Thought Experiment Revealing a Contradiction in the Special Theory of Relativity

In the thought experiment in this paper, we considered inertial frames M and A moving at a constant velocity relative to each other. A light signal emitted from inertial frame A, when time of a clock in inertial frame A was 1(s), arrived at inertial frame M when time of a clock in inertial frame M was 2(s). In this paper, the time in inertial frame A when the time in inertial frame M was 2(s) was predicted by observers in inertial frames M and A by applying the special theory of relativity (STR). Predictions of the two observers did not match. Einstein regarded all inertial frames as equivalent, but there are cases where a velocity vector is attached to some inertial frame. Einstein overlooked this fact, and thus a discrepancy appeared in the values predicted by the two observers. It is not the case that all inertial frames are equivalent. This paper concludes that the STR is a theory incorporating a contradiction which must be corrected.


Introduction
STR is not just a single theoretical system.It is composed of two theories of different types.The first is a theory derived from Lorentz transformations which has full symmetry, and the second is Einstein's energy-momentum relationship which holds in free space.
Consider a rod A (inertial frame A) and rod B (inertial frame B) moving at constant velocity relative to each other.First, let us regard inertial frame A as a stationary system, and treat inertial frame B as a moving system in motion at constant velocity v in the x-axis direction of inertial frame A.
According to the STR, when length in the direction of motion of rod B, moving at constant velocity, is measured from inertial frame A, the rod contracts in the direction of motion.Also, the time which elapses on clock B in inertial frame B is delayed compared to the time which elapses on clock A in inertial frame A.
If, conversely, inertial frame A is measured from inertial frame B, rod A contracts in the direction of motion, and the time which elapses on clock A is delayed.
According to Einstein's "principle of relativity," the two inertial frames are equivalent, and thus the same results are obtained no matter which inertial frame measurement is carried out from.The essence of STR is the symmetry of the theory.
Theoretically, there is no problem with the STR, as indicated below: 1) It is mathematically complete.
2) It can be explained using Minkowski diagrams.
It is also thought that the correctness of the STR has been demonstrated based on the following two types of experiments: 1) Extended life of elementary particles.
2) When the velocity of a moving object increases, the mass (or energy) of the object increases.
Experiment (1) is recognized even by physicists who have doubts about the STR.However, to demonstrate the correctness of the STR, one must observe lengthening of the life of stationary elementary particles from a moving system.Experiments carried out thus far have not demonstrated the symmetry of time delay.
Next is Einstein's energy-momentum relationship, which holds in free space. (1) Here, 2 0 m c is the rest mass energy of a particle or object, 2 mc is the relativistic energy, and p is the momentum.According to Equation (1), when the velocity of a moving object increases, the mass (or energy) of the object also increases.However, even if physical quantities of a stationary system are measured from a moving system, the STR does not assert that the same results are obtained.That is, there is no symmetry in Equation ( 1).Even if we assume that an increase in the mass of a moving object has been detected, that does nothing more than demonstrate the correctness of Equation (1).
Incidentally, Equation ( 1) is not applicable in the atom where potential energy exists (Suto, 2011: Suto, 2014: Suto, 2015).However, the equation definitely holds in free space.The STR which this paper views as a problem is the former theory which has perfect Lorentz symmetry.

Thought Experiment Indicating a Contradiction in the Special Theory of Relativity
Thought experiment: Rocket A is moving at a constant velocity of 3c/5 in the x-axis direction of "Stationary system."(In the following, "Stationary system may be indicated as S , and the coordinate system of rocket A as A .

S′ )
There is an observer M at the origin O of the x-axis of S , and M has a stopwatch W. In addition, there is an observer A at the origin A O′ of the A x′ -axis of A S′ , and A has a stopwatch W A .(In the following "stopwatch W" may be abbreviated as W, and "stopwatch W A " as W A .) Now, when rocket A passes in front of observer M in S , observer M starts W, and observer A starts W A .
According to the STR, an observer in S , finds the following relationship between the time t which elapses on W and the time A Here, when 1(s) is substituted for t, A 4 (s).5 t′ = (3) Here, this thought experiment is explained using Minkowski diagram 1 (see Figure 1).Oe is the value when an observer in S measures the distance OE′, and Oe′ is the value when the distance OE is measured by an observer in A S′ .However, Ee′ is parallel to the ct-axis, and eE′ is parallel to the A ct′ -axis.
Therefore, the relationship between OE，OE′，Oe and Oe′ is as follows.
Oe Oe 1 , Here, when the position of the point E is determined, it is possible to determine the positions of the points e′，e and E′ based on the relationship in Equation ( 4).Furthermore, if a point is plotted on the ct-axis at a distance equal to OE from O, that is the point event M 1 for O at 1(s) t = .Also, if a point is plotted on the A ct′ -axis at a distance equal to OE′ from O, that is the point event A 1 for A O′ at A 1(s) t′ = . Now, how should we find the relationship between the times which elapse in the stationary system and in the coordinate system of rocket A?
To find that, it is enough to compare the times when the straight line parallel to the x-axis intersects with the ct-axis and A ct′ -axis.
For example, among the lines which pass through M 1 , the straight line parallel with the x-axis intersects the A ct′ -axis at point A 4/5 , and this is the point event of W A when 1(s) t = . Therefore A t′ matches with Equation (3).Now when W in S is at 1(s), a light signal is emitted from O to A O′ in A S′ .That light propagates isotropically with respect to O. Then it arrives at A O′ when W A on rocket A is 2(s).(This light signal corresponds to the world line M 1 A 2 .) In the inverse case, when W A on rocket A is 1(s), a light signal is emitted from A O′ to O.That light arrives at O when W of the stationary system is 2(s).(This light signal corresponds to the world line A 1 M 2 .) These results also seem to show there is symmetry between the two inertial systems.In this paper, the propagation situation of the two light signals (M 1 A 2 and A 1 M 2 ) is expressed as follows.
A 1 (s) 2 (s), Now, are the two inertial systems truly equivalent, as claimed by the STR?Next let's try having observer M and A predict the time of W A on rocket A when W is 2(s).

Discussion
A. Prediction of observer M (prediction based on the STR) (see Figure 2 (a)) In this paper, the moving object is taken to be rocket A, which has passed through an acceleration stage.Therefore, the time A t′ of W A can be found from Equation (2).
To find A t′ when 2(s) t = , it is enough to substitute 2(s) for t and 3c/5 for v in Equation ( 2).This yields: ( ) The observer in S concludes that the time A t′ of W A when the time of W is 2(s) is 1.6(s).
B. Prediction of observer A (prediction based on the STR) (see Figure 2 (b)) The observer in rocket A regards his own coordinate system as the stationary system.With the STR, the observer in A S predicts the value of the time A t of W A as follows when the time of W in S′ is 2(s).
( ) The observer in A S concludes that the time A t of W A when the time of W is 2(s) is 2.5(s).
In the end, if observers M and A predict the time of W A at a certain instant by applying the STR, different values are obtained.
Point O indicates both origins: 0 x = , 0 t = and A 0 x′ = , A 0 t′ = .The point event M 0 of the point light source O and the point event A 0 of the point light source A O′ are at the origin O. (Here, the subscripts " 0 " of the point events M 0 and A 0 mean, respectively, t＝0 and A 0 t′ = .)Thex-axis indicates the x-axis of the inertial frame S when t＝0.In addition, the A x′ -axis indicates the A x′ -axis of the inertial frame A S′ when A 0 t′ = .The ct-axis is the path for x＝0.Put another way, it is the world line of the origin of S .The A ct′ -axis is the world line of the origin of A S′ .In addition, the straight line extending at a 45° angle from the origin O indicates the light signal emitted from the two light sources at the instant that O and A O′ pass by each other.OE is the distance over which the light signal emitted from O propagates in the x-axis direction while 1(s)elapses on the stopwatch W in S .OE′ is the distance over which the light signal emitted from A O′ propagates in the A x′ -axis direction while 1(s) elapses on the stopwatch W A in A S′ .

Figure 1 .
Figure 1.Minkowski diagram 1: This diagram corresponds to thought experiment