A Violation of the Special Theory of Relativity Demonstrated Using the Correlation between Two Pair-Generated Photons

When Einstein developed the special theory of relativity (STR), he assumed the principle of relativity, i.e. that all inertial frames are equivalent. Einstein thought it was impossible to differentiate inertial frames into classically stationary frames where light propagates isotropically, and classically moving frames where light propagates anisotropically. However, the author has previously pointed out that classically moving frames have a velocity vector attached, and presented a thought experiment for determining the size of that velocity vector. The author has already shown a violation of the STR, but this paper presents a violation of the STR using different reasoning. More specifi-cally, this paper searches for a coordinate system where light propagates anisotropically. This is done by using the correlation of two photons pair-generated from a photon pair generator. If the existence of such a coordinate system can be ascertained, it will constitute a violation of the STR.


Introduction
At the end of the 19 th century, most physicists were convinced of the existence of ether as a medium that propagates light. Further, they thought ether to be "absolutely stationary". Michelson and Morley attempted to detect Earth's motion relative to the luminiferous ether, i.e. the absolute velocity. However, they failed to detect the ef-How to cite this paper: Suto, K. (2020) A Violation of the Special Theory of Relativity Demonstrated Using the Correlation between Two Pair-Generated Photons. Journal of Applied Mathematics and Physics, 8, fect they had expected [1]. In order to explain why they failed to detect the effect they had expected, Michelson concluded that the ether was at rest relative to the earth in motion (i.e. it accompanied the earth).
On the other hand, Lorentz was convinced of the earth's motion relative to the "preferred frame". He made a stopgap solution by proposing a hypothesis that a body moves through space at the velocity v relative to the ether contracted by a factor of ( ) 2 1 v c − in the direction of motion [2].
Michelson believed that light emitted from a laboratory on earth propagated isotropically, while light propagated anisotropically in the interpretation of Lorentz.
However, in his special theory of relativity (STR) published in 1905, Einstein insisted that physics did not require an "absolutely stationary frames" provided with special property, and that there be no such things as "specially-favoured" coordinate systems to occasion the introduction of the ether-idea [3].
As a physical theory representing the 20 th century, the STR has held sway in the world of physics for more than a century. During this time, the STR has fended off challenges and counterarguments from many physicists [4].
When Einstein developed the STR, he assumed the principle of relativity, i.e. that all inertial frames are equivalent. Einstein thought it was impossible to differentiate inertial frames into classically stationary frames cl S where light propagates isotropically, and classically moving frames cl S′ where light propagates anisotropically. However, the author has previously pointed out that classically moving frames have a velocity vector attached, and presented a thought experiment for determining the size of that velocity vector [5] [6] [7] [8]. The author has already shown a violation of the STR, but this paper presents a violation of the STR using different reasoning. The thought experiment discussed here strictly distinguishes between classically stationary frames and classically moving frames.
In the STR, the Cartesian coordinate system and oblique coordinate system of the Minkowski diagram are equivalent, and the two can be interchanged. However, in this paper, the argument is developed by placing classically stationary frames into correspondence with the Cartesian coordinate system of the Minkowski diagram, and placing classically moving frames into correspondence with the oblique coordinate system of that diagram.
Einstein assumed the following two principles when developing the STR. 1) Principle of relativity; 2) Principle of constancy of light speed. First, Einstein explained the principle of relativity as follows [9]. "The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion".
In addition, he explained the principle of constancy of light speed as follows [3].
"Light is always propagated in empty space with a definite velocity c which is K. Suto Journal of Applied Mathematics and Physics independent of the state of motion of the emitting body".
Light is always propagated at a constant velocity c, regardless of the velocity of the source emitting the light. In this paper, this principle is called the "principle of constancy of light speed I" (principle I) (However, note that Einstein himself did not classify the principle of constancy of light speed).
Einstein also said the following [3].
"These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies".
However, the STR cannot be developed with these two assumptions alone (principle of relativity and principle I). Einstein also explained the principle of constancy of light speed as follows [10].
"Let a ray of light start at the 'A time' A t from A towards B, let it at the 'B time' B t be reflected at B in the direction of A, and arrive again at A at the 'A In agreement with experience we further assume the quantity to be a universal constant-the velocity of light in empty space". In Formula (2), when the distance covered by light making a round trip over the interval AB is divided by the time needed for the round trip, light speed becomes c. This principle will be called the "principle of constancy of light speed II" (principle II).
Incidentally, even if principle II holds, light on the outward path does not necessarily propagate isotropically. However, if the light source is in a classically stationary frame, then by definition light on the outward path propagates isotropically (In this case, light speed on both the outward and return path is c). This principle will be called the "principle of constancy of light speed O" (principle O).
Einstein claimed that all inertial frames are equivalent from the standpoint of the principle of relativity. However, among the coordinate systems regarded as inertial frames, this paper defines a coordinate system in which principle O holds to be a classically stationary frame. Also, a coordinate system in which principle II holds but O doesn't is defined to be a classically moving frame (Principle O is a special case of principle II).
In frame cl S , light propagates isotropically, and in frame cl S′ , light propagates anisotropically. The reason why light cannot propagate isotropically in frame cl S′ is the velocity vector attached to frame cl S′ (Appendix).
The author has already presented a thought experiment demonstrating the existence of that velocity vector. The author has also pointed out the contradictions of the STR in another paper [11]- [18].

Journal of Applied Mathematics and Physics
The author has already demonstrated a violation of the STR, but this paper demonstrates a violation of the STR using a different thought experiment that is easier to understand. If a coordinate system in which light propagates anisotropically can be found amount coordinate systems regarded as inertial frames, that will constitute a violation of the STR.

Anisotropic Propagation of Light Demonstrated Using the Correlation between Two Pair-Generated Photons
In this section, an experiment is conducted in rocket A moving at constant velocity of 0.6c with respect to frame cl S . The A x -axis passing through the center of the room coincides with the direction of motion of rocket A, and with the x-axis of frame cl S is maintained. Now, let's observe, from frame cl S , the propagation of light emitted from a light source on the moving rocket.
The light speed of a moving frame does not depend on velocity of the light source (principle I). Light emitted from the moving frame propagates isotropically with respect to frame cl S .
Therefore, viewing from frame cl S , light arrives first at the rear wall of the rocket before the front wall (However, it is assumed that the distances from the light source to the front and back walls are equal).
This section presents a method for checking, through an experiment in the rocket, that light emitted from inside the rocket is propagated anisotropically.  Next, let's discuss this situation using a Minkowski diagram. The thought experiment discussed here strictly distinguishes between classically stationary frames and classically moving frames.

At the point
In the STR, the Cartesian coordinate system and oblique coordinate system of the Minkowski diagrams are equivalent, and the two can be interchanged [19] [20]. However, in this paper, the argument is developed by placing classically stationary frames into correspondence with the Cartesian coordinate system of the Minkowski diagrams, and placing classically moving frames into correspondence with the oblique coordinate system of that diagram (Figure 2).
In this paper, frame cl S is placed in correspondence with the Cartesian coordinate system of the Minkowski diagram. The ct-axis is the world line for  In the A A x ct ′ ′ − frame, the line parallel to the A x′ -axis is synchronous in this coordinate system [21]. Next, let's decide on a coordinate of the A x -axis corresponding to spacetime point B. If a point in these coordinates is determined, then the velocity of the moving frame can also be calculated. In this paper, that position is determined using two methods.
Method 1. When propagation of light in the rocket is observed from frame cl S , the following formula holds.
Here, a is defined as follows.
Here, the left side of Formula (3) is the value predicted using a clock in a stationary frame by an observer in frame cl S of the time required for photon 1 to arrive a polarizer 1. aL on the right side is the distance 1 l covered by photon 2 in the positive direction of the A x -axis when that time elapses. The formula contains γ because, viewing from frame cl S , the moving object contracts by 1/γ times in the direction of motion.
Solving Formula (3), the following values are obtained.
If the value of a is determined by measurement, then the velocity of the moving frame can be calculated from that.  OB . OC a = (8) In Figure 2, the A x′ -axis and A ct′ -axis are given by the following formulas. , Here, b is the inclination of the A x′ -axis.
Also, lines OA and OB for the two light signals can be expressed with the following formulas.
. ct x = − (11) . ct x = (12) Next, the world line for polarizer 1 is the line passing through spacetime points A and A′ . The coordinates of the intersection of this line and the x-axis are (−1 − b, 0). Since the slope of this line is 1/b, its equations is 1 .
In contrast, the line passing through spacetime points C and C′ is as follows because the x-coordinate of C′ is 1 + b 1 .
Next, the x-coordinates of spacetime points A and C are found from these equations.
First, to find the x-coordinates of spacetime point A, it is enough to solve the following simultaneous equations for x.

.
ct If the x found here is taken to be A x , Hence the x-coordinate B x of spacetime point B is Next, the following simultaneous equations are solved to find the x-coordinate of spacetime point C.

.
ct x If the x found here is taken to be C x , Hence, Also, using v = bc, In the above, it was possible to find a formula for the velocity of the moving frame using two methods.
In the thought experiment of this paper, v = 0.6c. Therefore if this is substituted into Formula (5), the following a is obtained.
In the end, photon 2 has a probability of passing through polarizer 3 placed in the positive direction of the x-axis when polarizer 3 is placed in this region.
In the STR, in contrast, light propagates isotropically in all inertial frames, and thus photon 2 has a probability of passing through polarizer 3 when polarizer 3 is placed in the following region.
When v = 0.6c, a difference arises between the predictions of this paper and the STR in the following region.
The prediction of the STR is that photon 2 has a probability of passing through polarizer 3 placed in the region of Formula (27), but the prediction of this paper is that photon 2 cannot pass through polarizer 3 placed in this region.

Conclusions
Einstein asserted that, from the standpoint of the principle of relativity, all inertial frames are equivalent. Therefore, Einstein was unable to accept that, within coordinate systems he regarded as inertial frames, there are classically stationary frames in which light propagates isotropically, and classically moving frames in which light propagates anisotropically.
If an experiment exists which can discriminate between these two types of coordinate systems, then the STR will no longer be a correct theory.
This paper has presented, for a second time, a thought experiment enabling discrimination between classically stationary frames and classically moving frames.
This means that the STR has been rejected.
The prediction of this paper is that photon 2 will not pass through polarizer 3 in the following region. This differs from the prediction of the STR, which states that photon 2 has a probability of passing through polarizer 3 in the region A 0 x L < < .
A difference arises between the predictions of this paper and the STR because the STR does not recognize the existence of the velocity vector attached to the frame cl S′ .
If the coordinate ( ) 1 l aL = can be determined by experiment, then it will be possible to find the velocity of the frame cl S′ from Formula (7). If the prediction of this paper is correct, then the author will have demonstrated a violation of the STR using two different methods.