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.
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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. It depends on the body’s inertia.
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.
The law of inertia is liable to breaks down when unknown forces acting on a system are not accounted for.
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.
Very small deviations from the law of inertia are observed that may be traced back to the “unaccounted for” rotation of earth.
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 unknown 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.
The experiments conducted on Earth, from which the mechanical laws are derived, assume a rigid bar and an ideal clock for the measurement of distance and time. Carefully performed, the experiments reveal discrepancies between theory and experiments due to the rotation of the 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.
We must have what we call some “frame of reference” to be able to determine the positions of bodies. This frame, to which we refer all our observations, constructed of rigid unchangeable bodies, is called the “co-ordinate system”.
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 coordinate system includes such properties as rotating or non-rotating. The classical mechanics does not identify its coordinate system. It assumes it to be 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.
“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.” –The Galilean relativity principle
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.
“Inertial systems” are those for which the laws of mechanics are valid.
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 relative velocities and positions 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 may measure the same material point in two different coordinate systems that are moving relative to each other at a certain velocity. We can always calculate the position of a particle in one c.s. if we know the position in the other. Here we assume the measure of time to be the same for both 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.
We can 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.
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, however, 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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We should be aware of the following definitions:
We must have what we call some “frame of reference” to be able to determine the positions of bodies. This frame, to which we refer all our observations, constructed of rigid unchangeable bodies, is called the “co-ordinate system”. “Inertial systems” are those for which the laws of mechanics are valid.
The classical mechanics does not identify its coordinate system. It assumes it to be the Earth.
The experiments conducted on Earth, from which the mechanical laws are derived, assume a rigid bar and an ideal clock for the measurement of distance and time. Carefully performed, the experiments reveal discrepancies between theory and experiments due to the rotation of the earth.
“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.” –The Galilean relativity principle
The transformation laws of classical mechanics may 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.
“Another factor that has been missing from the law of inertia is that the natural uniform motion of a body is not arbitrary. It depends on the body’s inertia.” –Vinaire’s observation
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