18. DILATATION (CONTRACTION) OF TIME
 
   Classical physics, with Newton at the head of considers time to be the absolute value which flows "continually, evenly and independently of anything else". In 1895 Lorentz introduced the concept of local time into physics, and in 1905 Einstein gave this a completely new interpretation.
   While working on his transformation, Lorentz came to the conclusion that the hypothesis of space contraction was not sufficient, and in 1895 he offered another which was as amazing as the previous one: "In systems which move with uniform translation a new measure for time is necessary". The new hypothesis was necessary so that electromagnetic phenomena in the moving systems would be the same as in the ether. Both hypotheses indicate that space and time have to be measured in different ways in the quiescent ether and in systems which move relative to that ether. In this way time was relativised, changing at transition from one system to another . Lorentz called the new time local time, and treated it as an auxiliary mathematical magnitude, not as absolute time.
   Einstein asserts that there is no means by which make possible to determine the existence of absolute time and its differentiation from the infinite number of local times in systems of reading which move relatively. According to him, time is connected to space, to bodies, and flows differently in different systems, in some places slower and in others faster. How time will flow depends on the relative speed of motion. It flows slower in motion and faster at rest. An important conclusion of the theory of relativity is that time dilation occurs with motion.
   As there were some remarks on the old way of determining the contraction of space there are also criticisms of the method of determining the dilatation of time. Before approaching a determination of the contraction of time in a new way, we will carry out an analysis of the determination of the time dilatation according to the special theory of relativity and the scientific literature. As before, we will use the equations of four chosen coordinate transformations for analysis.
 
18.1 Dilatation of time according to the special theory of relativity
 
   In the special theory of relativity [6] on the subject of time dilatation, Einstein says the following:
   Quotation: "Let us observe the clock which shows seconds and which is always at rest at the origin ( = 0) of the system . Let = 0 and = 1 be two successive strikes of this clock. For both these strikes the first and fourth equation of the Lorentz transformation give

(18.1)

   If it is measured in the system , then the clock moves at a speed , and between its two strikes, measured from the same reference body, passes not one but

seconds, thus a somewhat longer time. In consequence of motion, the clock runs more slowly than when it is at rest. In this case also velocity likewise plays the role of the unattainable speed." End of quotation.
   This is all there is about the dilatation of time in the special theory of relativity.
   The first and the fourth equations of the Lorentz transformation (10.11), solved for yield

(18.2)

   Einstein first takes in Eq. (18.2) that = 0 and = 0 which is correct, and then he takes = 0 and = 1 which must not be used, because he himself demands by Eq. (15.3) that . He says the same in deriving the Lorentz transformation in Eq. (10.2). So, when =0 must be =0.
   If Einstein had kept to the conditions under which he derived the Lorentz transformation, and if he had taken into consideration that the time in system , expressed by means of coordinates and of the system is given by the Eq. (18.2) and that it is, according to the second principle of the theory of relativity, always he would have had to derive the coefficient of the dilation of time in this way

(18.3)

   From this it necessarily follows that, between two strikes of the clock, in system pass not one but

seconds, and not as Einstein asserts

seconds.
   Einstein's derivation of the proof of the relativity of time and the magnitude of the dilation of time contradicts his assertion that the time in system depends on the coordinate as well, that is on the position of the clock in the system . So, if one chooses not to abide by the principles of the theory of relativity and not to respect the conditions under which the Lorentz transformation is derived, then he can, following Einstein's way, derive a "proof" that between two strikes of the clock which is in motion, any number of seconds may pass in system , while in system , where the clock is at rest, only one second passes.
   For example, let us assume that the clock is not at the origin of system but at some point . Then in system , according to Einstein's procedure cited above

seconds would pass between two strikes of the clock instead of

seconds, when the clock is at the origin of system , that is when = 0.
   In this way, by choosing the position of the clock in system , in effect by choosing the value of constant , that is , it can be proved that, in system , in which the clock is moving, any number of seconds pass between two strikes of the clock.
   From the above it follows that Einstein's derivation of the relativity of time and its extent (the coefficient of dilation) is incorrect. The relationship between the time , which passes in the system at rest and the corresponding time which passes in the moving system is given by the relation

(18.4)

   Before going further, some explanations are necessary.
   The condition of time and space (dilatation or contraction) in any coordinate system are independent of the fact whether and from where someone is observing them.
   In the system during the motion of the light wave, there is neither contraction nor dilatation, of both space and time. However, in the system which "pursues" the light wave, the contraction of space and time appears, but only in a mathematical sense. If that system were to reach that wave, that is, if the speed of the system were equal to the velocity of light, then the space would disappear, or more exactly put, the interstice between the origin and the wave front would disappear. Then the time disappears as well. This has been explained earlier on. For all that, nothing has happened in system . Therefore it is more reasonable to observe the condition in the system than in the system .
   Einstein analyses phenomena in relation to system . As a result the dilation of time is, according to him, in system , instead of contraction of time in system , where it in fact occurs, at least in a mathematical sense.
 
18.2 Dilatation of time according to the scientific literature
 
   In case of the Lorentz transformation further presentation of the contraction - dilatation of time is based on the literature [10], where we must keep in mind that the procedure and final result is the same with the other authors.
 
18.2.1 Dilatation of time according to the Lorentz transformation
 
   Here is how the dilatation of time has been treated in the literature [10]:
   Quotation: "Einstein's explanation of the Lorentz transformation for time, shows that time flows differently in different coordinate systems, in some places faster, and in some place slower, because absolute time does not exist. It is easy to show this by taking the corresponding relation for time

(18.5)

   In order to determine the time interval in the inertial system , which moves with uniform translation relatively to the system , we will take a certain process which is of course realistic. Let the beginning of the process in the system be at the moment , and the end of the process at the moment . Then the process in the system has lasted for time . This interval in the system corresponds to a certain interval in the system . Since the moment in the system corresponds to the moment in the system and moment corresponds to the moment , that this process observed from the system will last for time . But, since according to Einstein time depends on the position and not only on the speed, as is seen in Eq. (18.5), it can be taken that, observed from , the beginning of the event has happened in the point of the abscissa of the system , and the end in the point of the abscissa . In the system the process takes place in one place. Then it is clear that between the distance , and the time interval , there is a relation

because the body (), in which the process takes place, has been moved for that distance at speed observed from .
   According to Eq. (18.5) will be

and from there

So,

(18.6)

   This important relation shows that

(18.7)

that is, the time interval in the system which is connected to the process, whose duration is measured, is smaller than the time interval for the same process whose duration is measured from another system with mutual motion. It can be seen that one second in the system corresponds to seconds in system .
   This means that the process is slower in the system than in the system . From this we reach a conclusion about the clock, that is, the time flow register. It turns out that the clock functions more slowly in the system in relation to which the clock moves, that is, the clock functions slower when it is moving, than when it is at rest. In other words, time, connected to a body, flows slower in motion than at rest. Motion causes the dilatation of time. This is a very important conclusion of Einstein's theory of relativity." End of quotation.
   In the following three transformations the same procedure will be applied in order to determine the dilatation of time, without comments in detail.
 
18.2.2 Dilatation of time according to transformation No. 2
 
   In this case of transformation time is given, in the coordinate system , by Eq. (12.22) as follows

(18.8)

so

and

As in the previous case, that is after substitution we obtain

(18.9)

Since then from Eq. (18.9) we have

(18.10)

   The dilatation of time is in the system , as in the case of the Lorentz transformation, but magnitudes of these dilatations are different. So, for example, if then the dilatation coefficient in case of Lorentz transformation is , but in case of this transformation . As can be seen the difference is big.
 
18.2.3 Dilatation of time according to transformation No. 4
 
   For this transformation, time in the system is given by Eq. (12.24), as follows

(18.11)

and from there

So, by subtraction we obtain

(18.12)

which results

(18.13)

   The dilatation of time appears in the system also, as in the two previous cases, but the dilatation coefficient is considerably larger. For example, if then the dilatation coefficient is 20.
 
18.2.4 Dilatation of time according to transformation No. 5
 
   For this transformation, time in the system is given by Eq. (12.25) as follows

(18.14)

so

After substitution and subtraction yields

that is

(18.15)

so

(18.16)

   The dilatation of time is the same as in the previous case.
 
18.3 Checking the correctness of determining the contraction of space and dilatation of time
 
   Earlier on, it was said that the correctness of the method of determining the contraction of space and dilatation of time, would be checked. This checking is done by dividing the length interval with the corresponding time interval in the corresponding coordinate system. If the method of determining the contraction and dilatation is correct, then the quotient will have to be the velocity of light because the theory of relativity is based upon it.
   The checking is done for all four treated transformations.
 
18.3.1 Checking in case of the Lorentz transformation
 
   The interval of the length is given by Eq. (17.6)

and the interval of time by Eq. (18.6)

so, by division we have

(18.17)

18.3.2 Checking in case of transformation No. 2
 
   The length interval given by Eq. (17.14) is

and time interval by Eq. (18.9)

so, by division we have

(18.18)

18.3.3 Checking in case of transformation No. 4
 
   The length interval given by Eq. (17.19) is

and time interval by Eq. (18.12)

so, by division we obtain

(18.19)

18.3.4 Checking in case of transformation No. 5
 
   The length interval given by Eq. (17.21) is

and time interval by Eq. (18.15)

so, by division we obtain

(18.20)

   Finally, we can say that in all four cases of transformations is proved that the quotient which is obtained by division of the length interval by the corresponding time interval is not equal to the velocity of light, which is explicitly required by the theory of relativity, because this theory is based on that. The only possible conclusion is that the way of calculating the contraction of space and dilatation of time is not correct.
 
18.4 A new way of determining the contraction of time
 
   Here we say contraction of time instead of dilatation of time, because, as we said earlier on, the contraction of time in a mathematical sense really arose, but in the moving coordinate system .
   All four transformations will also be treated here as in the previous case. The new way of determining the contraction of time in the system is based on the substitution of , that is and . The old way as we saw is based on the substitution of , that is and what is contrary to the second principle of the theory of relativity.
 
18.4.1 Contraction of time according to the Lorentz transformation
 
   In this transformation time in the system is given by the equation

(18.21)

and from there

After substitution we obtain

that is

(18.22)

so

(18.23)

18.4.2 Contraction of time according to transformation No. 2
 
   Time in system is given by equation

(18.24)

that is

From there and after substitution and we have

(18.25)

and

(18.26)

18.4.3 Contraction of time according to transformation No. 4
 
   Time in system is given by equation

(18.27)

and from there

After substitution and and subtraction we obtain

(18.28)

so

(18.29)

18.4.4 Contraction of time according to transformation No. 5
 
   In this case time in system is given by equation

(18.30)

and from there

After substitution and and subtraction we have

(18.31)

so

(18.32)

   So, the contraction of time for all four used transformations, in a new way of determining the contraction, happen in the system . However all these contractions are different.
   Let us we see now, what will happen if we express by means of and and after that let us find out where the contraction of time is. Therefore we shall take the time from the Lorentz transformation using Eq. (12.20)

that is

and after substitution and and subtraction we obtain

This equation is the same as Eq. (18.22), that is we obtain the same result as when we expressed by means of and .
   Finally we can conclude that the time contraction always appears in system independently of the type of coordinate transformation and does not depend on where the system is being observed from, like the contraction of space, which is quite logical, and confirms the correctness of a new way of determining the contraction of time. But, here we must remember that the magnitudes of time contraction also depend on the type of the transformation.
 
18.5 Checking the correctness of the new way of determining the contraction of space and time
 
   Since the procedure is already known, a shortened procedure of checking has been given for each transformation.
 
18.5.1 Checking in case of the Lorentz transformation
 
   According to Eq. (17.25) is

and according to Eq. (18.22) is

so, by division we obtain

(18.33)

18.5.2 Checking in case of transformation No. 2
 
   According to Eq. (17.29) is

and according to Eq. (18.25) is

so, by division we obtain

(18.34)

18.5.3 Checking in case of transformation No. 4
 
   According to Eq. (17.33) is

and according to Eq. (18.28) is

so, by division we obtain

(18.35)

18.5.4 Checking in case of transformation No. 5
 
   According to Eq. (17.37) is

and according to Eq. (18.31) is

so, by division we obtain

(18.36)

   These checks show that the new way of determining the contraction of space and time is correct, because in all four cases of the transformation treated, by dividing the length interval with the time interval, both in system and , the velocity of light was obtained.
   At the end, in regard to time contraction it should be said that even in the new correct procedure of determining the contraction of space and time different values for the different coordinate transformations are obtained at the same speed of the system relatively to the system . From this it can be concluded that time contraction cannot be connected to the duration of some realistic physical process or state. The real duration of some process cannot depend on the mathematical procedure of the coordinate transformation. Neither can time depend on it. So the time we obtain can only be some conditional or local time as Lorentz called it.
   The contraction of time is a mathematical concept related to the motion of the light wave or acoustic wave which is followed from two inertial systems, under the condition that the speed of the wave in both systems is equal to the velocity of light, or, to the speed of sound, when sound is in question.
   If the coordinate system is the system of reference, where time passes normally, then the countdown of time ("ticking of a clock") in system is slowed relatively to the countdown of time in system . Because of this we should rather talk about time contraction in system than the dilatation of time in the system . Simply said, we can talk about time and space contraction in the coordinate system which moves with uniform translation relatively to the other coordinate system and under the condition that the system moves in the direction of the light wave - ray motion.
   Up till now the event of contraction - dilatation of time and space has been considered only in the case of motion of the coordinate system in the same direction as the light wave. This was done because Lorentz did the same. In such an approach in the analysis it has been discovered that in system contraction of time and space occurs regularly, no matter if it is a plane or spherical wave, and what the transformation coordinate is.
   However, with motion of the system in the opposite direction of the light ray (or wave) motion, which has the same validity as the previous direction, a contrary state occurs. In system , instead of the earlier contraction we obtain the dilatation of time and space. This can be easily shown by the procedures already used, but on the basic of the new transformations No. 1 and 3. It must be born in mind that the new coordinate transformations also satisfy the requirement for invariability of equation for propagation of electromagnetic waves, same as Lorentz. As such they are equal to the Lorentz transformation, that is they have the same validity as the Lorentz transformation. Because of that, it is impossible to claim in advance what, and to what degree, will happen in motion, contraction or dilatation, not even in mathematical sense. This is even more evident with the application of the following transformation of coordinates, which also satisfy the requirement for invariability, as does the Lorentz transformation.

(18.37)

as well as

(18.38)