Content uploaded by Bahram Kalhor
Author content
All content in this area was uploaded by Bahram Kalhor on Aug 18, 2020
Content may be subject to copyright.
Comparing multi-dimensional energy–momentum equation and relativistic energy–
momentum equation, which equation is wrong?
Bahram Kalhor
1
, Farzaneh Mehrparvar1 , Behnam Kalhor1
Abstract
The paper compares the multi-dimensional energy-momentum equation with the relativistic
energy-momentum equation. We show that the relativistic energy-momentum equation is wrong
and unable to explain the mass-energy equivalence in the multi-dimensional real space.
Bertozzi’s experiment is good evidence for showing the problem of the relativistic energy-
momentum equation. Multi-dimensional energy-equation can explain the total energy of the
particles in the Bertozzi’s experiment and explain the relationship between the mass-energy
formula of the particles in the n-dimensional spacetime.
Introduction
Relativistic energy-momentum equation has been used in the past century by physicists:
or
where
,
At the speeds too less than the speed of light, and
Where E is the total energy of the particle, m is the rest mass, v is the velocity, p is the
momentum, and c is the speed of light [1-8].
On the other hand, we have introduced a multi-dimensional energy-momentum equation:
()
for three-dimensional particles and
for n-dimensional particles [9].
Differences between both equations not allowed, and one of them would be wrong. In this paper,
we will check the correctness of both equations. We try to present some evidence to distinguish
between them and find which one is correct.
The method of obtaining both formulas is the same. Flat spacetime or Minkowski spacetime has
used for obtaining the relativistic energy-momentum equation by physicists [10-13]. In the
Minkowski spacetime, a three-dimensional phenomenon and time dimension combine and make
a four-dimensional phenomenon. In the Minkowski spacetime, energy and momentum have used
as two components.
1
Independent researcher form Alborz, IRAN
Corresponding author. Email: bahram.Kalhor@kiau.ac.ir
On the other hand, Generalized Minkowski spacetime has used for obtaining the multi-
dimensional energy equation [9]. We have expected that both methods present the same result,
but different equations have obtained. Now, we investigate that using different components has
caused different results or defining the wrong relation between input parameters in the different
dimensions have made this fault.
The highest energy of the particles in the speed of light in both equations have used for
comparing both. We explain that the total energy of a particle in the speeds close to the speed of
light in Bertozzi’s experiment [14-18] could not be described by the relativistic energy-
momentum equation. Also, the mass-energy equivalence is investigated in the n-dimensional
spacetime. We show that relation between the total energy of a particle between two individual
dimensions has been explained by the multi-dimensional energy momentum equation very well,
while is not obtainable by the relativistic energy-momentum equation. Dimensional analysis is
another important tool for comparing these equations.
By using the multi-dimensional energy-momentum equation, we must redefine the kinetic energy
formula [19] and introduce new relation between the kinetic energy, potential energy, and the
total energy [20].
Dimensional analysis
In dimensional analysis, we investigate the relationship between the basic quantities such as
mass, length, electron charge, time and their unites of the measure in both sides of an equation.
We expect that both sides of any equation should have the same dimension. Different dimension
in different sides shows the equation is not dimensionally consistent and cannot possibly be a
correct statement of physical law.
Basic dimensions of the basic quantities are given by:
[Mass]=M
[Length]=L
[Time]=T
For comparing two multi-dimensional quantities in a multi-dimensional spacetime, using simple
dimensional analysis not allowed. We need orientation and use concepts of the frame of
reference or even dimension reference. For instance, Einstein’s general relativity equation which
is discussing in the multidimensional spacetime is not dimensionally consistent. Dimensional
analysis of Einstein’s fields equations in vacuum shows that [21]:
L = T (1)
We cannot assimilate time T to a length L in our three-dimensional universe. On the other hand.
accepting equality of the dimension of the length and dimension of the time in the highest
dimension (time dimension) in the multidimensional spacetime results no dimension for the
velocity in the time dimensions in the spacetime:
[v]=1 (2)
Where v is the velocity of the particle in the time dimensions.
On the other hand, in the three-dimensional real space, physicists use:
[E]= (3)
Now we can investigate both equations in dimensional analysis consistency.
The relativistic energy-momentum equation is given by:
(4)
And the multi-dimensional energy-momentum equation in a three-dimensional real space is
given by: (5)
By using the equation (2), the dimension of the relativistic energy-momentum equation is equal
to , while dimension of the multi-dimensional energy-momentum equation is equal to
. In the spacetime’s equations, c belongs to the real space, hence it has a normal
dimension so [c]= , while v belongs to the time dimension and [v]=1.
Although at the first view, we may think that the relativistic energy-momentum is correct and the
multi-dimensional energy-momentum would be wrong, using [v]=1 show that multi-dimensional
energy-momentum is correct.
Proposing dimensions reference instead of the frame reference
In the spacetime, for comparing two quantities in different dimensions, we suggest using
dimensions reference instead of the frame reference. In this case, by using equation (2) we omit
the effect of the velocity in the fourth dimension and compare quantities.
If we choose the first dimension as a dimension reference, we should change the effect of the
multiplying by the velocity in the higher dimensions and use the velocity of the light as a
constant number with no dimension. The equation is given by:
(6)
Where is the constant value in which its value is equal to the value of the c and []=1, also
[v]=1, hence .
The equation (6) shows that If we choose first real dimension as the dimension reference, the
unit of measuring the energy for the one-dimensional particle should be kilogram. and the
dimension of the energy for the one-dimensional particle would be equal to the M.
If we choose the third dimension as the dimension reference, equation (6) will be changed to:
(7)
where [v]=1, [c]= and [E]=
However, the equation (6) approximates a complete equation [22] of the total energy:
(8)
The equation (8) is not consistent, hence for using nth real dimension as the reference dimension
we need to make it consistent. The proposed equation is given by:
(9)
Where is the constant value which its value is equal to the value of the c and [] =1, also is
equal to one, and . This term is used for making consistency between different
dimension with reference dimension.
According to definition of the potential energy [20]:
(10)
And in the three-dimensional real space:
+ m (11)
Equation (11) illustrates that the total energy of a three-dimensional particle is equal to the
summation of its counts of the quanta masses in the first dimension (m) plus its counts of the
quanta masses in the second dimension ( plus its count of the quanta masses in the third
dimension ([22].
Comparing maximum energy
In both equations, when the particle reaches the speed of light, they reach their maximum total
energy. The amount of total energy that has obtained in both equations is different.
Equation of the relativistic energy-momentum equation is given by:
(12)
By using c instead of the v, , hence the energy of the particle at the peed of light is
infinite. In the special relativity, we can expect that we need infinite energy to increase the speed
of the particle to the speed of light, because:
(13)
if v=c (14)
and: (15)
or (16)
On the other hand, In the multidimensional energy-momentum equation, the energy is given by:
(17)
using c instead of the v (18)
or (19)
, hence (20)
In the multi-dimensional energy-momentum equation is given by [22]:
(21)
, hence (22)
using (21) and (22) (23)
Although the aim of the Bertozzi’s experiment was investigating the relativity mass, data of the
experiment shows that the potential energy of the electron when its velocity is close to the speed
of light is measurable and is too less than infinite. In this experiment, the energy of the electron
(P) at the speed of light is almost 30 times the. Data of the Bertozzi’s experiment is given in
table 1.
Table 1. Data and results of the Bertozzi’s experiment (
)
P MeV
0.5
1
2.60
1
2
2.73
1.5
3
2.88
4.5
9
2.96
15
30
3
In the Bertozzi’s experiment, energy of the electron at the speed of light is 30 times the rest
energy. Hence, it appears to be disagreement with the relativistic equations (12) and (16).
On the other hand, although 30 is too less than the expected amount in the multi-dimensional
energy-momentum equation that is mentioned in the equations (20) and (23), there is an
exception that can explain this difference. In the multi-dimensional energy-momentum, when the
particle tries to start oscillating in a new dimension, sometimes it would be unlabeled to do it,
hence starts a new path in the current dimensions and simulates its movement in the higher
dimension. This issue is explained in the [23]. In this case, the particle omits one dimension and
moves in parallel with it, hence particle needs too little energy to reach the speed of light. Hence,
we expect that the energy of the particle would be close to the energy of the previous dimension.
So, the expected energy for particles that simulates n-dimensional real space in a (n-1)
dimensional real space is close to the energy of the (n-1) dimensional real space and too less than
the energy of the n-dimensional real space. Therefor in the Bertozzi’s experiment, the energy of
the electron at the speed of light would be:
(24)
or
1< ( (25)
This agrees with multidimensional equation (23) with omitting one dimension.
Mass-energy equivalence formula
In the multi-dimensional spacetime, we assumed that a one-dimensional particle after reaching
the speed of light will be converted its kinetic energy to the potential energy and start oscillating
in the upper dimension. The generalize Minkowski spacetime is a good mathematical tool for
describing the energy-momentum equation in each spacetime.
According to the equation (16), at the speed of light the total energy of the particle in the
relativistic energy-momentum equation is infinite.
On the other hand, according to the equation (23) in the multi-dimensional energy-momentum
equation, at the end of each dimension, the total energy of the particles multiplied by the almost
c. Hence, total energy of the two-dimensional particle is equal to and total energy of the
three-dimensional particle is equal to . This means that the total energy of a three-
dimensional particle is times of the rest mass of its one-dimensional mass. It appears to be an
agreement with the mass-energy equivalence formula.
Conclusion
The energy-momentum equation that is used in the past century is wrong. Hence, the kinetic
energy, the potential energy, and the total energy formula that have used by physicists are wrong.
The energy-momentum equation is unable to describe the energy of a three-dimensional particle
based on the oscillating in the upper dimensions. Also, in Bertozzi’s experiment, the total energy
of the electron at the speeds close to the speed of light could not be explained by the energy-
momentum equation.
By the multi-dimensional energy-momentum equation, the total energy of the electron at the
speeds close to the speed of light could be described. Also, we can obtain more precision mass-
energy equivalence formula by using the multi-dimensional energy-momentum. The multi-
dimensional energy-momentum could change our formula in classic and quantum physics, and
precise the comparing the results of experiments with the theoretical prediction. We expect that a
new kinetic energy formula that has obtained by using the multi-dimensional energy-momentum
equation and a new relation between the total energy of the particle with the kinetic energy and
potential energy have introduced, would be more precise and valid in the more range of the
speeds and dimensions.
The Einstein’s mass-energy equivalence () approximates the total rest energy of the
three-dimensional particle (). Also, its equation for kinetic energy is not
correct, hence we could expect that the popular approximated formula for kinetic energy E= ½
mv2 is not correct.
The equation of the energy-momentum in the three-dimensional spacetime (
) is differ to relativistic energy-momentum () equation.
Although the energy equation of the multidimensional energy-momentum agrees that by
increasing the velocity of the object its energy would be increased, its equation is different from
Einstein’s equation. Hence, we could expect that Einstein’s relativistic mass is wrong and at the
speed of light mass of the object or momentum do not increase to infinite and just start
oscillating in a higher dimension.
There is a significant relation between mass-energy equivalence equations in the different
dimensions. In the spacetime [v]=1, hence relativistic energy-momentum is not consistent. And
multidimensional energy-momentum (
) is consistence. We
proposed to use dimension reference instead of the frame reference.
In all dimensions, the energy of the objects is a summation of its potential energy and kinetic
energy. We can use dimensions reference equation to respect dimensional analysis:
The kinetic energy is equal to the potential energy multiplied by the speed of the object here
velocity belong to the time dimension and [v]=1, hence maximum kinetic energy is equal to the
potential energy multiplied by the c.
The potential energy in the multidimensional mass-energy equivalence is not the same as the
Einstein ‘s mass-energy equivalence, also the kinetic energy is different.
All objects have made of quanta masses. The energy of the objects is total counts of its all quanta
masses multiplied by the k constant, while the mass of the objects is equal to the counts of its
quanta masses in the first dimensions multiplied by the k constant. Also, the count of the quanta
masses in the first dimension is equal to the frequency of the particle.
The k boxes are virtual boxes that explain the arrangement of the quanta masses. The capacity of
the full filled k boxes in the different dimensions is different while their mass is constant and is
equal to the k constant.
References
1. Rindler, Wolfgang. "Introduction to special relativity. 2." (1991).
2. Besancon, Robert. The encyclopedia of physics. Springer Science & Business Media, 2013.
3. Dennery, Philippe, and André Krzywicki. Mathematics for physicists. Courier Corporation, 2012.
4. Kleppner, Daniel, and Robert Kolenkow. An introduction to mechanics. Cambridge University
Press, 2014.
5. Dowker, John S., and Raymond Critchley. "Effective Lagrangian and energy-momentum tensor in
de Sitter space." Physical Review D 13.12 (1976): 3224.
6. Tolman, Richard C. "On the use of the energy-momentum principle in general relativity."
Physical Review 35.8 (1930): 875.
7. Ramos, Tomás, Guillermo F. Rubilar, and Yuri N. Obukhov. "Relativistic analysis of the dielectric
Einstein box: Abraham, Minkowski and total energy–momentum tensors." Physics Letters A
375.16 (2011): 1703-1709.
8. Cottingham, W. Noel, and Derek A. Greenwood. An introduction to the standard model of
particle physics. Cambridge university press, 2007.
9. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "Is it possible to obtain multi-
dimensional energy-momentum equation by using Minkowski spacetime?." Energy 100: P2. ,
2020.
10. Minkowski, Hermann. "Raum und zeit." JDMaV 18 (1909): 75-88
11. Schutz, John W. Independent axioms for Minkowski space-time. Vol. 373. CRC Press, 1997.
12. Ramos, Tomás, Guillermo F. Rubilar, and Yuri N. Obukhov. "Relativistic analysis of the dielectric
Einstein box: Abraham, Minkowski and total energy–momentum tensors." Physics Letters A
375.16 (2011): 1703-1709.
13. Thompson, Anthony C., and Anthony C. Thompson. Minkowski geometry. Cambridge University
Press, 1996.
14. W. Bertozzi; “Speed and Kinetic Energy of Relativistic Electrons", Am. J. Phys., 32 (1964), 551 –
555. Also online at:. http://spiff.rit.edu/classes/phys314/lectures/relmom/bertozzi.html
15. Bertozzi, Massimo, et al. "VIAC: An out of ordinary experiment." 2011 IEEE Intelligent Vehicles
Symposium (IV). IEEE, 2011.
16. Ireson, G. "Introducing relativistic mass: theultimate speed experiment'of William Bertozzi
revisited." Physics education 33.3 (1998): 182.
17. Huang, Young-Sea. "Has the Lorentz-covariant electromagnetic force law been directly tested
experimentally?." Foundations of Physics Letters 6.3 (1993): 257-274.
18. Semat, Henry. Introduction to atomic and nuclear physics. Springer Science & Business Media,
2012.
19. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "Does using multi-dimensional
energy-momentum equation change the kinetic energy formula?." Available at SSRN 3637222
(2020).
20. Kalhor, Bahram, and Farzaneh Mehrparvar, and Behnam Kalhor. "Introducing new equation for
total energy, potential energy, and kinetic energy in the multidimensional spacetime." Available
at SSRN 3650944 (2020).
21. Olivi-Tran, N. "Dimensional analysis of Einstein’s fields equations Adv." Studies Theor. Phys 3.1
(2009): 9-12.
22. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "How quantum of the mass, k box,
and photon make light and matter? " Available at SSRN 3667603 (2020).
23. Kalhor, Bahram, Farzaneh Mehrparvar, and Behnam Kalhor. "Do Stars in a Spiral Galaxy Simulate
4-dimensional Movement in a 3-dimensional Space" Available at SSRN 3613187 (2020).
24. Osiak, Zbigniew. "Energy in Special Relativity." Theoretical Physics 4.1 (2019): 23.
25. Kalhor, Bahram, and Farzaneh Mehrparvar. "k constant shows a mistake in wavelength equation
in the wave-particle duality and presents a new formula." Available at SSRN 3613198 (2020).
26. Jammer, Max. Concepts of mass in contemporary physics and philosophy. Princeton University
Press, 2009.
27. Šorli, Amrit Srečko. "Mass–Energy Equivalence Extension onto a Superfluid Quantum Vacuum."
Scientific reports 9.1 (2019): 1-9.
28. Wesson, Paul S., and J. Ponce de Leon. "Kaluza–Klein equations, Einstein’s equations, and an
effective energymomentum tensor." Journal of mathematical physics 33.11 (1992): 3883-3887.
29. Callan Jr, Curtis G., Sidney Coleman, and Roman Jackiw. "A new improved energy-momentum
tensor." Annals of Physics 59.1 (1970): 42-73.
30. Kalhor, Bahram, and Farzaneh Mehrparvar. "k constant and energy." Available at SSRN 3613201
(2020).
31. Eshelby, John D. "Energy relations and the energy-momentum tensor in continuum mechanics."
Fundamental contributions to the continuum theory of evolving phase interfaces in solids.
Springer, Berlin, Heidelberg, 1999. 82-119.
32. Einstein, Albert. Relativity: The Special and the General Theory-100th Anniversary Edition.
Princeton University Press, 2019.
33. Sorli, Amrit, Magi Mageshwaran, and Davide Fiscaletti. "Energy-Mass-Gravity Theory." American
Journal of Modern Physics 5.4-1 (2016): 20-26.