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The purpose of the research in this article is the examination of the agreement of the hypothesis of the absolute reference system with the results of experiments that have been implemented in the past in order to confirm the special theory of relativity. To achieve this goal, we have chosen for discuss ing a theoretical topic of electromagnetism, that of electromagnetic mass calculation, and some experiments, some of which concern the transverse Doppler effect in a rotated system, two experiments that concern the kinetic energy measurement of accelerated electrons, one of which is the well known Bertozzis experiment, one experiment that concern s the propagation of Coulomb fields and one more experiment that concern s the effect of annihilation. The basic principles of the hypothesis of the absolute reference system, and the electromagnetic theory derived from these principles, are used to explain the experimental results. In these examples, the hypothesis of the absolute reference system is confirmed, since the experimental results agree with the predictions of this hypothesis. Also, in the discussion of calculation of electromagnetic mass is addressed the difficulty of solving this problem, when someone tr ies to solve this according to the energy-mass relation of the theory of relativity.

In all the experiments carried out in order to examine the agreement of the special theory of relativity with the corresponding experimental results, the theoretical predictions that are commented on in relation to this agreement are those of the theory of relativity and the Newtonian physics. This means that the theories examined in all these experiments are the special theory of relativity and Newtonian physics. Bertozzi’s experiment is one such example. What is being proven in this experiment is that the speed of the accelerating electrons can never reach the speed of light in vacuum. However, as can be seen from a closer look at this article, the expected relativistic kinetic energy values resulting from the measured velocities of the electrons deviate greatly from the measured kinetic energies, which were measured by a thermodynamic method. But because this deviation for the corresponding Newtonian energies was much greater, this was enough to convince the scientific community of the correctness of the theory of special relativity.

Of course, Newtonian physics deviates significantly from experimental reality as particle speed and energy increase. This article examines a new theory in terms of its agreement with these experimental results, and this theory is expressed in the newly introduced hypothesis of the absolute reference system. The previous comparison, therefore, concerns in this case the predictions of the special theory of relativity in comparison with the predictions of the hypothesis of the absolute reference system, regarding these experimental results.

It was deemed appropriate in this article to first consider a theoretical issue, that of the problem of incompatibility of the electromagnetic mass with the relation E = m c 2 of the special theory of relativity, because this incompatibility is eliminated by applying the principles of hypothesis of the absolute reference system for the electromagnetic interactions. This is a much-discussed problem, but we will refer specifically to the relevant analysis of this issue by Feynman, as it is examined in [

In the study of the experiment of positron-electron annihilation process, it seems that the theoretical results of the hypothesis of the absolute reference system are the same as those of the special relativity. In all other experiments studied in this article, the agreement of the newly introduced hypothesis with the experimental results is examined, because there is no such agreement of the experimental results and the corresponding relativistic theoretical predictions (for example, the experiment for Measuring Propagation Speed of Coulomb Fields, in [

The attempt to interpret natural phenomena on the basis of the hypothesis of an absolute reference system does not based on the attempt to interpret the phenomena using the relativistic conception, which was formulated in Galilean relativity and subsequently in Einstein’s theory of relativity. The relativistic notion is based on the evaluation of physical quantities by the observer of the natural phenomenon at the observation point. On the basis of the absolute reference system, the space-time is Newtonian and the physical quantities such as momentum, force and kinetic energy are not determined on the basis of the relative velocity of the observed object by the observer’s reference system, but on the basis of the object’s reference system and the inertial reference system of the source of the field that affects the state of the object.

Electromagnetic interactions based on this hypothesis are carried out by force carriers, which are real photons with some peculiar behavior, unknown to this day, although this theoretical result is derived from classical electromagnetic theory. Also as explained in [

The very old problem of electromagnetic mass in relation to the equation of mass and energy in the special theory of relativity is the first issue that we will consider in this paper. We will begin first with the study of this subject according to classical electromagnetic theory.

We consider a particle of charge e moving at velocity v with respect to the inertial laboratory reference system. We will calculate the energy of the electromagnetic field in space between two concentric spheres of radii a and b. We consider that the center of the spheres is the center of mass of the charged particle. We will then calculate the momentum and mass of the electromagnetic field in the space between the two spheres1. Assuming a < b the electromagnetic energy is:

U e l e c = 1 2 e 2 4 π ϵ o ( 1 a − 1 b ) (1)

when the speed v is much less than the speed of light in the vacuum, the momentum of that field is:

p = 2 3 e 2 4 π ϵ o c 2 v ( 1 a − 1 b ) (2)

The corresponding electromagnetic mass is:

m e l e c = 2 3 e 2 4 π ϵ o c 2 ( 1 a − 1 b ) = 4 3 U e l e c c 2 (3)

According to the absolute reference system hypothesis the previous equation is fully justified. It turns out that, on the basis of this hypothesis, the force carriers of Coulomb electric field are real photons. It also turns out that part of the energy ℏ ω of these photons is the energy of an elementary mass. The amount of energy of this elemental mass, as shown by the relevant calculations2, is equal to 2 3 ℏ ω . If we denote by m p h the mass of this photon, the energy equation of this photonic mass is m p h c 2 = 2 3 ℏ ω .

This result has been obtained by making use of Maxwell’s equations and the Coulomb field spherical symmetry. Also, according to the same hypothesis, the total kinetic energy of all the photons of the field in the space between the two spheres, for N photons, is U e l e c = 1 2 ℏ ∑ i = 1 N ω i . Therefore, in accordance with all the foregoing, which derive on the basis of the absolute reference system hypothesis, the electromagnetic mass is:

m e l e c = ∑ i = 1 N m p h i = 2 3 ℏ c 2 ∑ i = 1 N ω i = 4 3 U e l e c c 2 (4)

The origin of this problem lies in Walter Kundig’s 1963 experiment entitled “Measurement of the Transverse Doppler Effect in an Accelerated System’’, in [

The experiment was repeated in 2009 by the same scientific team, and the result of the experiment, exposed to [

One attempt to explain this experimental result on the basis of theory of relativity, in 2015, in [

The experiment has been repeated several times. In 2011 by A. L. Kholmetskii, T. Yarman and O. V. Missevitch. (Int. J. Phys. Sci. 6, 84 (2011)), the experimental result yields the coefficient k = 0.66 ± 0.03 within the expression frame for the relative energy shift between emission and absorption lines Δ E / E = − k v 2 / c 2 .

We will then use the theory of the absolute reference system to address this problem. According to this hypothesis in the Coulomb field of an atomic nucleus of the absorber material, the energy ℏ ω of the absorbed photons is divided into two components of energy, due to the two components of their helical motion within this field, as explained in [

In the case of the experiment in [

E a r = 1 γ ℏ ω ′ r = ℏ ω r (5)

The amount of energy ℏ ω v , characterized as energy of mass, obeys the kinetic energy conservation relation of an elementary photon mass in the source-absorber system:

1 2 ℏ ω v = 1 2 ℏ ω v a + E p h k i n (6)

where E p h k i n is the kinetic energy of the photonic mass due to the rotary motion of the absorber and the angular frequency of the photonic mass, measured by the clock of the instantaneous inertial system of the absorber, is denoted by ω v a . If the absorbed photon mass is denoted by m v a , the previous relation becomes:

1 2 ℏ ω v = 1 2 m v a c 2 + 1 2 m v a γ 2 u 2 = 1 2 m v a c 2 ( 1 + γ 2 β 2 ) = 1 2 γ 2 ℏ ω v a (7)

where m v a = ℏ ω v a / c 2 and β = u / c . The angular frequency of the absorbed mass according to the last relation is ω v a = ω v / γ 2 . The change in absorption rate due to the Doppler effect is canceled by the time contraction as in the case of calculating the radial motion energy ℏ ω r , previously mentioned. The energy supplied by the source will be given by the relation:

E a v = ℏ ω v a = 1 γ 2 ℏ ω v (8)

Therefore, the total absorbed photon energy is E a = E a r + E a v , while the total energy of a source photon is E s = ℏ ω and only an amount of kinetic energy, equal to E p h k i n = 1 2 m v a γ 2 u 2 , is offered by the rotated absorber, whereby, the equation of the relative energy shift between emission and absorption lines, is:

E a − E s E s = ℏ ω r + 1 γ 2 ℏ ω v − ℏ ω ℏ ω = 1 3 ℏ ω + 1 γ 2 2 3 ℏ ω − ℏ ω ℏ ω = − 2 3 β 2 (9)

This result is in excellent agreement with the above experimental results.

In order to experimentally test the correctness of the absolute reference system hypothesis, we will study two experiments measuring the kinetic energy of electrons, which simultaneously measure the beam electron velocity in a linear accelerator. The peculiarity of these experiments is that the velocity is measured by measuring the flight time of the electrons at a given travel length. One of these two experiments is the historical experiment of Bertozzi, described in [

According to the absolute reference system hypothesis, kinetic energy which is transferred to the detection device, which is the target of accelerated electrons, when the kinetic energy E a of the electrons is high enough so that the scattering of the force carriers by atoms of target to be no longer elastic, equals E a / 2 . The remaining half of the kinetic energy of the electrons is converted into mass absorbed by the atoms of the target. This absorbed mass is derived from the photon mass of the electromagnetic field force carriers of the accelerated electrons. As we have mentioned, this electromagnetic mass of the force carriers are real photons that exhibit such behavior under the given conditions in the Coulomb field.

A first estimate of this phenomenon in the Bertozzi experiment is made by studying the experimental results presented in [

Measured energy E m , MeV | Electron velocity u , × 10 8 m / sec | Relativistic kin. energy E k , MeV | Detectable kin. energy E d e t , MeV |
---|---|---|---|

1.6 | 2.88 | 1.3 | 1.5 |

4.8 | 2.96 | 2.7 | 4.8 |

The transmitted kinetic energy of the beam is equal to the increase in target heat. This energy per electron, measured by the thermal arrangement of the Bertozzi experiment, is denoted by E m in

The transfer of kinetic energy in the form of heat to the target is accomplished by the transfer of the kinetic energy of the electromagnetic mass of the force carriers of the electron beam electromagnetic field of the linear accelerator. The electromagnetic mass m e l , passing through the active scattering cross section of the force carriers with the bound electron of an atom in the target material, transfers all the kinetic energy of the beam electron and is equal to the mass of this electron from which it is derived, that is, m = m e l . In order for the scattering to be completely inelastic, it must be possible to incorporate half the kinetic energy of the beam’s electron, that is energy equal to E a / 2 , as photonic mass in the atoms of the target material. There is a limitation to be implemented this completely inelastic interaction. Assuming that N force carriers transfer the kinetic energy of a beam electron, the corresponding electromagnetic mass, according to the relation (4), is m e l = 2 3 ℏ c 2 ∑ i = 1 N ω i . Therefore for fully inelastic scattering the energy E a / 2 should be greater than ℏ ∑ i = 1 N ω i , which is the total energy of the Coulomb field in the area of the specific electromagnetic mass m e l . Therefore the threshold for a fully inelastic scattering is determined by the relation:

E a 2 = 1 4 ( γ 2 − 1 ) m c 2 = ℏ ∑ i = 1 N ω i = 3 2 m e l c 2 (10)

Since m = m e l , the last relation gives ( 1 / 4 ) ( γ 2 − 1 ) = 3 / 2 and therefore this relation is satisfied for γ values greater than a critical value of γ which is γ c = 7 and a corresponding critical speed u c given by the relation u c = 6 7 c . The corresponding kinetic energy is E c = 1 4 ( γ c 2 − 1 ) m c 2 = 3 2 m c 2 . Therefore the criterion for completely inelastic scattering is γ ≥ 7 or equivalently 1 2 E a ≥ 3 2 m c 2 .

For γ < 7 an amount of energy equal to 1 4 ( γ 2 − 1 ) m c 2 is not sufficient to fully incorporate all the electromagnetic mass m e l , but only one part of it, which is equal to a m e l , where the coefficient a is equal to the ratio of the energy values ( E a / 2 ) / E c , that is, a = ( γ 2 − 1 ) / ( γ c 2 − 1 ) . In this case the amount of kinetic energy incorporated as a mass is equal to 1 4 ( γ 2 − 1 ) a m e l c 2 . Therefore the detectable kinetic energy transferred to the target will be:

E d e t = 1 2 ( γ 2 − 1 ) m c 2 − 1 4 ( γ 2 − 1 ) γ 2 − 1 6 m c 2 = ( γ 2 − 1 2 − ( γ 2 − 1 ) 2 24 ) m c 2 (11)

According to the latter relation, at low speeds, that is, u ≪ c and γ → 1 , the detectable value of energy tends to the Newtonian value of kinetic energy, that is, E d e t → 1 2 m u 2 .

The electron beam measured energy values of the experiment by M. Lund and U. I. Uggerhøj are in excellent agreement with the values of the energy curve E d e t in the two aforementioned speed ranges, that is, for γ ≥ 7 , where E d e t = E a / 2 , and for γ ≤ 7 , where the relation (11) applies, that is, the experimental data in [

In the case of very high energies of electrons, for example energies of the order of several tens of TeV, due to the very high value of the coefficient γ , the actual physical contraction of the particle contributes to the decomposition of the particle, and, so, the particle no longer behaves as a solid body. This has the effect of hitting the target by a photon beam instead of the particle beam and these photons of the total mass of the particles have frequencies denoted by ν i l , measured in the laboratory, which are γ times smaller than the corresponding measured frequencies in the particle reference system.

It should be noted that according to the hypothesis of the absolute reference system, the Lorentz contraction is a real physical contraction and not a geometric contraction, in Minkowski’s space-time, which concerns the observer’s inertial system, as inferred from the special theory of relativity.

In this case the kinetic energy is E a = ( 1 / 2 ) γ h ∑ i ν i l = ( 1 / 2 ) ( γ 2 − 1 ) m c 2 and the total energy transferred to the target is equal to h ∑ i ν i l ≃ m γ c 2 .

The theoretical calculations in this section concern the measurement of the potential difference that results from the effect of the maximum transverse electric field on the detector of the experiment in [

According to the following equation ( [

[ E ] r e t = e 4 π ϵ o [ n ^ − β γ 2 ( 1 − β ⋅ n ^ ) 3 R 2 ] r e t (12)

the transverse component of the electric field, in the unit system SI, at a constant hight y = b , with the beam extending along the Z-direction, considering that the point M is the origin of the axes ( [

E 2 = e 4 π ϵ o b γ 2 ( b 2 + z 2 + β z ) 3 (12)

and the maximum transverse component of the electric field is:

E 2 m a x = e 4 π ϵ o γ b 2 (13)

We first assume that a free electron beam propagates in a linear range of about 1000 meters, with the sensor in the middle of this distance. In

Concerning the electron beam of the accelerator it is mentioned “… the

electron beam produced at the DAΦNE Beam Test Facility (BTF) [

Therefore the length of a portion of the beam coming from a burst is l = 10 ns × u ≃ 3 m . We assume that the sensor is above the middle of this segment and that ϕ is the angle formed by R ( t ′ ) defining the position of the electron giving the signal, and by the linear segment OM ( [

According to the relation R ( t ′ ) − ( 1 / c ) R ( t ′ ) ⋅ u = R ( t ) 1 − ( u 2 / c 2 ) sin 2 ( θ ( t ) ) ( [

E ( t ) = e 4 π ϵ o R ( t ) R ( t ) 3 1 − u 2 / c 2 ( 1 − u 2 c 2 sin 2 ( θ ( t ) ) ) 3 2 (15)

The differential component derived from an elementary charge d q = λ d z , where λ is the linear charge density of the segment of the beam in time t, taking into account that y = b = R ( t ) cos ( ϕ ( t ) ) , is:

d E y ( t ) = λ 4 π ϵ o γ 2 cos ( ϕ ( t ) ) d z R ( t ) 2 ( 1 − u 2 c 2 sin 2 ( θ ( t ) ) ) 3 2 (16)

Since R ( t ) cos ( ϕ ( t ) ) = y and d tan ϕ ( t ) = d z ( t ) / y = d ϕ ( t ) / cos 2 ( ϕ ( t ) ) , the following relation applies:

d z ( t ) R ( t ) 2 = d ϕ ( t ) y (17)

The angles ϕ and θ are complementary, so:

d E y ( t ) = λ 4 π ϵ o γ 2 y d sin ϕ ( 1 γ 2 + u 2 c 2 sin 2 ( ϕ ) ) 3 2 (18)

For convenience, we set β sin ϕ = x , where β = u / c , and so:

d E y ( t ) = λ 4 π ϵ o γ 2 y 1 β d x ( 1 γ 2 + x 2 ) 3 2 (19)

In a linear infinite distribution, because of the symmetry, the integration gives a transverse component of the electric field, at a height y, at time t, equal to:

E y ( t ) = 2 λ 4 π ϵ o γ 2 y 1 β ∫ 0 β d x ( 1 γ 2 + x 2 ) 3 2 = λ 2 π ϵ o γ 2 y 1 β [ x 1 γ 2 x 2 + 1 γ 2 ] 0 β = λ 2 π ϵ o y (20)

Since the length of this beam segment is finite ( l ≃ 3 m ), the maximum angle ϕ m a x is 79˚ and the corresponding sine, at a height of y = 30 cm , is sin ϕ m a x = 0.98 . Therefore, the term in square brackets, of the previous relation, becomes:

1 β [ x 1 γ 2 x 2 + 1 γ 2 ] 0 0.98 β ≃ 0.99999998 γ 2 ≃ γ 2 (21)

Therefore, this finite length portion of the beam gives a transverse electric field component equal to that which is given by an infinite length beam.

The maximum potential difference at the ends of the sensor y 1 = y , y 2 = y + 14 cm is:

V m a x = λ 2 π ϵ o ∫ y 1 y 2 d y y = λ 2 π ϵ o ln ( y + 14 cm y ) (22)

If we multiply the second member of the previous relationship by a calibration coefficient η , we get the relation (8) of the bibliographic reference [

The transverse component of the electric field derived from a single electron, based on the Equation (15), and the relation R ( t ) = y / sin θ , is given by the relation:

E 2 ( t ) = e 4 π ϵ o sin 3 θ y 2 1 − u 2 / c 2 ( 1 − u 2 c 2 sin 2 ( θ ( t ) ) ) 3 2 (23)

The ratio E 2 ( t ) / E 2 m a x is therefore given by the relation:

E 2 ( t ) E 2 m a x = sin 3 ( θ ( t ) ) γ 3 ( 1 − u 2 c 2 sin 2 ( θ ( t ) ) ) 3 2 (24)

In

As can be seen from

In order to better understand this, we will calculate the relation between z ( t ′ ) and z t = z ( t ) . From the known relation z ( t ) = z ( t ′ ) + β R ( t ′ ) = z ( t ′ ) + β b 2 + z ( t ′ ) 2 , by setting z = z ( t ′ ) , z t = z ( t ) , we obtain the following equation:

z 2 − 2 γ 2 z t z + γ 2 z t 2 − β 2 γ 2 b 2 = 0 (25)

The resulting values of z are z = γ 2 z t ± β γ γ 2 z t 2 + b 2 . The value with (+) is rejected, since for z t = 0 we have the relation z = − β γ b . This solution shows that for very short distances from M we have large absolute values of z and that is why we take the curve of

However, since in the laboratory of this experiment the electronic beam extends at distances of less than 10 m, the sensor should receive signals departing from this region, and so the transverse component of the electric field should be much smaller relative to the maximum we calculated in the previous relation.

In ^{−4}, and such a signal is not detectable by the experimental device. Therefore, the measured signal is many orders of magnitude larger than the predicted by relativistic theory.

From the Equation (25), if we consider b = y , we obtain an acceptable solution z t = z + β z 2 + y 2 . Assuming that the left end of the portion of the electron beam is in the initial position where the charges generating the field are not be shielded by conductors and it is possible to emit an electromagnetic signal, then the signal will arrive at the sensor at time t and this end of the beam will be at the corresponding position z t 1 ( t ) . The right end of the beam at the same time t will be at the position z t 2 = z t 1 + 3 m . The limits of integration of the relation (20) are determined by the relation x i = β sin ( arctan ( z t i / y ) ) .

The calculation of the ratio of the relativistic maximum electrical potential differences to measured maximum electrical potential at three different positions of the sensor in the direction of Z, at 1720 mm, 3295 mm, and 5525 mm, but at a constant height from the lower end of sensor end at 30 cm, gives us:

V 1720 V m a x = 4.6914 e − 05

V 3295 V m a x = 1.6953 e − 04

V 5525 V m a x = 4.7477 e − 04

where V 1720 , V 3295 , V 5525 , the calculated electrical potential differences according to the special theory of relativity, that is, at the height y = 30 cm at which we consider that is the location of the sensor, the measured V m a x is many orders of magnitude higher than the one would expect from the calculations of relativistic electric potential differences.

In

However, the measurements of this experiment show that the maximum electric potential differences are in agreement with the relation (22). Therefore, as shown in

In the abstract of my published work entitled “The Physics of an Absolute Reference System’’ ( [

In the present experiment of the reference [

We will now calculate the electric field derived from a portion of the electron beam with a linear density λ in the beam reference system, according to the hypothesis of the absolute reference system. The maximum value of the calculated electric field at a height of y, at the observation point, that is, at the sensor, will be when the sensor is above the mean of the linear charge distribution. Since the sensor is stationary in the laboratory, the calculation of electric field is derived from Maxwell’s equations. So, it is calculated according to the relation:

E M = λ 2 π ϵ o y j ^ (26)

where the index M means that the electric field is calculated according to Maxwell’s equations and j ^ is a unit vector in the Y-direction.

The electric potential difference between the y 1 and y 2 sensor edges is calculated as follows:

V m a x = λ 2 π ϵ o [ ln y ] y 1 y 2 (27)

so for y 1 = y and y 2 = y + 14 cm this relation becomes:

V m a x = λ 2 π ϵ o ln ( y + 14 cm y ) (28)

An experiment implemented in order to experimentally investigate relativistic kinematics in an undergraduate student laboratory is the article in [

In this article we give the theoretical background for this process using the kinematics of the hypothesis of the absolute reference system. We consider a positron with kinetic energy E and momentum p that annihilates with an electron at rest with the emission of two photons as shown in

The total kinetic energy of the positron is E = ( 1 / 2 ) m γ 2 c 2 and because the frequencies of the bound photons measured with the clock of the inertial reference system of the laboratory are γ times smaller than the corresponding ones measured in the reference system of the positron, the contribution of positron to energy of the emitted photons is equal to E l = E / γ = ( 1 / 2 ) m γ c 2 , while the total kinetic energy of the electron that is at rest in the laboratory is equal to ( 1 / 2 ) m c 2 . Conservation of momentum and energy give

p = p 1 + p 2 (29)

E l + 1 2 m c 2 = 1 2 E 1 + 1 2 E 2 (30)

Squaring both sides of the Equation (29) we get

p 2 = p 1 2 + p 2 2 + 2 p 1 p 2 cos θ (31)

Using the previous relations and the relation ( m γ c 2 ) 2 = p 2 c 2 + m 2 c 4 , which is valid in relativistic kinematics and in kinematics of hypothesis of absolute reference system, and after some simple algebraic calculations, we get the following relation,

1 E 1 + 1 E 2 = 1 − cos θ m c 2 (32)

which is the same as that derived from relativistic kinematics. Also, the last equation is the main result that was experimentally verified in [

In all experiments studied in this article on the basis of the hypothesis of an absolute reference system, it appeared unequivocally that the experimental results are in excellent agreement with the theoretically predicted results of this hypothesis. There are further similar issues to be discussed, but the most of them have already been discussed in the initial publications of the hypothesis of the absolute reference system. In the present study some issues discussed in order to test the validity of this hypothesis. However, as much as possible experimental research is suggested, mainly taking as many as possible experimental data on the speed and energy values of the Bertozzi’s experiment, as the results of this experiment have a one-sided interpretation in the scientific world, in order for us to reach scientific conclusions, fully accepted by the scientific community, regarding to the agreement of the hypothesis of absolute reference system with this experiment.

The author declares no conflicts of interest regarding the publication of this paper.

Patrinos, K. (2020) The Confirmation of Hypothesis of the Absolute Reference System. Journal of Applied Mathematics and Physics, 8, 999-1015. https://doi.org/10.4236/jamp.2020.85078