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Is Einstein’s special relativity wrong?
Bahram Kalhor
1
, Farzaneh Mehrparvar1, Behnam Kalhor1
Abstract
The paper investigates physical parameters and the path of the movement of the particles in the
spacetime and shows that special relativity is incomplete. The special relativity does not explain
the correct relationship between the energy and momentum in the spacetime, so then the energy-
momentum equation is incorrect. We present the relationship between the momentum and
potential energy in all dimensions in spacetime and compare the extracted equations with the
mass-energy equivalence. The potential energy of the three-dimensional particles is good
evidence of the correctness of the multi-dimensional energy-momentum equation. For covering
these problems, we suggest a new principle law to explain the behavior of the particle, path of
the movement, and the relationship between the physical parameters in the different dimensions.
Introduction
The special relativity has been based on two principle lows:
The first principle of the relativity implies that in the all frame of references that move with
invariant speed to each other, the lows of the physic are invariant. Einstein has written, "The
states of physical systems alter are independent of the alternative, to which of two systems of
coordinates, in uniform motion of parallel translation relatively to each other" [1].
The second principle of the relativity implies that in a vacuum, regardless of the movement of
the frames, directions, observers, and source of the light, the speed of light is invariant. The
principle of the constancy of the velocity of light has been contained in Maxwell's equations. He
has written that reaching to the speed of light is not accessible: “from a composition of two
velocities which are less than c, there always results a velocity less than c“ and “the velocity of
light c cannot be altered by composition with a velocity less than that of light” [1].
According to the second law, Einstein has investigated the behavior of the bodies at the speed of
the light. In the special relativity, “Velocities greater than that of light have no possibility of
existence.” and the work that we need to increase velocity more than the speed of light is
infinite”, reaching the velocity of the body to the speeds more than light is impossible.
The mathematical core of the special relativity is Lorentz transforms. By using the Lorentz
transforms, physicists would be able to compare or transform the mathematical and physical
parameters of two different observers in two different frames [2-5]. Also, Minkowski spacetime
has used to investigate the special relativity, as the Lorentz transforms [6-9]. In the special
relativity, the Minkowski four-vectors has used for extracting energy and momentum in 4-
dimensional spacetime. Nowadays, this equation has called the energy-momentum equation:
1
Independent researcher form Alborz, IRAN
Corresponding author. Email: Kalhor_bahram@yahoo.com
or Where
. At the speeds too less than
the speed of light, and . In the energy-momentum equation, the
momentum and the total energy have been used as two components in the Minkowski spacetime
and extract the energy equation.
On the other hand, it proved that if we use the potential energy and the total energy as two
different components in the Minkowski spacetime we would extract different equation which is
called multidimensional energy-momentum equation [10]: .Reaching to
the two different results with two different inputs is not regular, while both of them have used the
Minkowski spacetime.
Both methods have used complicated mathematical methods for extracting the equation of the
relativistic energy-momentum in the special relativity. Using the complex methods in two frames
with different dimensions increases the chance of mistakes and obtaining fault equation. In this
paper, we show that this fault has done in the energy-momentum equation. In the past century,
all results and predictions that have made based on the energy-momentum equation theory are
wrong. One of the most famous wrong predictions in this theory is the infinite mass of the
particles at the speed of the light, though nowadays, physicists believe that this is relativistic
mass and the momentum increases to the infinite.
In this paper, Firstly, we investigate the both methods and find the source of the mistake. We
propose a new correction in the energy-momentum equation while we explain the reason of the
problem. We explain that the root of the problem belongs to the special relativity, where the
theory does not explain what would happen at the speed of the light. Secondly, we propose some
correction to the special relativity principle lows by adding a new principle. This new law clears
the behavior of the particles at the speed of light and predicts their path at the higher dimension.
Finally, according to the proposed principle low, we find the relation between the mass, total
energy, kinetic energy, potential energy, and the momentum of the particle, respectively.
Comparing energy-momentum equation and multidimensional energy-momentum
equation
Physicists use the momentum of the particle and the total energy as two components of the
Minkowski spacetime, for obtaining the energy-momentum equation:
(1)
where E is the total energy of the particle, is the Minkowski metric, is the inner
product of the momentum in the real space, c is the speed of the light, and P is the momentum in
the spacetime.
(2)
(3)
using (2) and (3) =
(4)
hence (5)
(6)
If (7)
The equation (7) is the expanded energy-momentum equation in the four-dimensional
Minkowski spacetime when the speed of the particle is too less than the speed of light.
On the other hand, in the multi-dimensional energy-momentum equation, the potential energy,
and the total energy have used as two components in the Minkowski spacetime.
(8)
so (9)
(10)
Hence, (11)
and (12)
The equation (12) is the expanded multi-dimensional energy-momentum equation in the four-
dimensional Minkowski spacetime.
Differences between equations (6 and 7) and (12) not allowed, hence one of them is incorrect.
Investigating the problem of the energy-momentum equation and Einstein’s special
relativity
In the equation (1), the momentum which is used in the , refers to the momentum of the
particle in the real space, while the momentum P which is used on the other side of the equation
refers to the momentum of the particle in the Minkowski spacetime.
In the equation (1) it is assumed that P and , while on the other side of the
equation it is assumed that P=mv and it is used in the equation (5). The equation
is a condition of the Minkowski spacetime. This means that we should have a three-dimensional
particle () that has reached its limit in the velocity in the third dimension. Hence, reaching to
the speed of light in the third dimension for starting the oscillating in the 4th dimension is an
essential condition for using Minkowski spacetime.
Unfortunately, in the special relativity, no statement or law explains what would be happened to
the particle at the speed of the light. Meanwhile, at the time dimension, acts like a
relativistic mass and the momentum of the particle in the 4th dimension is equal to . The
equation is given by: (13)
In the 4th dimension is the momentum of the particle in the four-dimensional
spacetime. Hence, regardless of the 4th time dimension or the 4th real space dimension, we can
say that the momentum of the particle in the higher dimension is equal to the c times of the
momentum of the particle in the previous dimension.
Now, let us use a new concept of the momentum in the higher dimension in the Minkowski
spacetime and obtain a new energy-momentum equation.
(14)
where E is the total energy of the particle, is the Minkowski metric, is the inner
product of the momentum in the real space, c is the speed of the light, and is the
momentum of the particle in the spacetime.
(15)
(16)
using (15) and (16) =
(17)
hence
(18)
(19)
or (20)
so (21)
The equations (20,21) are not relativistic, the equation (20) is the newly expanded energy-
momentum equation in the four-dimensional Minkowski spacetime and is the same as the
multidimensional energy-momentum equation where n=3.
In the special relativity, changing the physical parameters at the speed of light did not consider.
We show that the behavior of the particles at the speed of light in the special relativity is too
vogue. Restricting the velocity of the particle to the speed of light restricts the validation of the
special relativity’s results to move in one dimension. Therefore, when physicists have used an
additional time dimension for obtaining the energy-momentum equation in the spacetime, the
effect of the new dimension on the physical parameters did not have considered.
After reaching the speed of the light, particle oscillates in the new dimension or simulates it,
where the new direction is perpendicular to previous dimensions [11]. Hence, regardless of
choosing the real 4th dimension or time dimension as the 4th dimension, the particle tries to
choose a path that is perpendicular to previous dimensions or simulate it. Although at the low
speeds, the particle has some freedom to move in any direction, at the higher speeds, it moves
perpendicular to the previous dimension or simulates it. This is one reason that validation of
some popular equation such as the kinetic energy (
) is in the speeds too less than the
speed of light and in the higher speeds we need a new formula () [12].
The rest mass of the three-dimensional particle does not change at the speed of the light. In the
three-dimensional real space when the speed of the particle reaches the speed of the light, the
total kinetic energy will be converted to potential energy and the potential energy increases.
Here, the meaning of increasing the potential energy is not equal to increasing the mass of the
particle. In this situation, the particle saves its movement direction with the speed of light with
the unchanged rest mass, while it starts oscillating in a new direction. Hence, saving movement
direction with the speed of light and stating new movement is equal to increasing the energy of
the particle.
If we investigate this concept according to work, we would say that: in the special relativity
while the speed of the particle is less than the speed of light doing more work on the particles
increases the velocity of the particle, and at the speed of light doing additional work on the
particle causes to start oscillating in the new dimension while particle saves its movement with
the speed of light in the previous movement path. Hence, moving path of the particle would be a
combination of the previous directions with the speed of the light, and new direction with the
velocity v. Fig.1.(a). shows how a combination of movement in perpendicular directions make a
2-dimensional sinusoidal path and Fig.1.(b). illustrates how the combination of one 2-
dimensional sinusoidal movement and oscillating in the 3rd dimension makes a 3-dimensional
spring path in the real space.
Fig.1: Combination of movement in perpendicular directions and making a new path: (a) After reaching the
speed of a one-dimensional particle to the speed of the light, it starts oscillating in a new dimension that is
perpendicular to the previous direction and makes a sinusoidal path. (b) A combination of the sinusoidal two-
dimensional movement with a new perpendicular direction in the higher dimension makes a spring path.
After reaching the speed of light and starting movement in the new dimension, physical
parameters such as the potential energy, and the momentum formula would be changed depends
on the speed of the light, while the velocity and the kinetic energy of the particle in the new
dimension resets to zero. Table 1. illustrates the relationship between the potential energy and the
momentum of the particle with the rest mass (m) in different spacetime dimensions in the
Generalized Minkowski spacetime.
Table 1. Potential energy and momentum of the particle in the Generalized Minkowski spacetime
Real dimension
number.
(n dimensional
particle)
Potential
energy
Minkowski
spacetime
(n+1)
Momentum in the
Generalized Minkowski
spacetime
1
m
2
2
mc + m
3
(mc + m)v
3
+ mc + m
4
Finally, the potential energy in each dimension is given by:
(22)
Where is the potential energy and n is the number of dimensions.
And the momentum of the particle in each dimension is given by:
(23)
Where is the momentum of the particle in the latest dimension,
We did not find differences between 4th time dimension and using 4th regular dimension at the
high speeds. Hence, we believe that we can extract the energy-momentum equation and then
investigate the special relativity equation without using Minkowski spacetime. Also, according
to the multidimensional energy-momentum equation, we would have redefined a new equation
for the kinetic energy and its approximation and finally, we would have expected that the general
relativity equation would be changed.
To conclude this section, the potential energy is invariant in the movement of the particle while
its velocity is less than the speed of the light. By increasing the velocity and reaching to the
speed of the light, the kinetic energy will be converted to the potential energy and add to
previous potential energy. Also, the momentum of the particle in the higher dimension increases
almost by multiplying to the speed of the light.
Does special relativity need more principle law?
For solving the problem of the special relativity and explaining the behavior of the particle we
suggest a law:
• After reaching the velocity of the particle to speed of the light, total kinetic energy will be
converted to the potential energy, and particle starts oscillating in a new dimension that is
perpendicular to the previous dimensions or simulates it.
This new law explains the behavior of the particle at the speed of the light, according to this
definition, particle left moving in the previous direction, and follows a new path that is a
combination of the new direction and previous path. Also, the new work on the particle causes to
increase its velocity in the new dimension.
New law implies that there is no increase of the rest mass in the higher dimensions, while the
multiplication of the mass by the speed of light in each dimension almost makes the total
potential energy of the particle. Hence, when we investigate a three-dimensional particle, we
would expect that a one-dimensional particle with mass (m) at the beginning of the second
dimension has started its movement, after increasing its velocity to the speed of the light, its
kinetic energy in the second dimension (mv) has been converted to the potential energy, while it
has been saved its movement with the speed of light in the second dimension.
Now in the third dimension where the kinetic of the particle in the third direction is zero, its
potential energy is equal to (mc), and its momentum would be equal to (mv). After reaching the
speed of light in the third dimension a three-dimensional particle will be made with the potential
energy ().
While the velocity of the particle is less than the speed of the light, it is moving in four-
dimensional spacetime. Mass-energy equivalence formula is good evidence of creating three-
dimensional particles according to this law.
Table 2. The approximate formulas of the potential energy and the momentum of the particle in the Generalized
Minkowski spacetime
Real dimension number.
(n dimensional particle)
Potential
energy at
Minkowski
spacetime
(n+1)
Momentum in the
Generalized Minkowski
spacetime
1
m
2
2
mc
3
mv
3
4
mvc
Conclusion
Einstein’s special relativity is incomplete. It is vogue at the speed of light and does not present a
correct value of the momentum in the spacetime. Hence, all formulas that have been extracted in
the last century should be reviewed. We have shown that the energy-momentum equation (
) that has used in the past century is wrong, We presented a new amount of the
potential energy in different dimensions. We need to redefine the relationship between the
kinetic energy, potential energy, total energy, and momentum of the particle to each other
according to the number of the dimension. We introduced a new law that explains the behavior
of the particle at the speed of the light.
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