection between the Newtonian principles of kinetic energy and the
relativistic principles of kinetic energy including the relationships of both to the principles of
force over distance that they are based upon.
2. The Newtonian Force and Distance Formulas
In Newtonian physics we are given the mathematical definition for inertial force as
1 where F is the force responsible for mass m0 being accelerated at a constant rate of acceleration
a. Note: For consistency with the mathematical convention of relativistic physics, the symbol m
normally used for mass in Newtonian physics has been replaced with the symbol m0 that
represents rest mass in Einstein’s Special Theory of Relativity5. With that understood we then
continue with
1
2
2 where D is the distance traveled by an object under constant acceleration a over a time period T.
Then, since the constant rate of acceleration a is defined by
3 where v is the speed reached during time period T, we obtain by way of substitution with
distance formula (2)
1
2
4
that reduces to
1
2
5 2
where D is the distance traveled while accelerating to speed v during time period T.
3. The Newtonian Kinetic Energy Formula
Now that we have the required definitions for constant acceleration force F and constant
acceleration distance D we can define kinetic energy by
6 where K is the kinetic energy involved in acceleration force F over distance D. Through the
process of substitution with the right sides of force formula (1) and distance formula (5) with
regard to formula (6) we then obtain
1
2
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which by further substitution using the right side of formula (3) for acceleration rate a, gives
1
2
8
for the developing kinetic energy formula. Simplification of the just derived formula then gives
1
9
2
that is then rearranged into the standard form of
1
2
10 where K is the Newtonian definition for kinetic energy. As can be seen through the just
conducted derivation process, this formula represents the kinetic energy involved in mass m0
accelerating at a constant rate a over the distance D in which speed v is reached. In analyzing the
overall behavior represented by this formula it is important to understand the nature of the
fraction, ½. Referring back to distance formula (5) it can be deduced that since vT represents the
distance traveled at a constant speed v during time period T, the distance traveled during constant
acceleration rate a, to reach speed v during the same time period T is only half the constant speed
distance. I.e., if an object is accelerated from a state of rest to speed v during the same time
period that another object has been traveling at the same speed v for the entire time period, the
constant speed object will travel twice the distance of the accelerating object. Or, the
accelerating object will travel only half the distance of the constant speed object during that same
time period.
4. The Relativistic Kinetic Energy Formula
Now that we have a proper understanding of the Newtonian kinetic energy formula given
above as formula (10) we can directly apply the principles of relativistic physics to derive the
corresponding relativistic kinetic energy formula. To begin, we simply take formula (10) and
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replace the denominator 2 in the fraction with the sum of the numeral 1 and the following
Lorentz transformation factor
1
11 to obtain
1
1
12
1
in the first step of the derivation process. This change is needed to make the fraction conform to
the relativistic principle that nothing can exceed the speed of light. As can be seen from
examining the Lorentz factor, if speed v = 0, then v2/c2 = 0 and the Lorentz factor = 0 which
when added to the 1 at the left gives a denominator value of 2, just as in the Newtonian formula
(10). But if speed v reached the value of light speed c, we would have v = c and therefore v2/c2 =
c2/c2 = 1 that gives the Lorentz factor a total value of 0 resulting in a denominator of 1 in the
main fraction. This gives a final value of 1 for the main fraction thereby preventing any further
acceleration effect on the distance. I.e., the distance traveled will no longer represent the
constantly increasing distance per unit of time traveled by an accelerating object. That is, the
distance will increase at the same, non-changing, rate per unit of time attributed to that of an
object moving at a constant non-accelerating speed c in accordance with the principles of
relativistic physics. Now, let us proceed with the final relativistic modification.
By applying the relativistic gamma transformation factor given as
1
13 1
to the end of equation (12) we obtain
1
1
1
1
14 1
that when applied directly to the rest mass term m0 gives
1
1
15 1
where mass m represents the increased mass of the accelerating object due to its increase in
speed v as defined by
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1
16 1
where m is the moving mass, m0 is the rest mass, and the fraction is the gamma factor previously
given as expression (13).
The just derived relativistic versions of the kinetic energy formula (14) and (15) give
exactly the same results as Einstein’s relativistic kinetic energy formula for all values of v from 0
to c with the minor exception that Einstein’s formula becomes slightly inaccurate at very low
Newtonian speeds. This functional problem with Einstein’s formula was discovered early on
during the author’s research process6 and will be discussed further in this paper after covering
Einstein’s indirect treatment of the relativistic kinetic energy formula.
5. Einstein’s Relativistic Kinetic Energy Formula
As is well documented in the scientific literature, Einstein’s treatment of relativistic
kinetic energy is dependent upon his famous E = mc2 formula for total energy. That is, given
17 where E is the total energy of a system with mass m moving at light speed c, we have
1
18
1
when mass m is replaced with its rest mass value m0 and the gamma transformation factor as
given in formula (16). For kinetic energy K we then obtain
1
19 1
where the last term at the right is the internal energy of the rest mass m0. I.e., when the rest mass
energy is removed from the total energy of a system, we are left with the kinetic energy of the
system. This definition for relativistic kinetic energy could also be represented by its simpler
version given by
20 where the rest mass m0 in the first term to the right of the = sign is replaced with it motion value
m.
As stated earlier, when these versions, (19) and (20), of the relativistic kinetic energy
formula are directly compared with the author’s versions, (14) and (15), in an advanced
mathematical computer application, it is found that the author’s versions give the exact same
results as Einstein’s versions. That is, except for very low Newtonian speeds v. As determined
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by the author, this is simply the result of Einstein’s version reaching the limits of the decimal
places used to round out the computational results when they are not exact. I.e., this does not
mean that the affected formula is mathematically incorrect. It means only that it is not given in
the most ideal form for computational purposes. For example, if the comparison of the author’s
formulas to Einstein’s and Newton’s formulas is made at 16 decimal places for a mass speed of v
= 30 km/s, the following results are obtained:
4.493775893684087
10 21 4.493775893684122
10 22 ′ 10 23 ′ 4.608
Note: For these computations, the following values were used: c = 299792458 m/s, m0 = 102 kg,
and v = 0.0000001c which is equivalent to v = 29.979 m/s at 3 decimal places. The Mathcad
computer application worksheet that was used for the above comparison along with another that
gives the results for v = 0.999999c m/s are included as appendixes at the end of this paper.
6. Conclusion
Whereas the Newtonian treatment of kinetic energy is based upon the mechanics of
physics dealing with the acceleration of an object through space, Einstein’s treatment is indirect
and involves the energy that remains when the rest mass energy is deducted from the total energy
of an object or system in motion. Due to this vastly different treatment and the overall mystique
of relativistic principles, a direct relationship between Newtonian and relativistic kinetic energy
has evaded us until now. Only after 15 years and eight months of research covered in ten
different previous papers7 was the author finally able to solve this problem. It is now clear as to
just exactly how the relativistic version of the kinetic energy formula relates to the Newtonian
version derived from the mechanical principles of physics. This finding resulted from a far more
complicated effort involving the principles of acceleration and their relationship to the relativistic
principles of physics that will be covered in a future paper.
Appendixes
Appendix 1 - Comparison of Rybczyk’s, Newton’s, and Einstein’s Kinetic Energy Formulas at
Low Newtonian Speed
Appendix 2 - Comparison of Rybczyk’s, Newton’s, and Einstein’s Kinetic Energy Formulas at
Near Light Speed
REFFERENCES
1
Joseph A. Rybczyk, Time and Energy, (2001), Available from the Millennium Relativity Website at,
http://www.mrelativity.net
2
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
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3
Sir Isaac Newton, Mathematical Principles of Natural Philosophy, first published in 1687. Note: 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.[4][5]
4
Joseph A. Rybczyk, Newtonian Mechanics Solution to E = mc2, (2015), Available from the Millennium Relativity
Website at, http://www.mrelativity.net
5
Albert Einstein, the Special Theory of Relativity, originally published under the title, On the Electrodynamics of
Moving Bodies, Annalen der Physik, 17, (1905)
6
Joseph A. Rybczyk, Time and Energy, (2001), Available from the Millennium Relativity Website at,
http://www.mrelativity.net
7
Joseph A. Rybczyk, Time and Energy, (2001), Time and Energy, Inertia and Gravity, (2002), Relativistic Motion
Perspective, (2003), Using Special Relativity’s Gamma to Derive the Millennium Relativity Kinetic Energy
Formula, (2003), Directly Deriving the Millennium Relativity Acceleration Distance Formula, (2003),
Corroboration of Research Findings Letter, (2003), The Relationship between E = Mc2 and F = ma, (2007), A
Mathematical Discrepancy in Special Relativity, (2009), Most Direct Derivation of Relativistic Kinetic Energy
Formula, (2010), Newtonian Mechanics Solution to E = mc2, (2015), All available from the Millennium Relativity
Website at, http://www.mrelativity.net
The Ultimate Clarification of Relativistic Kinetic Energy
Copyright © 2016
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
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Appendix 1 – Comparison of Rybczyk’s, Newton’s, and Einstein’s Kinetic Energy Formulas at
Low Newtonian Speed
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Appendix 2 – Comparison of Rybczyk’s, Newton’s, and Einstein’s Kinetic Energy Formulas at
Near Light Speed
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