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Reference: Evolution of Physics

This paper presents Chapter III, section 6 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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Ether and Motion

The Galilean relativity principle is valid for mechanical phenomena. The same laws of mechanics apply to all inertial systems moving relative to each other. Is this principle also valid for non-mechanical phenomena, especially for those for which the field concepts proved so very important? All problems concentrated around this question immediately bring us to the starting-point of the relativity theory.

The Galilean relativity principle applies to a narrow range of velocities of matter. In this range, the velocities are small and inertia is very large (compared to light). Change in inertia corresponding to change in velocity is virtually undetectable. The laws of mechanics are based on this invariability of inertia (mass). All inertial frames exist in this narrow range of velocities of matter.

We remember that the velocity of light in vacuo, or in other words, in ether, is 186,000 miles per second and that light is an electromagnetic wave spreading through the ether. The electromagnetic field carries energy which, once emitted from its source, leads an independent existence. For the time being, we shall continue to believe that the ether is a medium through which electromagnetic waves, and thus also light waves, are propagated, even though we are fully aware of the many difficulties connected with its mechanical structure.

The principle of field applies to a broad spectrum of electromagnetic waves. In this spectrum, the inertia (represented by frequencies) is extremely small, but the velocity is very large. Change in velocity corresponding to change in inertia (frequency) is virtually undetectable. The electromagnetic field carries energy which, once emitted from its source, leads an independent existence.

We are sitting in a closed room so isolated from the external world that no air can enter or escape. If we sit still and talk we are, from the physical point of view, creating sound waves, which spread from their resting source with the velocity of sound in air. If there were no air or other material medium between the mouth and the ear, we could not detect a sound. Experiment has shown that the velocity of sound in air is the same in all directions, if there is no wind and the air is at rest in the chosen c.s.

Let us now imagine that our room moves uniformly through space. A man outside sees, through the glass walls of the moving room (or train if you prefer), everything which is going on inside. From the measurements of the inside observer he can deduce the velocity of sound relative to his c.s. connected with his surroundings, relative to which the room moves. Here again is the old, much discussed, problem of determining the velocity in one c.s. if it is already known in another.

The observer in the room claims: the velocity of sound is, for me, the same in all directions.

The outside observer claims: the velocity of sound, spreading in the moving room and determined in my c.s., is not the same in all directions. It is greater than the standard velocity of sound in the direction of the motion of the room and smaller in the opposite direction.

These conclusions are drawn from the classical transformation and can be confirmed by experiment. The room carries within it the material medium, the air through which sound waves are propagated, and the velocities of sound will, therefore, be different for the inside and outside observer.

We can draw some further conclusions from the theory of sound as a wave propagated through a material medium. One way, though by no means the simplest, of not hearing what someone is saying, is to run, with a velocity greater than that of sound, relative to the air surrounding the speaker. The sound waves produced will then never be able to reach our ears. On the other hand, if we missed an important word which will never be repeated, we must run with a speed greater than that of sound to reach the produced wave and to catch the word. There is nothing irrational in either of these examples except that in both cases we should have to run with a speed of about four hundred yards per second, and we can very well imagine that further technical development will make such speeds possible. A bullet fired from a gun actually moves with a speed greater than that of sound and a man placed on such a bullet would never hear the sound of the shot.

Sound lies in the narrow range of velocities of matter where inertia is very large. Because of high inertia, the wind is substantial. Sound is carried with the wind, so its velocity is higher in the direction of the wind.

All these examples are of a purely mechanical character and we can now formulate the important questions: could we repeat what has just been said of a sound wave, in the case of a light wave? Do the Galilean relativity principle and the classical transformation apply to optical and electrical phenomena as well as to mechanical? It would be risky to answer these questions by a simple “yes” or “no” without going more deeply into their meaning.

In the case of the sound wave in the room moving uniformly, relative to the outside observer, the following intermediate steps are very essential for our conclusion:

1. The moving room carries the air in which the sound wave is propagated.
2. The velocities observed in two c.s. moving uniformly, relative to each other, are connected by the classical transformation.

The corresponding problem for light must be formulated a little differently. The observers in the room are no longer talking, but are sending light signals, or light waves in every direction. Let us further assume that the sources emitting the light signals are permanently resting in the room. The light waves move through the ether just as the sound waves moved through the air.

Is the ether carried with the room as the air was? Since we have no mechanical picture of the ether, it is extremely difficult to answer this question. If the room is closed, the air inside is forced to move with it. There is obviously no sense in thinking of ether in this way, since all matter is immersed in it and it penetrates everywhere. No doors are closed to ether. The “moving room” now means only a moving c.s. to which the source of light is rigidly connected. It is, however, not beyond us to imagine that the room moving with its light source carries the ether along with it just as the sound source and air were carried along in the closed room. But we can equally well imagine the opposite: that the room travels through the ether as a ship through a perfectly smooth sea, not carrying any part of the medium along but moving through it. In our first picture, the room moving with its light source carries the ether. An analogy with a sound wave is possible and quite similar conclusions can be drawn. In the second, the room moving with its light source does not carry the ether. No analogy with a sound wave is possible and the conclusions drawn in the case of a sound wave do not hold for a light wave. These are the two limiting possibilities. We could imagine the still more complicated possibility that the ether is only partially carried by the room moving with its light source. But there is no reason to discuss the more complicated assumptions before finding out which of the two simpler limiting cases experiment favours.

We shall begin with our first picture and assume, for the present: the ether is carried along by the room moving with its rigidly connected light source. If we believe in the simple transformation principle for the velocities of sound waves, we can now apply our conclusions to light waves as well. There is no reason for doubting the simple mechanical transformation law which only states that the velocities have to be added in certain cases and subtracted in others. For the moment, therefore, we shall assume both the carrying of the ether by the room moving with its light source and the classical transformation.

If I turn on the light and its source is rigidly connected with my room, then the velocity of the light signal has the well-known experimental value 186,000 miles per second. But the outside observer will notice the motion of the room, and, therefore, that of the source—and, since the ether is carried along, his conclusion must be: the velocity of light in my outside c.s. is different in different directions. It is greater than the standard velocity of light in the direction of the motion of the room and smaller in the opposite direction. Our conclusion is: if ether is carried with the room moving with its light source and if the mechanical laws are valid, then the velocity of light must depend on the velocity of the light source. Light reaching our eyes from a moving light source would have a greater velocity if the motion is toward us and smaller if it is away from us.

If our speed were greater than that of light, we should be able to run away from a light signal. We could see occurrences from the past by reaching previously sent light waves. We should catch them in a reverse order to that in which they were sent, and the train of happenings on our earth would appear like a film shown backward, beginning with a happy ending. These conclusions all follow from the assumption that the moving c.s. carries along the ether and the mechanical transformation laws are valid. If this is so, the analogy between light and sound is perfect.

But there is no indication as to the truth of these conclusions. On the contrary, they are contradicted by all observations made with the intention of proving them. There is not the slightest doubt as to the clarity of this verdict, although it is obtained through rather indirect experiments in view of the great technical difficulties caused by the enormous value of the velocity of light. The velocity of light is always the same in all c.s. independent of whether or not the emitting source moves, or how it moves.

Light lies in the broad spectrum of electromagnetic waves where inertia is extremely small. The “wind” due to moving light source is not substantial at all, and the velocity is extremely large. Therefore, no change in velocity is detected.

We shall not go into detailed description of the many experiments from which this important conclusion can be drawn. We can, however, use some very simple arguments which, though they do not prove that the velocity of light is independent of the motion of the source, nevertheless make this fact convincing and understandable.

In our planetary system the earth and other planets move around the sun. We do not know of the existence of other planetary systems similar to ours. There are, however, very many double-star systems, consisting of two stars moving around a point, called their centre of gravity. Observation of the motion of these double stars reveals the validity of Newton’s gravitational law. Now suppose that the speed of light depends on the velocity of the emitting body. Then the message, that is, the light ray from the star, will travel more quickly or more slowly, according to the velocity of the star at the moment the ray is emitted. In this case the whole motion would be muddled and it would be impossible to confirm, in the case of distant double stars, the validity of the same gravitational law which rules over our planetary system.

Let us consider another experiment based upon a very simple idea. Imagine a wheel rotating very quickly. According to our assumption, the ether is carried by the motion and takes a part in it. A light wave passing near the wheel would have a different speed when the wheel is at rest than when it is in motion. The velocity of light in ether at rest should differ from that in ether which is being quickly dragged round by the motion of the wheel, just as the velocity of a sound wave varies on calm and windy days. But no such difference is detected! No matter from which angle we approach the subject, or what crucial experiment we may devise, the verdict is always against the assumption of the ether carried by motion. Thus, the result of our considerations, supported by more detailed and technical argument, is:

1. The velocity of light does not depend on the motion of the emitting source.
2. It must not be assumed that the moving body carries the surrounding ether along.

We must, therefore, give up the analogy between sound and light waves and turn to the second possibility: that all matter moves through the ether, which takes no part whatever in the motion. This means that we assume the existence of a sea of ether with all c.s. resting in it, or moving relative to it. Suppose we leave, for a while, the question as to whether experiment proved or disproved this theory. It will be better to become more familiar with the meaning of this new assumption and with the conclusions which can be drawn from it.

The analogy between sound and light in terms of wind and velocity does not exist. If there is any analogy, it is in terms of a balance between inertia and velocity.

There exists a c.s. resting relative to the ether-sea. In mechanics, not one of the many c.s. moving uniformly, relative to each other, could be distinguished. All such c.s. were equally “good” or “bad”. If we have two c.s. moving uniformly, relative to each other, it is meaningless, in mechanics, to ask which of them is in motion and which at rest. Only relative uniform motion can be observed. We cannot talk about absolute uniform motion because of the Galilean relativity principle. What is meant by the statement that absolute and not only relative uniform motion exists? Simply that there exists one c.s. in which some of the laws of nature are different from those in all others. Also that every observer can detect whether his c.s. is at rest or in motion by comparing the laws valid in it with those valid in the only one which has the absolute monopoly of serving as the standard c.s. Here is a different state of affairs from classical mechanics, where absolute uniform motion is quite meaningless because of Galileo’s law of inertia.

What conclusions can be drawn in the domain of field phenomena if motion through ether is assumed? This would mean that there exists one c.s. distinct from all others, at rest relative to the ether-sea. It is quite clear that some of the laws of nature must be different in this c.s., otherwise the phrase “motion through ether” would be meaningless. If the Galilean relativity principle is valid, then motion through ether makes no sense at all. It is impossible to reconcile these two ideas. If, however, there exists one special c.s. fixed by the ether, then to speak of “absolute motion” or “absolute rest” has a definite meaning.

We really have no choice. We tried to save the Galilean relativity principle by assuming that systems carry the ether along in their motion, but this led to a contradiction with experiment. The only way out is to abandon the Galilean relativity principle and try out the assumption that all bodies move through the calm ether-sea.

The laws of mechanics based on Galileo’s Relativity principle apply to a very small range of velocities for near constant inertia. Absolute velocities do not exist because the condition of absolute rest is not recognized in this range. But in the wider range, there can be absolute rest for a body of infinite inertia. An example would be a black hole at the center of the galaxy.

The next step is to consider some conclusions contradicting the Galilean relativity principle and supporting the view of motion through ether, and to put them to the test of an experiment. Such experiments are easy enough to imagine, but very difficult to perform. As we are concerned here only with ideas, we need not bother with technical difficulties.

Again we return to our moving room with two observers, one inside and one outside. The outside observer will represent the standard c.s., designated by the ether-sea. It is the distinguished c.s. in which the velocity of light always has the same standard value. All light sources, whether moving or at rest in the calm ether-sea, propagate light with the same velocity. The room and its observer move through the ether. Imagine that a light in the centre of the room is flashed on and off and, furthermore, that the walls of the room are transparent so that the observers, both inside and outside, can measure the velocity of the light. If we ask the two observers what results they expect to obtain, their answers would run something like this:

The outside observer: My c.s. is designated by the ether-sea. Light in my c.s. always has the standard value. I need not care whether or not the source of light or other bodies are moving, for they never carry my ether-sea with them. My c.s. is distinguished from all others and the velocity of light must have its standard value in this c.s., independent of the direction of the light beam or the motion of its source.

The inside observer: My room moves through the ether-sea. One of the walls runs away from the light and the other approaches it. If my room travelled, relative to the ether-sea, with the velocity of light, then the light emitted from the centre of the room would never reach the wall running away with the velocity of light. If the room travelled with a velocity smaller than that of light, then a wave sent from the centre of the room would reach one of the walls before the other. The wall moving toward the light wave would be reached before the one retreating from the light wave. Therefore, although the source of light is rigidly connected with my C.S., the velocity of light will not be the same in all directions. It will be smaller in the direction of the motion relative to the ether-sea as the wall runs away, and greater in the opposite direction as the wall moves toward the wave and tries to meet it sooner.

Thus, only in the one c.s. distinguished by the ether-sea should the velocity of light be equal in all directions. For other c.s. moving relatively to the ether-sea it should depend on the direction in which we are measuring.

The crucial experiment just considered enables us to test the theory of motion through the ether-sea. Nature, in fact, places at our disposal a system moving with a fairly high velocity: the earth in its yearly motion around the sun. If our assumption is correct, then the velocity of light in the direction of the motion of the earth should differ from the velocity of light in an opposite direction. The differences can be calculated and a suitable experimental test devised. In view of the small time-differences following from the theory, very ingenious experimental arrangements have to be thought out. This was done in the famous Michelson-Morley experiment. The result was a verdict of “death” to the theory of a calm ether-sea through which all matter moves. No dependence of the speed of light upon direction could be found. Not only the speed of light, but also other field phenomena would show a dependence on the direction in the moving c.s., if the theory of the ether-sea were assumed. Every experiment has given the same negative result as the Michelson-Morley one, and never revealed any dependence upon the direction of the motion of the earth.

The situation grows more and more serious. Two assumptions have been tried. The first, that moving bodies carry ether along. The fact that the velocity of light does not depend on the motion of the source contradicts this assumption. The second, that there exists one distinguished c.s. and that moving bodies do not carry the ether but travel through an ever calm ether-sea. If this is so, then the Galilean relativity principle is not valid and the speed of light cannot be the same in every c.s. Again we are in contradiction with experiment.

Michelson-Morley’s experiment simply demonstrated that the motion of Earth through space does not affect the speed of visible light. This result may be interpreted in different ways. But it consistent with almost zero inertia of light and aether. Light is a slightly condensed form of aether. Matter is an extremely condensed form of aether.

More artificial theories have been tried out, assuming that the real truth lies somewhere between these two limiting cases: that the ether is only partially carried by the moving bodies. But they all failed! Every attempt to explain the electromagnetic phenomena in moving c.s. with the help of the motion of the ether, motion through the ether, or both these motions, proved unsuccessful.

Thus arose one of the most dramatic situations in the history of science. All assumptions concerning ether led nowhere! The experimental verdict was always negative. Looking back over the development of physics we see that the ether, soon after its birth, became the enfant terrible of the family of physical substances. First, the construction of a simple mechanical picture of the ether proved to be impossible and was discarded. This caused, to a great extent, the breakdown of the mechanical point of view. Second, we had to give up hope that through the presence of the ether-sea one c.s. would be distinguished and lead to the recognition of absolute, and not only relative, motion. This would have been the only way, besides carrying the waves, in which ether could mark and justify its existence. All our attempts to make ether real failed. It revealed neither its mechanical construction nor absolute motion. Nothing remained of all the properties of the ether except that for which it was invented, i.e. its ability to transmit electromagnetic waves. Our attempts to discover the properties of the ether led to difficulties and contradictions. After such bad experiences, this is the moment to forget the ether completely and to try never to mention its name. We shall say: our space has the physical property of transmitting waves, and so omit the use of a word we have decided to avoid.

According to Einstein, aether is an invented concept. All we can say is that our space has the physical property of transmitting waves. Einstein then expresses the idea of space mathematically only.

The omission of a word from our vocabulary is, of course, no remedy. Our troubles are indeed much too profound to be solved in this way!

Let us now write down the facts which have been sufficiently confirmed by experiment without bothering any more about the “e——- r ” problem.

(1) The velocity of light in empty space always has its standard value, independent of the motion of the source or receiver of light.

(2) In two c.s. moving uniformly, relative to each other, all laws of nature are exactly identical and there is no way of distinguishing absolute uniform motion.

There are many experiments to confirm these two statements and not a single one to contradict either of them. The first statement expresses the constant character of the velocity of light, the second generalizes the Galilean relativity principle, formulated for mechanical phenomena, to all happenings in nature.

In mechanics, we have seen: If the velocity of a material point is so and so, relative to one c.s., then it will be different in another c.s. moving uniformly, relative to the first. This follows from the simple mechanical transformation principles. They are immediately given by our intuition (man moving relative to ship and shore) and apparently nothing can be wrong here! But this transformation law is in contradiction to the constant character of the velocity of light. Or, in other words, we add a third principle:

(3) Positions and velocities are transformed from one inertial system to another according to the classical transformation.

The contradiction is then evident. We cannot combine (1), (2), and (3).

The classical transformation seems too obvious and simple for any attempt to change it. We have already tried to change (1) and (2) and came to a disagreement with experiment. All theories concerning the motion of “e——- r” required an alteration of (1) and (2). This was no good. Once more we realize the serious character of our difficulties. A new clue is needed. It is supplied by accepting the fundamental assumptions (1) and (2), and, strange though it seems, giving up (3). The new clue starts from an analysis of the most fundamental and primitive concepts; we shall show how this analysis forces us to change our old views and removes all our difficulties.

Einstein’s assumptions may be criticized as follows: (1) Light is a substance of almost zero inertia; whereas, the source of light is considered to be a material object of extremely high inertia. The velocity of light is so high that variations in it due to changes in frequency cannot be detected in the framework of matter. (2) The idea of c.s. is limited to a narrow range of velocities for which inertia is so large that changes in it corresponding to changes in velocity cannot be detected. Based on inertia we can say that the velocity of light intuitively applies to “sunlight relative to the sun” and not to “sun relative to the sunlight.” (3) The laws of classical mechanics are a very small subset of the laws of nature under conditions as described above. The classical viewpoint does not consider the balance in nature between velocity and inertia.

Velocities can be differentiated as being greater or lesser on an absolute basis by comparing the inertia of the particles.

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Final Comment

According to the Galilean relativity principle of mechanical phenomena, it is impossible to say which particle is at rest relative to the other. But we have no such difficulty in intuitively recognizing that the Sun is at rest relative to the sunlight and not the opposite. This intuition has its basis in the concept of inertia.

The problem with the laws of classical mechanics is that they are limited to the assumption that the only substance is matter that has a fixed level of inertia (mass). Classical mechanics does not recognize that from matter to aether, there is a spectrum of substance with a gradation of inertia, and that the relative velocity is a manifestation of that inertia. The lesser is the inertia, the greater is the inherent velocity of substance.

Science assumes that all material systems have the same inertia (mass) for different velocities. It then makes another assumption that all electromagnetic systems have the same velocity for different levels of inertia (frequencies). Both assumptions are approximations for two extremes. Einstein’s theory predicts velocities for situations close to these extremes by extrapolating between these two extremes. But that extrapolation does not predict velocities correctly for situations farther away from these extremes.

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Reference: Evolution of Physics

This paper presents Chapter III, section 5 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

.

The Mechanical Scaffold

On reaching this stage of our story, we must turn back to the beginning, to Galileo’s law of inertia. We quote once more:

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Once the idea of inertia is understood, one wonders what more can be said about it. Although this problem has already been thoroughly discussed, it is by no means exhausted.

A factor that has been missing from the law of inertia is that the natural uniform motion of a body is not arbitrary. If inertia resists the uniform motion from being changed, then it will also try to restore the uniform motion after it is changed.

Imagine a serious scientist who believes that the law of inertia can be proved or disproved by actual experiments. He pushes small spheres along a horizontal table, trying to eliminate friction so far as possible. He notices that the motion becomes more uniform as the table and the spheres are made smoother. Just as he is about to proclaim the principle of inertia, someone suddenly plays a practical joke on him. Our physicist works in a room without windows and has no communication whatever with the outside world. The practical joker installs some mechanism which enables him to cause the entire room to rotate quickly on an axis passing through its centre. As soon as the rotation begins, the physicist has new and unexpected experiences. The sphere which has been moving uniformly tries to get as far away from the centre and as near to the walls of the room as possible. He himself feels a strange force pushing him against the wall. He experiences the same sensation as anyone in a train or car travelling fast round a curve, or even more, on a rotating merry-go-round. All his previous results go to pieces.

Our physicist would have to discard, with the law of inertia, all mechanical laws. The law of inertia was his starting-point; if this is changed, so are all his further conclusions. An observer destined to spend his whole life in the rotating room, and to perform all his experiments there, would have laws of mechanics differing from ours. If, on the other hand, he enters the room with a profound knowledge and a firm belief in the principles of physics, his explanation for the apparent breakdown of mechanics would be the assumption that the room rotates. By mechanical experiments he could even ascertain how it rotates.

Inertia is the centeredness of a body because of natural rotation of its substance at atomic level. The more centered the body is the lesser is its linear velocity. This is the proper understanding of the law of inertia. 

Why should we take so much interest in the observer in his rotating room? Simply because we, on our earth, are to a certain extent in the same position. Since the time of Copernicus we have known that the earth rotates on its axis and moves around the sun. Even this simple idea, so clear to everyone, was not left untouched by the advance of science. But let us leave this question for the time being and accept Copernicus’ point of view. If our rotating observer could not confirm the laws of mechanics, we, on our earth, should also be unable to do so. But the rotation of the earth is comparatively slow, so that the effect is not very distinct. Nevertheless, there are many experiments which show a small deviation from the mechanical laws, and their consistency can be regarded as proof of the rotation of the earth.

There are many experiments which show a small deviation from the mechanical laws, and their consistency can be regarded as proof of the rotation of the earth. That means, the heavens are being used as the absolutely still background for the mechanical laws.

Unfortunately we cannot place ourselves between the sun and the earth, to prove there the exact validity of the law of inertia and to get a view of the rotating earth. This can be done only in imagination. All our experiments must be performed on the earth on which we are compelled to live. The same fact is often expressed more scientifically: the earth is our co-ordinate system.

Our co-ordinate system happens to be the Earth, and the Earth’s rotation happens to be the implicit and unacknowledged part of it.

To show the meaning of these words more clearly, let us take a simple example. We can predict the position, at any time, of a stone thrown from a tower, and confirm our prediction by observation. If a measuring rod is placed beside the tower, we can foretell with what mark on the rod the falling body will coincide at any given moment. The tower and scale must, obviously, not be made of rubber or any other material which would undergo any change during the experiment. In fact, the unchangeable scale, rigidly connected with the earth, and a good clock are all we need, in principle, for the experiment. If we have these, we can ignore not only the architecture of the tower, but its very presence. The foregoing assumptions are all trivial and not usually specified in descriptions of such experiments. But this analysis shows how many hidden assumptions there are in every one of our statements. In our case, we assumed the existence of a rigid bar and an ideal clock, without which it would be impossible to check Galileo’s law for falling bodies. With this simple but fundamental physical apparatus, a rod and a clock, we can confirm this mechanical law with a certain degree of accuracy. Carefully performed, this experiment reveals discrepancies between theory and experiment due to the rotation of the earth or, in other words, to the fact that the laws of mechanics, as here formulated, are not strictly valid in a co-ordinate system rigidly connected with the earth.

Mechanical laws assume absolute rigidity of matter, space and time. This assumption is incorporated in Einstein’s measuring rod and clock. This reveals that the laws of mechanics are not strictly valid in a co-ordinate system rigidly connected with the rotating earth.

In all mechanical experiments, no matter of what type, we have to determine positions of material points at some definite time, just as in the above experiment with a falling body. But the position must always be described with respect to something, as in the previous case to the tower and the scale. We must have what we call some frame of reference, a mechanical scaffold, to be able to determine the positions of bodies. In describing the positions of objects and men in a city, the streets and avenues form the frame to which we refer. So far we have not bothered to describe the frame when quoting the laws of mechanics, because we happen to live on the earth and there is no difficulty in any particular case in fixing a frame of reference, rigidly connected with the earth. This frame, to which we refer all our observations, constructed of rigid unchangeable bodies, is called the co-ordinate system. Since this expression will be used very often, we shall simply write c.s.

The coordinate system (c.s.) is the frame of reference, constructed of rigid unchangeable bodies, that enables us to determine the positions of material points at some definite time. The laws of mechanics use the coordinate system provided by the fixed configuration of stars in the heavens.

All our physical statements thus far have lacked something. We took no notice of the fact that all observations must be made in a certain c.s. Instead of describing the structure of this c.s., we just ignored its existence. For example, when we wrote “a body moves uniformly. . .” we should really have written, “a body moves uniformly, relative to a chosen c.s. …” Our experience with the rotating room taught us that the results of mechanical experiments may depend on the c.s. chosen.

If two c.s. rotate with respect to each other, then the laws of mechanics cannot be valid in both. If the surface of the water in a swimming pool, forming one of the co-ordinate systems, is horizontal, then in the other the surface of the water in a similar swimming pool takes the curved form similar to anyone who stirs his coffee with a spoon.

When formulating the principal clues of mechanics we omitted one important point. We did not state for which c.s. they are valid. For this reason, the whole of classical mechanics hangs in mid-air since we do not know to which frame it refers. Let us, however, pass over this difficulty for the moment. We shall make the slightly incorrect assumption that in every c.s. rigidly connected with the earth the laws of classical mechanics are valid. This is done in order to fix the c.s. and to make our statements definite. Although our statement that the earth is a suitable frame of reference is not wholly correct, we shall accept it for the present.

The classical mechanics does not identify its coordinate system. We have kind of assumed that the laws of classical mechanics are valid in every c.s. rigidly connected with the earth.

We assume, therefore, the existence of one c.s. for which the laws of mechanics are valid. Is this the only one? Suppose we have a c.s. such as a train, a ship or an aeroplane moving relative to our earth. Will the laws of mechanics be valid for these new c.s.? We know definitely that they are not always valid, as for instance in the case of a train turning a curve, a ship tossed in a storm or an aeroplane in a tail spin. Let us begin with a simple example. A c.s. moves uniformly, relative to our “good” c.s., that is, one in which the laws of mechanics are valid. For instance, an ideal train or a ship sailing with delightful smoothness along a straight line and with a never-changing speed. We know from everyday experience that both systems will be “good”, that physical experiments performed in a uniformly moving train or ship will give exactly the same results as on the earth. But, if the train stops, or accelerates abruptly, or if the sea is rough, strange things happen. In the train, the trunks fall off the luggage racks; on the ship, tables and chairs are thrown about and the passengers become seasick. From the physical point of view this simply means that the laws of mechanics cannot be applied to these c.s., that they are “bad” c.s.

This result can be expressed by the so-called Galilean relativity principle: if the laws of mechanics are valid in one c.s., then they are valid in any other c.s. moving uniformly relative to the first.

A “good” c.s. is one in which the laws of mechanics are valid. The so-called Galilean relativity principle is, “If the laws of mechanics are valid in one c.s., then they are valid in any other c.s. moving uniformly relative to the first.”

If we have two c.s. moving non-uniformly, relative to each other, then the laws of mechanics cannot be valid in both. “Good” co-ordinate systems, that is, those for which the laws of mechanics are valid, we call inertial systems. The question as to whether an inertial system exists at all is still unsettled. But if there is one such system, then there is an infinite number of them. Every c.s. moving uniformly, relative to the initial one, is also an inertial c.s.

“Good” co-ordinate systems, that is, those for which the laws of mechanics are valid, we call INERTIAL SYSTEMS.

Let us consider the case of two c.s. starting from a known position and moving uniformly, one relative to the other, with a known velocity. One who prefers concrete pictures can safely think of a ship or a train moving relative to the earth. The laws of mechanics can be confirmed experimentally with the same degree of accuracy, on the earth or in a train or on a ship moving uniformly. But some difficulty arises if the observers of two systems begin to discuss observations of the same event from the point of view of their different c.s. Each would like to translate the other’s observations into his own language. Again a simple example: the same motion of a particle is observed from two c.s. the earth and a train moving uniformly. These are both inertial. Is it sufficient to know what is observed in one c.s. in order to find out what is observed in the other, if the relative velocities and positions of the two c.s. at some moment are known? It is most essential, for a description of events, to know how to pass from one c.s. to another, since both c.s. are equivalent and both equally suited for the description of events in nature. Indeed, it is quite enough to know the results obtained by an observer in one c.s. to know those obtained by an observer in the other.

It is quite enough to know the results obtained by an observer in one inertial system to know those obtained by an observer in another inertial system, provided their positions and relative velocities at some moment are known.

Let us consider the problem more abstractly, without ship or train. To simplify matters we shall investigate only motion along straight lines. We have, then, a rigid bar with a scale and a good clock. The rigid bar represents, in the simple case of rectilinear motion, a c.s. just as did the scale on the tower in Galileo’s experiment. It is always simpler and better to think of a c.s. as a rigid bar in the case of rectilinear motion and a rigid scaffold built of parallel and perpendicular rods in the case of arbitrary motion in space, disregarding towers, walls, streets, and the like. Suppose we have, in our simplest case, two c.s., that is, two rigid rods; we draw one above the other and call them respectively the “upper” and “lower” c.s. We assume that the two c.s. move with a definite velocity relative to each other, so that one slides along the other. It is safe to assume that both rods are of infinite length and have initial points but no end-points. One clock is sufficient for the two c.s., for the time flow is the same for both. When we begin our observation the starting-points of the two rods coincide. The position of a material point is characterized, at this moment, by the same number in both c.s. The material point coincides with a point on the scale on the rod, thus giving us a number determining the position of this material point. But, if the rods move uniformly, relative to each other, the numbers corresponding to the positions will be different after some time, say, one second. Consider a material point resting on the upper rod. The number determining its position on the upper c.s. does not change with time. But the corresponding number for the lower rod will change. Instead of “the number corresponding to a position of the point”, we shall say briefly, the coordinate of a point. Thus we see from our drawing that although the following sentence sounds intricate, it is nevertheless correct and expresses something very simple. The co-ordinate of a point in the lower c.s. is equal to its co-ordinate in the upper c.s. plus the co-ordinate of the origin of the upper c.s. relative to the lower c.s. The important thing is that we can always calculate the position of a particle in one c.s. if we know the position in the other. For this purpose we have to know the relative positions of the two co-ordinate systems in question at every moment. Although all this sounds learned, it is, really, very simple and hardly worth such detailed discussion, except that we shall find it useful later.

It is worth our while to notice the difference between determining the position of a point and the time of an event. Every observer has his own rod which forms his c.s., but there is only one clock for them all, Time is something “absolute” which flows in the same way for all observers in all c.s.

We assume that two rigid rods (c.s.) are moving with a definite velocity relative to each other, so that one slides along the other. One clock is sufficient for the two c.s., for the time flow is the same for both. We observe that The co-ordinate of a point in the lower c.s. is equal to its co-ordinate in the upper c.s. plus the co-ordinate of the origin of the upper c.s. relative to the lower c.s.

Now another example. A man strolls with a velocity of three miles per hour along the deck of a large ship. This is his velocity relative to the ship, or, in other words, relative to a c.s. rigidly connected with the ship. If the velocity of the ship is thirty miles per hour relative to the shore, and if the uniform velocities of man and ship both have the same direction, then the velocity of the stroller will be thirty-three miles per hour relative to an observer on the shore, or three miles per hour relative to the ship. We can formulate this fact more abstractly: the velocity of a moving material point, relative to the lower c.s., is equal to that relative to the upper c.s. plus or minus the velocity of the upper c.s. relative to the lower, depending upon whether the velocities have the same or opposite directions. We can, therefore, always transform not only positions, but also velocities from one c.s. to another if we know the relative velocities of the two c.s. The positions, or co-ordinates, and velocities are examples of quantities which are different in different c.s. bound together by certain, in this case very simple, transformation laws.

The positions, or co-ordinates, and velocities are examples of quantities which are different in different c.s. bound together by certain transformation laws.

There exist quantities, however, which are the same in both c.s. and for which no transformation laws are needed. Take as an example not one, but two fixed points on the upper rod and consider the distance between them. This distance is the difference in the co-ordinates of the two points. To find the positions of two points relative to different c.s., we have to use transformation laws. But in constructing the differences of two positions the contributions due to the different c.s. cancel each other and disappear, as is evident from the drawing. We have to add and subtract the distance between the origins of two c.s. The distance of two points is, therefore, invariant, that is, independent of the choice of the c.s.

The distance between two points is independent of the choice of the c.s.

The next example of a quantity independent of the c.s. is the change of velocity, a concept familiar to us from mechanics. Again, a material point moving along a straight line is observed from two c.s. Its change of velocity is, for the observer in each c.s., a difference between two velocities, and the contribution due to the uniform relative motion of the two c.s. disappears when the difference is calculated. Therefore, the change of velocity is an invariant, though only, of course, on condition that the relative motion of our two c.s. is uniform. Otherwise, the change of velocity would be different in each of the two c.s., the difference being brought in by the change of velocity of the relative motion of the two rods, representing our co-ordinate systems.

The change of velocity is also independent of the choice of the c.s., on condition that the relative motion of the two c.s. is uniform.

Now the last example! We have two material points, with forces acting between them which depend only on the distance. In the case of rectilinear motion, the distance, and therefore the force as well, is invariant. Newton’s law connecting the force with the change of velocity will, therefore, be valid in both c.s. Once again we reach a conclusion which is confirmed by everyday experience: if the laws of mechanics are valid in one c.s., then they are valid in all c.s. moving uniformly with respect to that one.

The forces acting between two material points, which depend only on the distance, are also independent of the choice of the c.s., on condition that the relative motion of the two c.s. is uniform.

Our example was, of course, a very simple one, that of rectilinear motion in which the c.s. can be represented by a rigid rod. But our conclusions are generally valid, and can be summarized as follows:

(1) We know of no rule for finding an inertial system. Given one, however, we can find an infinite number, since all c.s. moving uniformly, relative to each other, are inertial systems if one of them is.

(2) The time corresponding to an event is the same in all c.s. But the co-ordinates and velocities are different, and change according to the transformation laws.

(3) Although the co-ordinates and velocity change when passing from one c.s. to another, the force and change of velocity, and therefore the laws of mechanics are invariant with respect to the transformation laws.

The above are laws of transformation.

The laws of transformation formulated here for coordinates and velocities we shall call the transformation laws of classical mechanics, or more briefly, the classical transformation.

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Final Comment

According to Postulate Mechanics, a body has a natural uniform velocity balanced by its inertia. The greater is body’s inertia, the smaller is its  velocity. The inertia cannot be changed arbitrarily. If the velocity of the body is changed by external forces it is restored when the force is removed. 

The ideal frame of reference shall be infinite inertia with zero velocity. This is approximated by the background of the stars in the heavens. This frame of reference, or co-ordinate system, shall apply equally to matter, subatomic particles and radiation, as they all have inertia as well as velocity. From matter to radiation we have the full spectrum of inertia.

The path of a matter particle is very much curved with its greater inertia and slower velocity. As inertia lessens, the velocity increases and the path becomes increasingly straighter. Mechanical laws, in their simplicity, apply to matter particles only. Here we have classical mechanics. The inertial systems fall in this narrow band of high inertia. 

In this band, a material object can be approximated as a point particle called “center of mass.” This approximation, however, becomes increasingly questionable as inertia decreases and the centeredness of the particle is lost. The particle can no longer be located at a point location. This has generated a lot of confusion in science.

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