# Mathematical solution to relativity problems

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

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

It must be recognized that not all researchers agree with the theory of relativity by A. Einstein, and basically they are supporters of the existence of the luminiferous ether. They believe that, nevertheless, the influence of ether was present in the experiments and that relativists ignored this fact at one time.

This analysis provides an alternative view, revealing the error of Michelson and Morley experience, which may contribute to a review of the views of both parties.

The authors R. Feynman, R. Leighton, M. Sands in their work provide data on the experience of Michelson and Morley. But, unfortunately, the authors overlooked the error in the addition of speeds. Describing the path of the beam from a translucent mirror (

Perhaps they did not notice that these mathematical expressions do not completely cover the present speeds in the experiment. There is no Earth velocity with the interferometer. The speed of the Earth around the Sun, about 30 km / s, is reliably known, not counting the speed of the solar system around the center of the galaxy. Given this speed, the previous expressions should look like:

The result of the experiment (the experimenters did not receive the expected speed of the Earth relative to the “immobile" ether) confused the researchers and consider the Hendrick Lorenz transformation to be a way out. On the basis of the Lorentz transformation lies the idea that ... "all material bodies contract during movement, only in the direction of motion."

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. In this case, air plays the role of ether. But in the end, it comes to the result that there is no ether, which is true. In this case, the flight path of the bird is considered only relative to the cage, but in fact it was necessary in both cases to be considered in the coordinate system of the moving wind.

Another author A. Pais notes that "... the same applies to H. Lorentz, who found a mistake in the theory of A. Michelson’s experiment and questioned the interpretation of his result. H. Lorenz’s distrust and Rayleigh’s urges urged Michelson to repeat his experience, this time in collaboration with Edward Williams Morley, a chemist from nearby Western Riverserv University. Based on the idea of the Potsdam Michelson experiment conducted in 1881, they built a new interferometer, the design of which paid special attention to minimizing external influences.

In August 1887, Michelson wrote to Rayleigh [Lord Rayleigh or 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 experiment was published in November of that year ”[AMERICAN JOURNAL OF SCIENCE, november 1887“ On the Relative Motion of the Earth and the Luminiferous Ether. ”]

“ It is clear that a negative result upset not only the authors of the experiment, but and Kelvin [Lord Kelvin, (William Thomson), Rayleigh (John William Strutt) and H. Lorenz. And yet, in spite of everything, this experimental result was considered reliable. Therefore, the error lies in the theory of the experiment of Michelson and Morley.

In 1892 Lorenz asked Rayleigh: “Could it have happened that some moment in Mr. Michelson’s theory of experience was overlooked? In a lecture given by the Royal Association on April 27, 1900, Kelvin referred to this experiment as being carried out with the utmost care, ensuring the reliability of the result.

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

Michelson and Morley assembled the installation with an interferometer, the device and the principle of its operation are described in many textbooks. The accuracy of measurements and the impeccability of taking into account all side effects were taken into account with particular scrupulousness.

It is necessary to pay attention to the features of the theoretical part of the experiment, aimed at processing the results of the experiment. It is based on the course of solving the problem and the scheme of experience from the literature.

Path parallel to movement:

Way back

The total time is:

Perpendicular Ray Path:

Given round-trip traffic

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

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

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 - instrument-Earth system) relative to an imaginary “fixed” point (relative to the “ether”) is not taken into account. If the paper on which the experimental diagram is drawn (Fig. 1) is considered motionless together with the ether, then this is exactly the picture. This is a projection of experience on a certain plane, resting also like ether. Experience was designed that way.

Pay attention to these mathematical expressions:

It seems that all speeds are taken into account - there are both s and u. But these expressions are described not by speed, but by 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$$with speed u. The left side of the expressions suggests that the way$$ and $$ passed by the beam in time $$ accordingly with speed $$.

This means that Michelson and Morley initially assumed that the speed of light is independent of the speed of the object emitting the beam.

Beam path parallel to movement:

The way back:

the time spent by the beam back and forth:

The transit time of the beam in the perpendicular direction:

In

It should be noted here that Michelson and Morley mistakenly took the hypotenuse of a right-angled triangle for the path of the beam with speed c. In fact, it is an extension of the path and this path is described by the expression:

here for convenience we used the expression:

Where

As well as shoulder length $$

If we squared the sum of two speeds $$ would be confused, in fact, the physical essence of this expression is a concrete path, measured in our case in meters, therefore the quantity $$.

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

Further: $$; $$; $$

Then there and back:

Compare formulas (3) and (6).

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

There is a difference in the paths traveled. In the perpendicular direction, the beam path is longer than in the parallel direction.

In order to determine by what magnitude the path in the perpendicular direction is greater, we turn to the calculations.

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

Because$$ we can find the path difference in the form:

Analyzing the design of the interferometer, I came to the conclusion that whatever the relative speed, speed 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 mirror of the shoulders of the interferometer and vice versa at the same time. The relative motion of the device-Earth system, (so far, at least with respect 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 the 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. Hence, the speed of light depends on the movement of the object emitting the beam. In fig.$$more from $$ by the amount $$.

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

Michelson and Morley at one time did not pay attention to the fact that they initially accepted the speed of light independent of whether the object was moving or not, and as a result got what they started from.

Some authors, such as 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 tilted, and the mirrors of the Michelson and Morley interferometer were perfectly perpendicular.

1. The mathematical apparatus of the theory of experiment of Michelson and Morley is erroneous, the principle of addition of velocities is incorrectly used.

2. The experience of Michelson and Morley confirms 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 the amount:

3. Here are two postulates of Einstein:

1) No experiment can detect absolute peace or uniform motion.

2) Regardless of the movement of the source, light always moves through empty space at the same speed$$.

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. In this regard, an increase in the speed of light in the vertical section is understandable.

In this work, I analyzed the theoretical basis of the Michelson and Morley experiment. Comparing the formulas of scientists, I found that the main idea of the experiment to determine the speed of the Earth was erroneous due to the incorrect statement 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$$Earth’s speed u is used only to determine the distance in time t; there is no principle of addition of speeds. And also the hypotenuse of a right-angled triangle is not accepted as the resulting velocity of the velocity components$$, but 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 the Michelson and Morley experiment has so far been incorrect due to an unfortunate error in the formulation of 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.

## I. Introduction

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

It must be recognized that not all researchers agree with the theory of relativity by A. Einstein, and basically they are supporters of the existence of the luminiferous ether. They believe that, nevertheless, the influence of ether was present in the experiments and that relativists ignored this fact at one time.

This analysis provides an alternative view, revealing the error of Michelson and Morley experience, which may contribute to a review of the views of both parties.

## II. Literature review

The authors R. Feynman, R. Leighton, M. Sands in their work provide data on the experience of Michelson and Morley. But, unfortunately, the authors overlooked the error in the addition of 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:$$

Perhaps they did not notice that these mathematical expressions do not completely cover the present speeds in the experiment. There is no Earth velocity with the interferometer. The speed of the Earth around the Sun, about 30 km / s, is reliably known, not counting the speed of the solar system around the center of the galaxy. Given this speed, the previous expressions should look like:

$$

The result of the experiment (the experimenters did not receive the expected speed of the Earth relative to the “immobile" ether) confused the researchers and consider the Hendrick Lorenz transformation to be a way out. On the basis of the Lorentz transformation lies the idea that ... "all material bodies contract during movement, only in the direction of motion."

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. In this case, air plays the role of ether. But in the end, it comes to the result that there is no ether, which is true. In this case, the flight path of the bird is considered only relative to the cage, but in fact it was necessary in both cases to be considered in the coordinate system of the moving wind.

Another author A. Pais notes that "... the same applies to H. Lorentz, who found a mistake in the theory of A. Michelson’s experiment and questioned the interpretation of his result. H. Lorenz’s distrust and Rayleigh’s urges urged Michelson to repeat his experience, this time in collaboration with Edward Williams Morley, a chemist from nearby Western Riverserv University. Based on the idea of the Potsdam Michelson experiment conducted in 1881, they built a new interferometer, the design of which paid special attention to minimizing external influences.

In August 1887, Michelson wrote to Rayleigh [Lord Rayleigh or 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 experiment was published in November of that year ”[AMERICAN JOURNAL OF SCIENCE, november 1887“ On the Relative Motion of the Earth and the Luminiferous Ether. ”]

“ It is clear that a negative result upset not only the authors of the experiment, but and Kelvin [Lord Kelvin, (William Thomson), Rayleigh (John William Strutt) and H. Lorenz. And yet, in spite of everything, this experimental result was considered reliable. Therefore, the error lies in the theory of the experiment of Michelson and Morley.

In 1892 Lorenz asked Rayleigh: “Could it have happened that some moment in Mr. Michelson’s theory of experience was overlooked? In a lecture given by the Royal Association on April 27, 1900, Kelvin referred to this experiment as being carried out with the utmost care, ensuring the reliability of the result.

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

Michelson and Morley assembled the installation with an interferometer, the device and the principle of its operation are described in many textbooks. The accuracy of measurements and the impeccability of taking into account all side effects were taken into account with particular scrupulousness.

*Figure 1*It is necessary to pay attention to the features of the theoretical part of the experiment, aimed at processing the results of the experiment. It is based on the course of solving the problem and the scheme of experience from the literature.

Path parallel to movement:

$$

Way back

$$

The total time is:

$$

Perpendicular Ray Path:

$$

Given round-trip traffic

$$

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

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

## III. The theoretical part of the experiment

### 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 - instrument-Earth system) relative to an imaginary “fixed” point (relative to the “ether”) is not taken into account. If the paper on which the experimental diagram is drawn (Fig. 1) is considered motionless together with the ether, then this is exactly the picture. This is a projection of experience on a certain plane, resting also like ether. Experience was designed that way.

Pay attention to these mathematical expressions:

$$

It seems that all speeds are taken into account - there are both s and u. But these expressions are described not by speed, but by 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$$with speed u. The left side of the expressions suggests that the way$$ and $$ passed by the beam in time $$ accordingly with speed $$.

This means that Michelson and Morley initially assumed that the speed of light is independent of the speed of the object emitting the beam.

**Since the light source moves with the platform, we must add the speed of the system u to the speed of light c, otherwise it will not be possible to explain a certain slope of the beam in the perpendicular direction.**This is the error of experience, which has led modern science to error.Beam path parallel to movement:

$$

The way back:

$$

the time spent by the beam back and forth:

$$

The transit time of the beam in the perpendicular direction:

In

*Figure 1*shows the ray path diagram 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 beam with speed c. In fact, it is an extension of the path and this path is described by the expression:

$$

here for convenience we used the expression:

$$

Where

$$

As well as shoulder length $$

If we squared the sum of two speeds $$ would be confused, in fact, the physical essence of this expression is a concrete path, measured in our case in meters, therefore the quantity $$.

*Figure 2*Inserting the value from formula (5) into expression (4) we obtain:

$$

Further: $$; $$; $$

$$

Then there and back:

$$

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

There is a difference in the paths traveled. In the perpendicular direction, the beam path is longer than in the parallel direction.

In order to determine by what magnitude the path in the perpendicular direction is greater, we turn to the calculations.

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

Because$$ we can find the path difference in the form:

$$

*Figure 3*Analyzing the design of the interferometer, I came to the conclusion that whatever the relative speed, speed 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 mirror of the shoulders of the interferometer and vice versa at the same time. The relative motion of the device-Earth system, (so far, at least with respect 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 the 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. Hence, the speed of light depends on the movement of the object emitting the beam. In fig.$$more from $$ by the amount $$.

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

Michelson and Morley at one time did not pay attention to the fact that they initially accepted the speed of light independent of whether the object was moving or not, and as a result got what they started from.

Some authors, such as 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 tilted, and the mirrors of the Michelson and Morley interferometer were perfectly perpendicular.

## IV. conclusions

1. The mathematical apparatus of the theory of experiment of Michelson and Morley is erroneous, the principle of addition of velocities is incorrectly used.

2. The experience of Michelson and Morley confirms 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 the amount:

$$

3. Here are two postulates of Einstein:

1) No experiment can detect absolute peace or uniform motion.

2) Regardless of the movement of the source, light always moves through empty space at the same speed$$.

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. In this regard, an increase in the speed of light in the vertical section is understandable.

## V. Conclusion

In this work, I analyzed the theoretical basis of the Michelson and Morley experiment. Comparing the formulas of scientists, I found that the main idea of the experiment to determine the speed of the Earth was erroneous due to the incorrect statement 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$$Earth’s speed u is used only to determine the distance in time t; there is no principle of addition of speeds. And also the hypotenuse of a right-angled triangle is not accepted as the resulting velocity of the velocity components$$, but 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 the Michelson and Morley experiment has so far been incorrect due to an unfortunate error in the formulation of 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