24. ON SIMULTANEITY AND RELATIVITY OF LENGTH AND TIME INTERVAL
 
   The main subject of the special theory of relativity are three concepts and they are: simultaneity, relativity of lengths and relativity of time intervals. Einstein began his work on the theory of relativity by defining and explaining these concepts in the first and second paragraph of his first paper in that field [2].
   The relativistic way of treating time, simultaneity and space is the subject of many discussions in different scientific spheres, from physics to philosophy.
 
24.1 Einstein's determination of simultaneity and relativity of length and time interval
 
   With regard to the importance of the mentioned concepts and the originality in their treatment, it is best if the reader gets first hand information on Einstein's exposition. For that purpose we shall quote here both paragraphs from his first paper on relativity, and then give our commentary on the quoted material.
   Quotation: "§1 Determining simultaneity
   Let us take a coordinate system in which are valid the equations of Newton's mechanics. For the purpose of distinguishing it from later introduced coordinate systems and for the purpose of defining terminology let us name this coordinate system an "unmoving system".
   If a material point is at rest relatively to this coordinate system, then its position relatively to that system can be determined by the methods of Euclid's geometry with the help of solid ruler and expressed in Descartes coordinates.
   If we want to describe a motion of some material point, we set the values of its coordinates in the function of time. Thereby we should bear in mind that such a mathematical description has physical meaning only then when it is previously clarified what is meant by the concept of "time". We should focus our attention to the fact that in all our judgements, in which time plays some role, the judgment about simultaneity always appears. If I, for example, say: "That train arrives here at 7 o'clock." That, for example, means the following: "The small hand on my watch showing seven o'clock and the arrival of the train are simultaneous events." [Here will not be considered an inaccuracy in the conception of the simultaneity of two events, which originate (approximately) in the same place, which would also be overcome by the help of some abstraction.]
   It can be shown that all difficulties in connection with the determining "time" can be overcome if, instead of the word "time", I write "the position of the small hand on my clocks". Such a decision really is sufficient only in case when we determine the time for the particular place in which the clocks are just situated. However, that decision is already insufficient when we should connect, from the point of view of time, two series of events, one another, which flowing in different places. In one word, it would determine the time of events which occur in places distant from clocks.
   If we want to determine the time of events, we could, of course, satisfy ourselves by compelling an observer, who is standing with a watch at the origin, to compare corresponding positions of the watch hands with every light signal coming to him through vacuum and informing him of the registered event. However, that comparison is connected with the difficulties that we know from experiments. Namely, it will not be independent of the place where the observer is standing with the clock. We shall come up with a far more practical determination by means of the following reasoning.
   If a clock is placed at point in space, then the observer, standing at point , can determine the time of events in the immediate vicinity of point through the simultaneous observation of these events and the position of the clock hands. If at another point of space there is also a clock (we add: "The same clock as at the point ") then it is also possible for an observer at point to assess the time of events in the immediate vicinity of . However, it is impossible to compare, from the point of view of time, some event at with an event at without making further assumptions. For now we shall only determine " - time" and " - time", but not the general "time" for and . The latter can be determined by introducing the definition that the "time" needed for the passage of light from to equals the "time" needed for the passage of light from to . At a moment by " - time" let a ray of light come out of towards , let it reflect at the moment by " - time" from to and return to at the moment by " - time". The clocks in and will, according to the definition, run in a synchronized manner if

(24.1)

We believe that the determining of simultaneity can be given in an un-contradictory manner and for an arbitrary number of points and that the following claims are true:
   1) If the clock at runs synchronized with the clock at then the clock in runs synchronized with the clock at .
   2) If the clock in runs synchronized with the clock at , as well as with the clock at , then the clocks at and run synchronized relatively to each other.
   In this manner, by using some physical thought experiments, we have determined what should be understood by synchronized clocks, which are at rest in different places and owing to that we have, obviously, obtained the definition of the concepts: "simultaneity" and "time". The "time" of events - that is simultaneously with events indication of clocks at rest, which are placed at the place of the events and which run synchronized with a certain number of clocks at rest.
   In accordance with the experiment we shall also assume that the magnitude

(24.2)

is an universal constant (the speed of light in vacuum).
   Having in mind that we determined time with the help of clocks at rest in the system at rest, then we shall name the time belonging to the system at rest the "time in the system at rest".
   §2 On relativity of length and time interval
   Further thinking relies on the principle of relativity and the principle of the constancy of the speed of light. We formulate both principles in the following way:
   1) The laws by which the states of physical systems change, do not depend from that on which of the two systems, moving with uniform translation relatively to each other, these changes of state refer to.
   2) Every ray of light moves in the "unmoving" system of coordinates at a definite speed , independently of whether that ray of light is emitted by an unmoving body or a moving body.
   Thereby we have

whereat "time interval" should be understood in the sense of the definition in §1.
   Let us take a solid piston at rest and let its length be measured with a ruler, which is also at rest. Now let us imagine that the piston, whose axis is directed by the -axis of the unmoving coordinate system, is pushed into gradual motion (at a speed ) uniformly and translatory in the direction of the growth of value . Let us now question the length of the piston in motion, which we are intending to determine with the help of two following operations.
   a) The observer is moving together with the said ruler and with the measured piston and measures the length of the piston directly by resting the ruler against the piston, the same as if the measured piston, observer and measuring device were at rest.
   b) With the help of separate unmoving clocks in the unmoving system, which are synchronized, in the sense of §1, the observer determines in which points of the unmoving system the beginning and the end of the measured piston are at a certain time . The distance between these two points, measured by the said procedure, with the ruler at rest, is the length which can be marked as the "length of the piston".
   In accordance with the principle of relativity, the length determined by the operation "a", which we shall call the "length of the piston in the moving system" should be equal to the length of the piston at rest.
   The length determined by the operation "b", which we shall call "the length (in motion) of the piston in an unmoving system" will be determined on the basis of our two principles and we shall find that it is different from .
   In the kinematics, which is usually applied, it is taken without objection that the lengths determined with the help of the two said operations are equal, or, in other words, that a solid body, which is moving, at a moment in geometrical relation can be completely substituted with the same body when it is at rest in a certain position.
   Let us imagine that clocks are fastened at both ends of the piston ( and ) which are synchronous with clocks in the unmoving system, that is, their indication respond to the "time in the unmoving system" in exactly those places in which these clock are situated; consequently these clocks are "synchronous in the unmoving system".
   Let us further imagine that by each clock there is an observer, moving with it, and that these observers apply on both clocks, as established in §1, the criteria of simultaneity in the working of the two clocks. At a time [The "time" here signifies the "time in the unmoving system" and together with the "positions of the hands of the moving clocks, which are situated in that place under discussion".] let a ray of light come out of , let it reflect at at a time and return to at a time moment . Taking into account the principle of constancy of the speed of light we find

(24.3)

where is the length of a moving piston, measured in an unmoving system. So, the observer who is moving together with the piston, will find that the clocks at points and do no run synchronized, whereas the observers, who are in the unmoving system would claim that the clocks were synchronized.
   So, we see that we should not give an absolute meaning to the concept of simultaneity. Two events which are simultaneous, when observed from one coordinate system, are not understood as such when observed from the system which is moving relatively to the given system." End of quotation.
 
24.2 Objections to Einstein's determination of simultaneity and relativity of length and time interval
 
   From the above quoted text the reader may have noticed Einstein's following claims.
   Every point of space has its time. There is no general time. Thus, for example, point has time , and point has time . The time in a coordinate system at rest differs from the time in the moving system, so there is "time in the system at rest" and "time in the moving system". His time is the position of the small hand on a clock.
   Simultaneity can exist only in one coordinate system, in the system which is at rest or in a moving system. Furthermore, the absolute meaning of time does not exist, since the events which are simultaneous at the observation from one system are not simultaneous at the observation from another system which is moving relatively to the given system.
   For measuring time and establishing of simultaneity of events clocks are used which work synchronized in the system at rest or in the system which is moving relatively to the system at rest. According to Einstein, they cannot work in synchronization in both systems at the same time. The synchronization of the clocks at and he conditions by the equality of time needed for a light ray to pass from to with the time needed for the same ray to pass from to , that is .
   He bases the negation of the existence of absolute time and simultaneity on the alleged impossibility of determining the existence of such time and simultaneity. In fact, this leads in essence to the assertion that something does not exist because I cannot determine its existence, thereby I do no take into account my ignorance or lack of equipment for the determination.
   With the following examples we can see the problem of determining time and simultaneity.
   Let us have a line of boats as in the Fig. 24.1.

Fig. 24.1

   When the boats are at rest, clocks on them can be synchronized in the following way. Let us place boat right in the middle and let us fire a shot from boat . The sound of that shot will be heard at the same time on boats and and it will be possible to synchronize all clocks to a set, agreed time, that is their telling of time will be synchronized. When that line of boats is moving, it is obvious that we can apply the same method again. Sailors who do not know that the boats are moving relatively to the air, will be convinced that they have synchronized the clocks in and . However, when the boats are moving then a signal from point will take longer time to reach boat than boat , because boat is going away from the source of sound, and the boat is coming towards the sound. That difference depends on the speed at which the line of boats is traveling. Therefore, it is impossible to synchronize the clocks by that procedure when a line of boats is moving. However, it would be completely wrong to claim that there are no other technical possibilities for synchronizing clocks in a given line of boats which is moving. For example, first the speed of the line of boats can be determined, then the time needed for the sound to travel form boat to the boats and . On the basis of this data a sound signal should be sent from the boat in the direction of each of them, which they will receive at the same time and synchronize their clocks by it. It is clear that thereby a signal sent in the direction should be delayed relatively to the signal sent in the direction . The delay will be the time difference between the time needed for the signal to reach boat , which is travelling towards the sound, and boat , which is going away from the sound.
   The precision in determining simultaneity, and thus the precision in synchronizing clocks, in first case, when the line of boats is at rest, will depend on the precision of determining the distances and . In the second case, when the line is moving, it will depend on the precision of determining these distances and also on the precision of determining the speed of motion.
   Whether two events are simultaneous or not does not depend on how we seen then and whether we see them at all. Our judgment whether something is or is not simultaneous does not depend only on our observation of the moment when a ray of light comes from the scene of an event, but also on our knowledge related to the event and the scene of the event. Thus, for example, two men are observing the explosion of a star through telescopes. One of them knows nothing about the distance to the star, and the other one is an astronomer. The fist one will think that the star explosion is happening at the same time as he is observing the star, while the other will know that it happened in a remote past, maybe even a million years ago, if the star is a million light years away from us. From this example we see that a subjective judgment of simultaneity is unreliable.
   With the development of social community, grew the need for common general time. Prehistoric man had no such need. For him the time of his zone of motion around a cave was sufficient. However, developed societies can not even be imagined with such segmented time.
   In principle, we measure time with the course of events. For example, for the ancient Egyptians the flooding of the Nile was such an event. It happened every year and so they could count years by it. With time man defined and measured time better and better.
   All determinations, both of position and time are relatively to something. Today, the whole world time is measured relatively to the moment of the sun passing above zero longitude. Moreover, relatively to that moment the earth is divided into 24 time zones. In each time zone all clocks, at the same moment relatively to the passage of the sun above zero longitude show in advance defined time. Thus our civilization has a general earth time in a wide and narrow sense. If there was a need for general galactic or cosmic time then we would have to find a possibility of connecting the zero time to some galactic that is cosmic event.
   The existence of general time on earth is imposed by the need to coordinate the activities of people all over the world. By using time, defined in this manner, we can, for example, bring about the simultaneity of two events in any two points in the world, at rest or moving, with a precision which equals the precision of registering the simultaneity of two events in the immediate vicinity. Such possibilities exist thanks to the agreed way of determining - measuring time, human knowledge and achieved technical capacities. If the determination of simultaneity and the measurement of time were as disputable and inaccessible as Einstein maintains, then modern systems of remote guidance, from various military systems to the systems for cosmic research would not exist.
   The way in which Einstein treats time and simultaneity, concerning knowledge of events and physical processes on which the judgment of time and simultaneity are based, is of poor quality. It is subjective and adjusted so that the reader reaches wrong conclusions determined beforehand, which will serve for the further derivation of new wrong conclusions. That this is really the case can be seen in the next chapter, number 2, in which relativity of lengths and time intervals is studied.
   When talking about the relativity of lengths and time intervals Einstein uses a piston length , which is at rest or it is moving at a constant speed along the -axis, so that the piston axis matches with the -axis. He also uses a ruler with which he measures the piston at rest and in motion. When the piston is at rest an observer measures the length of the piston by holding the ruler against the piston and in that way he determines that the piston's length equals . Then the observer moves with a ruler and the piston together (for example in a train). Then, again the observer in motion holds the ruler against the piston and determines again that the piston's length is . In that way the observer finds that the length of a piston at rest equals the length of a moving piston, when the measurement is performed by the observer who moves together with the piston. In short it means that the length of the piston at rest is equal to the length of the piston in motion, when that length is measured in a moving system in which the piston is at rest.
   The third measurement method is more complex, since the observer, who is at rest, should measure the length of a moving piston. That is the same as if the observer from the railway embankment measured the length of a wagon of a fast train, going past him. It is clear that in this case he cannot measure the length of the wagon by holding a ruler against the outer wall of the wagon. Therefore Einstein uses a different kind of measurement. In that measurement he uses light rays and clocks. And that is where the great deception in the construction of the theory of relativity begins - the deception on which this theory is based.
   In this experiment he uses two clocks, one of which fixed to the beginning of the piston at point , and the other to the end of the piston at point . He also puts the source of light at point , and a mirror at point which reflects light back to point . With the piston, which is at rest, thus equipped, he checks whether the clocks are synchronized, in the way that is described in the quoted text and the Eq. (24.1) on the equality of time intervals

where and are the times shown by the clock at point (beginning of the piston), and is the time shown by the clock at point (the end of the piston). The time interval is the time needed for a ray of light sent from point to reach point , and the time interval is the time needed for the same light ray, after being reflected from the mirror at point , to return to point . Since then the clocks will be synchronized if the equality of time interval given by the Eq. (24.1) is satisfied.
   In that manner he determines that the clocks are synchronized. On the basis of the measured time intervals and the light speed he finds that the piston's length is

(24.4)

   After making adjustments in this way, checking that the clocks are synchronized and determining the length of the piston, he puts the equipped piston into a state of motion at a constant speed and repeats the experiment to check whether the clocks are working in synchronization.
   A schematic representation of the experiment is given in the Figs. 24.2.1, 24.2.2 and 24.2.3. Fig. 24.2.1 gives the starting position of the piston, that is the state at the moment when a light ray starts from point (the beginning of the piston) towards point . In Fig. 24.2.2 the position of the piston at the moment when the ray arrives at the mirror at point (the end of the piston) is shown, and Fig. 24.2.3 gives the position of the piston at the moment when the ray reflected from the mirror at point arrives back at point . The starting position of the piston is given in full lines; the second position of the piston (when the ray arrives at point ) is given in interrupted lines and the third position (when the ray arrives back in the point ) in dotted lines.

Fig. 24.2.1	Fig. 24.2.2	Fig. 24.2.3

   As the pictures show, the ray passes from point towards point . The time (moment) of the start of the ray from point towards point is noted by an observer on the basis of the time shown by the clock at point . From that moment the ray moves towards point . During that time while the ray is moving at speed towards the mirror, the piston with the mirror is moving in the same direction so that the mirror is moving ahead by the length and arrives from point at point . Therefore, to reach the mirror, the ray had to cover the distance . As we know, if the piston had not moved, the ray would have covered only the distance which is equal to the length . This means that because of the piston's motion the ray had to cover a longer distance, and more time is needed for this, so

(24.5)

   Because of that, the time needed for the ray to arrive at point when the piston is at rest will differ from the time needed for the ray to arrive at point when the piston is moving. The observer will see that a time difference in the arrivals of the ray occurred, and Einstein would conclude, of course wrongly and probably intentionally, that the time shown by the clocks changed because, as a result of motion, the rhythm of the clock "ticking" changed, and not because the length of the path covered by the light ray changed.
   While the ray returns, after being reflected from the mirror, at point covers a distance shorter than the length of the piston because the beginning of the piston (point ) is coming towards the light ray at the speed , so

(24.6)

   The observer will notice that the time of the ray's return, according to the clock at when the piston is moving, differs from the time of the return of the ray when the piston is at rest. Einstein concludes that this clock also changed its "rhythm of ticking" because of its motion. However, it is clear that time intervals changed because of the change in the length of the ray's path, so that

(24.7)

And also

(24.8)

   As has already been said, Einstein deduces a conclusion, which is obviously wrong, that the clocks stop being synchronized as soon as they start moving and because of that the concept of simultaneity should not be given absolute meaning.
   Einstein's previous experiment with a piston can be made with sound instead of light. However, in that case, at the same length and the speed of piston motion, the disagreement between the clocks would go up by around 10¹² times, because the speed of sound is about 10⁶ times smaller than the speed of light. Naturally, with experiments where sound is used, the speed of piston motion must be less than the speed of sound.
   The clocks at rest can be synchronized even when they are far apart, by using the procedure and the requirement given by Eq. (24.1). Accordingly, a moving piston can be of any length, and still the clocks at its end would go on working in a synchronized manner.
   In the theory of relativity it is claimed that the de-synchronized function of the clocks which were synchronized while at rest occurs because of the motion of those clocks. However, it is not mentioned anywhere that the de-synchronization is also a function of the piston length, that is the distance between the clocks. De-synchronization is reduced with the reduction of the piston length, so the clocks, which are placed next to each other "tick" in rhythm, that is they are synchronized, independently of that how fast they move. The reason for this is clear from the explanation given in Figs. 24.2.1, 24.2.2 and 24.2.3, and which can be summarized thus: the greater the distance between the clocks, the greater the de-synchronization, because the light needs to travel not only the distance but also the additional distance , for which the piston moves while the light travels the distance . That move is proportional to the length and the speed at which the piston moves.
   The explanation given above of the different time taken by light rays to pass along the piston when it is at rest and when it is in motion is based on the real situation and is not in accordance with the theory of relativity; neither is Einstein's discussion of the synchronisation of the clock at rest and in motion. The fundamental principle of the theory of relativity is the constant velocity of light which will be the same in both systems, and . Also, according to this theory, the length of the piston is the same in all systems in which the piston is at rest.
   As a result, if the light source, the mirror and the clocks are fastened to the ends of the piston as Einstein describes in §2 quoted above, then, according to the theory of relativity, the time taken for the light rays to pass from the beginning to the end of the piston and vice versa must be the same, whether the piston is at rest in system or moves with system . In both cases, according to the theory, the speed of light relative to the piston is the same, and the length of the piston is the same too, since the piston is at rest in the system in which the measurement is made. Therefore, the observer who moves with the piston would not be able to perceive the change in the time taken for the rays to pass along the piston and would not be able to conclude that the clocks which are in motion do not work in the same rhythm as the clocks that are at rest. In reality the clocks will work in the same rhythm but they will show different times taken by the light rays to pass along the piston, for the reason explained before in Figs. 24.2.1 24.2.2 and 24.2.3.
   As a result Einstein's claim, that synchronized clocks while at rest lose synchronization when moving, is unfounded and that physical process in the given thought experiment with a piston and a clock in motion is incorrectly analyzed and interpreted in order to lead the reader astray and make him accept the claim that time and length change only because of motion.
   In the text quoted in §2, when assessing the synchronization of the clocks, Einstein says: "Taking into account the principle of the constancy of the speed of light, we find

(24.3)