Length and Time in Special Relativity

(First development : November 20^th, 1999 (in Spanish)
Last update: September 8^th, 2003)

In this page I will to try to approach to two typical consequences of the Special Relativity Theory. Namely, the concepts of time dilation and length contraction .

Time dilation in moving systems.

I begin with a classic system and later I will approach and focus on Special Relativity.

Let us suppose that we have got a clock that works by means of the bounce of sonorous waves between two flat boards (A and B) fixed to a rigid platform (fig. 1). If the distance between the planes A and B is 300 meters and the system is in rest with respect to the surrounding air, the sound will take 1 sec while crossing the distance between from one board to the other, and it will use another second while returning to the first board. In other words, every 2 seconds a complete period is made.

In this page I will try to approach to time dilation and lengths contraction concepts associated with Special Relativity Theory.

Let us suppose we have a clock working because the rebound of sound waves between two flat boards (A and B) fixed to a rigid platform (Fig. 1). If the distance between the boards A and B is 300 m and the system is at rest regarding the surrounding air, the sound will take 1 s in traveling the distance AB, and it will take another second returning from B to A. In other words, each 2 s an entire period is completed.

We can use this clock to measure events, and we take it as measuring standard in the stationary system.

Next, let us suppose the system moves at 200 m/s in the direction defined by the arrow going from A to B (Fig. 2). We also suppose that air remains stationary, meaning that, for an observer stationary on the platform ( at rest near the so called " clock ") the air moves at 200 m/s from B toward A.


*Fig. 1* - Clock at rest. The signal does the trip in 2 sec.	*Fig. 2* - Clock in movement: 2 sec in the moving system is equivalent to 3,6 sec at rest

Question : How much does it take the sound signal in traveling the round trip?

When the sound depart from A to B, it advances at 100 m/s to cover the distance separating both boards, since B goes away at 200 m/s from the signal moving at 300 m/s (the signal advances with opposite " wind ") The same reasoning leads to the 300 m distance from B to A being covered at 500 m/s (" wind " in favor) Conceptually, it is as a cyclist ascending and descending a hill: The ascending travel (with the opposite slope) is much slower than the descending trip (slope in favor)

And it leads to a total time round trip (measured from the stationary system) of 3 s + 0.6 s = 3.6 s

This clock in motion notably slows down in relation with the clock at rest. The mobile clock uses 3.6 s in doing the same thing that the stationary clock carries out in 2 s.

And in accordance with these results, the conversion factor to transform the intervals measured in the stationary clock to the values measured by the mobile clock is 2.0/3.6 = 0.5555...

The slow down is so remarkable that if the system AB were moving at 300 m/s, the sign would never reach the board B and therefore the mobile clock would be virtually stopped.

Well, to this point I believe that the concept is, more or less, simple. The clocks in movement slow down (they move slower) than clocks at rest. But in the previous example it is so because the clock has a mechanism that is affected by the displacement. The speed of air would not affect the operation of a battery clock.

For lengths the concept is a little more complex, but in the classic derivation of Special Relativity the same disorder of clocks, together with the concept of simultaneity already developed, leads straightforward to an alteration of lengths in moving systems. In order to visualize this alteration let us continue with the development of the example.

To be consistent, the observer in motion defines all its distances through the time used by sound in traveling each distance during a round trip itinerary. We will see, in particular, what happens when from the mobile system the distance separating two stationary objects is measured.

We already said that the mobile observer doesn't notice that its clock slows down. Then, it continues considering the period of its clock as 2 s (using 1 s for the going and 1 s while turning back), in spite of the fact that us (stationary) we know that the real period is 3.6 s (result of a 3 s trip in the going and of 0.6 s in the return)

However, to maintain the synchronism of its clocks, the mobile observer should make something that may seems very strange: He owes readjust its clocks. The reason is a consequence of considering that time used in the going is identical to the time lasting while turning back. If there is a clock (ClockA) placed in the board A and an identical clock (ClockB) placed in the board B, we hope that an acoustic signal leaving A when ClockA reads 10:00:00 (min: s: centimes of s), reaches B at 10:01:00 (ClockB reading) and return back to A when the reading (ClockA) is 10:02:00

There is not problem with ClockA times because although the process really takes 3.6 s, the clock slows down in such way that its own reading increases only 2 s during the experience.

The problem arises with ClockB. If it also reads 10:00:00 when the signal leaves ClockA, when receiving the signal it would read 3.0*0.5555... =1.666... more seconds.

Consequence: to maintain the synchronism, ClockB should read 0.6666 s less than ClockA at the moment (defined as simultaneous from the stationary system) that the signal departs from ClockA. (Fig. 3 and Fig. 4)


*Fig. 3* - Reading of clocks in the moving system (as viewed from the system at rest) when the signal makes the "going" trip.	*Fig. 4* - Reading of clocks in the moving system (as viewed from the system at rest) when the signal makes the "return" trip.

On the basis of the previous development, if the readings of the clocks of stationary and moving systems are compared when faced, (both systems having their clocks internally synchronized) , we are confronted with which in Special Relativity has been called "the mess of the clocks". If the stationary clock facing ClockA has the same reading than ClockA, then the stationary clock facing ClockB differs in 0.6666... s. And the farther from clock A are the faced clocks (mobile and stationary), bigger is the difference.

No !! it is easy to realize that the mobile observer altered its time readings to maintain the internal synchronism among its clocks, using the method suggested by Einstein.

Observation: What here looks like a crazy behavior of observers in moving system (alter its clocks intentionally "to pretend" that the time during the going of the signal is the same that the time consumed in the return), it is assumed by Einstein like representative of what it happens in the real world of physical interactions. This is the point that I have seen criticized in a harder way about Einstein developments. A lot of people affirm that this way to obtain the synchronism is capricious, and some other more logical ways exist to reach the internal synchronism. However, we are not trying to see which is the most logical or simpler method, but that one better representing the real interactions. And everything seems to indicate that the method proposed by Einstein coincides with how real systems behave.

And the real systems would use this method of synchronism in natural form (not forced by capricious observers) So natural, that a system that leaves the rest and becomes to be in movement would alter all its internal clocks so that they would not lose the Einstein's synchronism (we must remember that the internal clocks that govern the physical interactions are not independent entities, but are indissoluble with the interactions mechanism).

Nevertheless we must not forget that, in the development exposed in this page, it is assumed that the movable systems can be different from the systems at rest. Special Relativity assumes from the beginning that this differentiation is impossible. And this is certainly the origin of logical problems and paradoxes (author's remark ).

However we should not forget that, in the developments here presented, it is assumed that we can observe the difference between systems in motion and systems at rest. Inside relativity axioms it is impossible to achieve such differentiation.

And what happens with distances?

Let us suppose that we have two marks (A' and B'), separated by a certain distance in the stationary system, and we want to measure this distance from a mobile system. We will see that in this operation it is necessary to use clocks.

This form of measuring distance among systems in relative motion is the Einstein suggestion and it seems to be absolutely logical and natural. However we are already seeing that this methodology is based on the synchronism definition. And the synchronism used in the mobile system (to maintain the pretense of seeming stationary), leads to differences with the real stationary system. Let us remember that, in this example, we admit that air at rest defines the stationary system and allows the differentiation from system in motion.

If we identify the observers A and B (from mobile system) with those already well-known ClockA and ClockB, we know that ClockB reads 0.6666 .s permanently in defect from the reading of ClockA. In this way when ClockA reads 10:00:00 and it faces with A' (at the stationary system), ClockB doesn't still register 10:00:00. For this reason the system should follow its movement until ClockB also reads 10:00:00 (0.666... s own time), the moment in what B' from the stationary system, faces ClockB.

This mechanism leads to the stationary distance A'B', as an smaller one as seen from the mobile system. In other words, in accordance with the mobile system observers, the longitudes of stationary objects are smaller than those that the stationary observers declare.

Again it is easy to realize that the problem is originated by the mobile system for its seemingly capricious way of synchronizing clocks.

Discussion.

In the presented case the really stationary clocks and the really mobile clocks are clearly identified so, the mobile clocks really slow down in relation with those of the stationary system. Nevertheless, as analyzed in other pages, even although this it is the real situation, the observers of both systems believe that the other system clocks slow down and the others bars are shorter.

Special relativity makes this same outline but without allowing privileged systems. Therefore, not knowing which system is moving, it is necessary to accept that both systems are entitled to be considered stationary and therefore neither real measuring nor apparent measuring exist. Only the same degree of reality. According to Special Relativity only "own" realities exist rather than an underlying reality.

It is normal for an object be smaller than other, or slower and the conventional logic could accept each object seems smaller than the other (relative distance among observers effect) But both objects cannot be really smaller than the other, at the same time.

Objectively we know, based on consistent experimental results, that in real world clocks in motion slow down.

In fact, the equivalence of all the systems in uniform relative movement is a postulate of the Special Relativity Theory. Therefore it is not demonstrated. It is accepted and its consequences developed. And its consequences are the well-known times and distances equations (Lorentz transformations) And if equations work fine maybe it is not so "capricious" the election of the synchronism of Special Relativity.

In fact, I am convinced that this election shows by itself the Einstein great geniality. Establishing this methodology he was able to look objectively the behavior of the physical world and its interactions.

However, inevitably, the development of Einstein leads to paradoxical situations. And in other pages I demonstrate that could be derived the same transformation formulas accepting privileged systems and a physical universe structure that seems to mask the measurements. Let us remember the example developed in this page where the mobile observer accommodates the clocks to be able to feel himself as stationary.

In my own development of Lorentz transformations, I show how the Universe could do it automatic without responding to the observers' whims. To do that, it is necessary that speed of light in vacuum was something very special, bound to the intimate universe structure. In this way, directly affecting the speed of all clocks, it is impossible to detect the own movement in To and Fro interactions (the great majority of physical interactions).

1. Time dilation in motion systems does not seem to be a too complicated concept. It is only necessary to affect the clock structure by the movement.

3. If all clocks slow down consistently (mechanical, atomic and biological), should have something in its structures, which is affected in equal form by motion.

as the mark of the stationary system faced B when ClockB marks 10:00:00. But this mechanism gives rise to that the espaciamiento ÁB', sight from the stationary system is much greater than espaciamiento AB. Nevertheless, on the basis of the conventions used, sight from the movable system, AB is equal to ÁB' since it faced Á exactly 10:00:00 and B with B' also to 10:00:00. Again it is easy to realize that the problem originates the movable system by apparently " capricious " his way to synchronize Clockes. In the presented/displayed case, is understood the problem because the observation is reciprocal. Those of the stationary system see that the movable clocks are slow and those of the movable system see that the clocks stationary advance.

The problem with the RE is that it does this same one I raise but without allowing privileged systems. Therefore, when not knowing which of the systems is in movement it is necessary to accept that both systems are considered stationary with equality of rights and therefore both they observe that the clocks of the other system are slow and the longitudesa is shorter. Absurd. Si!!. sounds Again difficult to accept. Of course. It is normal that an object is minor who other, or slower, but each one cannot be smaller than the other simultaneously. The conventional logic can accept that each object seems smaller or slower than the other. But both they cannot be smaller at the same time. Or if. But the RE Funciona. Or it only seems to work. Objectively we know, on the basis of consistent experimental results, that in the real world the clocks in movement atrasan. In fact, the equivalence of all the systems in relative movement uniform is a postulate of the RE. Therefore it is not demonstrated. It is accepted and their consequences are developed. And their consequences are the formulas well-known (transformation of Lorentz) for times and distances. And perhaps if the formulas work is not so " capricious " the selection of the method of synchronism of the RE. In fact, I am convinced that this selection shows all the genius of Einstein. In order to establish this method he was able to objectively watch the behavior of the physical world and our methods of interaction with él.

Nevertheless the development of Einstein leads inevitably to paradoxical situations. And in the personnel I believe that the same formulas can be derived from transformation accepting that can exist privileged systems, and a physical structure of the universe that seems to mask the measurements. Let us remember the example developed in this page in which the movable observer accommodates the clocks to be able " to feel " stationary. In my own development of the transformations of Lorentz, I show how the Universe can do this in automatic form without responding " to whims " of the observers. For it it is necessary that the speed of the light in the emptiness is something very special, bound to the intimate structure of the universe and, therefore, affecting in direct form the march of all the clocks and preventing to detect the own movement in all the interactions of Roundtrip (the enormous majority of the physical interactions). As example of the expansion of the time for the systems in movement usually is mentioned the case of the muons that after forming in the superior layers of the atmosphere, These reach the terrestrial surface to a speed near " so quick cs ". mounes behave like possessors of an average life incredibly greater than those than form in conditions of laboratory, at moved away speeds of " c ". Is as if for muons in movement the time passed more slowly.

OK, but this does not demonstrate (as they affirm the enthusiastic ones of the RE) that the muons see us us as if our clocks retarded with respect to their own clocks everything what demonstrates east experiment is that the internal clocks of muons are slow with respect to ours, that are in rest with respect to our system of reference. Main conclusions of this page: The expansion of the time in systems in movement does not seem a too complicated concept. It is only necessary that the structure of the clock is affected by the movement. The expansion of the time leads naturally to the alteration of the lengths. If all the clocks are slow consistently (mechanical, atomic and biological clocks) it must have something in the structure of which is affected similarly by the movement. The conclusive experiments with respect to the expansion of the time in the systems in movement have been only observed from one only of the systems. The reciprocity of the observations has not been demonstrated.