Mathematical solution of the problems of relativity

This article analyzes the theoretical part of the Michelson and Morley experience. In particular, I am arguing about their annoying mistake made in the theoretical part of the experience. The value of the work done is to revise the results of the Michelson and Morley experiment, with the ensuing consequences.

I. Introduction

In 1887, a joint experiment of two American physicists - Albert Michelson and Edward Morley (hereinafter - the experience of Michelson and Morley) had to prove that the luminiferous ether actually exists. The result of this experience, one way or another, formed the basis of the theory of relativity by Albert Einstein, which is currently the fundamental theory in modern relativistic scientific research, although he does not directly refer to this experience in his works. However, in presenting the theory of relativity, rarely does the author bypass this topic.

It is necessary to recognize that not all researchers agree with A. Einstein's theory of relativity, and basically they are supporters of the existence of the luminiferous ether. They believe that the influence of the ether was present at the experiments and that the relativists at one time ignored this fact.
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This analysis provides an alternative view, identifying the error of the Michelson and Morley experience, which may contribute to the reconsideration of the views of both parties.

Ii. Literature review

Authors R. Feynman, R. Leighton, M. Sands in their work provide data on the experience of Michelson and Morley. But, unfortunately, the authors have missed the mistake of adding speeds. Describing the path of the beam from a translucent mirror ( Figure 1 ) B to E and also from B to C in the form:

$$

$$ct_1 = L + ut_1, (ct_3) ^ 2 = L ^ 2 + (ut_3) ^ 2$$

They may not have noticed that these mathematical expressions do not fully cover the present velocities in the experiment. There is no speed of the Earth together with the interferometer. The speed of the Earth around the Sun, around 30 km / s, is known for certain, not counting the speed of the solar system around the center of the galaxy. Given this speed, the previous expressions should look like:

$$

$$(c + u) t_1 = L + ut_1, (c + u) ^ 2t_3 ^ 2 = L ^ 2 + (ut_3) ^ 2$$

The result of the experiment (the experimenters did not get the expected speed of the Earth relative to the “fixed” ether) baffled the researchers and the Hendrick Lorentz transformation is considered to be the way out. On the basis of the Lorentz transformation lies the idea that ... "all material bodies shrink when moving, only in the direction of movement".

One of the followers of the idea of ​​H. Lorenz, E. Rogers gives an example with a bird flying along and across the wind inside the cage. At the same time the role of the ether is air. But in the end comes to the result that there is no ether, which is true. In this case, the trajectory of the bird's flight is considered only relative to the cage, but in fact it was necessary in both cases to consider the coordinate system of the moving wind.

Another author, A. Pais, notes that "... this also applies to H. Lorentz, who found an error in the theory of the experiment of A. Michelson and with doubt treated the interpretation of his result. H. Lorentz’s distrust and Rayleigh’s urgent appeals forced Michelson to repeat his experience, this time in collaboration with Edward Williams Morley, a chemist from the nearby University of Western Riserv. Based on the idea of ​​the Potsdam Michelson experiment conducted in 1881, they built a new interferometer, the design of which focused on minimizing external influences.

In August 1887, Michelson wrote to Rayleigh [Lord Rayleigh or John William Strutt (John William Strutt)] that a negative result was again obtained, the influence of the ether could not be detected. An article about the Michelson-Morley experience was published in November of the same year ”[AMERICAN JOURNAL OF SCIENCE, november 1887“ On the Relative Motion of the Earth and the Luminiferous Ether. ”]

“It is understandable that the negative result upset not only the authors of the experiment, but also Kelvin [Lord Kelvin, (William Thomson), Rayleigh (John William Strutt) and H. Lorenz. And yet, no matter what, this experimental result was considered to be reliable. Hence, the error lies in the theory of Michelson and Morley's experience.

In 1892 Lorenz asked Rayleigh: “Could it not have happened that some moment in the theory of experience of Mr. Michelson was overlooked? In a lecture given on April 27, 1900 in the Royal Association, Kelvin referred to this experience as performed with the greatest possible care, ensuring the reliability of the result obtained.

B. Hoffman in his book “Albert Einstein the Creator and Rebel” suggests that Lorenz was close to resolving the issue before making a final decision, but for some reason did not bring it to a logical end. He writes “... under the influence of criticism and advice from Poincare, Lorenz made systematic efforts in an attempt to reconcile Maxwell's equations with the results of the Michelson-Morley experiment and other experiments that have already been set or have not yet been conceived. By 1904, after hard work, he basically found a mathematical solution to the problem ... Lorenz used, among other things, a reduction in length. However, he failed to fully preserve the form of Maxwell's equations. ".

Michelson and Morley assembled the setup with an interferometer, the device and its principle of operation are described in many textbooks. Accuracy of measurements and impeccability of accounting for all side effects were taken into account with special scrupulousness.


Picture 1

It is necessary to pay attention to the features of the theoretical part of the experience, aimed at processing the results of the experiment. The basis is the course of solving the problem and the scheme of experience from the literature.
Path parallel to the movement:

$$

$$ct_1 = L + ut_1 or t_1 = L / (c-u)$$

Way back

$$

$$ct_2 = L-Ut_2 or t_2 = L / (C + U)$$

The total time is:

$$

$$t_1 + t_2 = 2 * Lc / (c ^ 2 + u ^ 2) or t_1 + t_2 = (2L / c) / (1-u ^ 2 / c ^ 2); (1)$$

The path of the perpendicular beam:

$$

$$(ct_3) ^ 2 = L ^ 2 + (ut_2) ^ 2 or t_3 = L / √ (c ^ 2-u ^ 2)$$

Taking into account the movement back and forth

$$

$$2t_3 = 2L / √ (1-u ^ 2 / c ^ 2); (2)$$

Comparing formulas (1) and (2), Michelson and Morley believed that there would be a difference in time: $$$t_1 + t_2$which is less $$$2t_3$. It remains only to accurately measure this difference. But the experimenters could not get an interference pattern indicating the movement of the Earth through the ether.

As suggested by modern science, the experience of Michelson and Morley could not achieve its result due to the lack of ether and due to the reduction in the length of objects along the direction of motion. The proposed Lorentz transformation later came into science as a fundamental expression for relativistic calculations of mass, time, path, etc.

Iii. Theoretical part of the experience

1) Error in theory

Incorrectly used the principle of addition of speeds. Although the light source moves with the platform, the speed u (the speed of the light source system — device-Earth) relative to an imaginary “fixed” point (relative to the “ether”) is not taken into account. If the paper on which the scheme of the experiment is drawn (Fig. 1) is assumed to be stationary together with ether, then such a picture will turn out. This is a projection of experience on a certain plane, resting as well as ether. The experience was designed that way.
Pay attention to these mathematical expressions:

$$

$$ct_1 = L + ut_1 and (ct_3) ^ 2 = L ^ 2 + (ut_3) ^ 2$$

It seems that all speeds are taken into account - there are both c and u. But these expressions are not described by speed, but by the way. In fact, the right side of the equation is actually the length L of both arms of the interferometer and plus the path over time $$$t_1 and t_3$with speed u. The left part of the expressions say that the way $$$L + ut_1$and $$$L ^ 2 + (ut_3) ^ 2$passed by the beam in time $$$t_1 and t_3$accordingly with speed $$$c$.

This means that Michelson and Morley initially accepted the speed of light, independent of the speed of the object, emitting a ray.

Since the light source moves with the platform, we must add the speed of the system u to the speed of light , otherwise it will be impossible to explain a certain inclination of the beam in the perpendicular direction. This is the error of experience that has led modern science to delusion.
The path of the beam is parallel to the movement:

$$

$$(c + u) t_1 = L + ut_1 or t_1 = L / (c + u-u) = L / c$$

Reverse path:

$$

$$(c-u) t_2 = L-ut_2 or t_2 = L / (c-u + u) = L / c$$

time spent by the beam back and forth:

$$

$$t_1 + t_2 = 2 * L / c; (3)$$

The time of passage of the beam in the perpendicular direction:
Figure 1 shows a diagram of the beam in the perpendicular direction.
It should be noted here that Michelson and Morley mistakenly took the hypotenuse of a right-angled triangle for the path of the ray at a speed c. In fact, it is an extension of the path and this path is described by the expression:

$$

$$L ^ 2 + u ^ 2 t_3 ^ 2 = c_1 ^ 2 t3 ^ 2; (4)$$

here, for convenience, use the expression:

$$

$$c1 ^ 2 t_3 ^ 2 = c ^ 2 t_3 ^ 2 + u ^ 2 t_3 ^ 2 (5)$$

Where

$$

$$c_1> c$$

And also shoulder length $$$L = ct_3$
If we squared the sum of two speeds $$$(c ^ 2 t_3 ^ 2 + u ^ 2 t_3 ^ 2) ^ 2$would be confused, in fact, the physical essence of this expression is a specific path, measured in our case in meters, so the value $$$c_1$.


Figure 2

Inserting the value from formula (5) into expression (4) we get:

$$

$$L ^ 2 + u ^ 2 t_3 ^ 2 = c ^ 2 t_3 ^ 2 + u ^ 2 t_3 ^ 2$$

Further: $$$L ^ 2 = c ^ 2 t_3 ^ 2$; $$$L ^ 2 = c ^ 2$; $$$t_3 = L / c$

$$

$$L = c t_3; t_3 = L / c; L = c t_3; t_3 = L / c$$

Then back and forth:

$$

$$2 t_3 = 2L / c; (6)$$

Compare formulas (3) and (6).

There is no difference in time, which was proved by the experience of Michelson and Morley.

2) Determination of the dependence of the speed of light on the movement of the source

The difference is traversed paths. In the perpendicular direction, the path of the beam is longer than in the parallel direction.

In order to determine which value the path in the perpendicular direction is greater, we turn to the calculations.

The time of passage of the rays in the perpendicular and parallel directions are the same, see (3) and (6).

Because $$$L = c t_3;$we can find the difference of the way in the form:

$$

$$L * = 2 c_1t_3-2ct_3;$$

Figure 3

Analyzing the design of the interferometer, I came to the conclusion that whatever the relative velocity, velocity of the beam or other object sent to the mirrors, the proportion of the circuit will not change. The rays split by a translucent mirror reach the mirrors of the interferometer arms and back simultaneously. The relative motion of the instrument-Earth system, (so far, at least relative to the Sun) which we cannot deny, only extends the path of the rays in parallel and perpendicular directions. Moreover, the path of the beam in the perpendicular direction is much larger than the path of the beam in a parallel direction. However, this does not affect the achievement of the rays of the translucent mirror at the same time, as evidenced by the absence of any noticeable interference rings. This means that the speed of light depends on the movement of an object emitting a ray. Figure 3 illustrates the graphical addition of the paths of the beam, which clearly demonstrates that the path of the beam in the perpendicular direction $$$2c_1t_3$more from $$$2 * ct_3 = 2 * L$by value $$$L *$.

This means that the device was initially unable to determine the speed of the Earth through space.

Michelson and Morley in their time did not pay attention to the fact that they initially accepted the speed of light regardless of whether the object is moving or not, and as a result they got what they started from.

Some authors, for example, E. Rogers and R. Feynman, R. Leighton, M. Sands mention aberration, believing that the tilt of the beam in the perpendicular direction is caused precisely by this phenomenon. But to compensate for the aberration, telescopes and other beam receivers were tilted, and the mirrors of the Michelson and Morley interferometers were perfectly perpendicular.

Iv. findings

1. The mathematical apparatus of the theory of experience of Michelson and Morley is erroneous, the principle of velocity addition is incorrectly used.
2. The experience of Michelson and Morley is a confirmation of the absence of ether, but at the same time he proved that the speed of light depends on the movement of the source. Moreover, using the example of the same experiment, it was proved that the speed of the beam in the perpendicular direction actually exceeded the speed of light relative to the coordinate system adopted by the authors by:

$$

$$c * = L * / t_3$$

3. Let us give two postulates of Einstein:
1) No experiment can detect absolute peace or uniform motion.
2) Regardless of the source movement, the light always moves through empty space at the same speed. $$$with$.

The first postulate is confirmed by the experience of Michelson and Morley. The second postulate is not true, since the path of light in the vertical direction is greater than in the horizontal direction. In this regard, it is quite understandable to increase the speed of light in the vertical section.

V. Conclusion

In this paper, I conducted an analysis of the theoretical basis of the Michelson and Morley experience. Comparing the formulas of scientists, I found that the main idea of ​​the experiment to determine the velocity of the Earth was erroneous because of the incorrect formulation of the problem of the theoretical part. Knowing for sure about the speed of the Earth at least around the Sun (30 km / s), as indicated above in the expressions $$$ct_1 = L + ut_1, (ct_3) ^ 2 = L ^ 2 + (ut_3) ^ 2$Earth velocity u is used only to determine the distance during time t, the principle of velocity addition is absent. And also the hypotenuse of a right triangle is not accepted as the resultant velocity of the velocity components. $$$(u + c) = c_1$as well as the speed of light c.

Using the principle of velocity addition correctly, I came to the conclusion that the interpretation of the result of Michelson and Morley’s experiment has hitherto been incorrect due to an annoying error in formulating the problem of the theoretical part, that in fact the speed of light depends on the speed of the object emitting light. Ether in no way affects the speed of light, which indicates its absence.

The author of this article is Zhunusov Zhakash Ilyasovich