1. Introduction

       Many scientific, as well as popular scientific, works are dealing with the imaginary twin paradox (clock paralogism). However, its real physical essence is not fully revealed in any of them. Usually this paradox is explained by the fact that one of the twins moves at a constant velocity all the time, while the other twin at particular points of time performs accelerated movements. Such explanation points out inequivalence of conditions of motion of the twins. It does not explain why the age of the second twin is always less than the age of the first one, independently on length of the path they have passed at a constant velocity and, consequently, independently on values of age differences accumulated in the IFR of every twin at the process of this uniform motion. In fact, identical finite differences in age of the twins are to appear in all "thought experiments" with identical world lines (WL) of accelerated motion of the second twin because of this accelerated motion. And the age differences, accumulated in the process of uniform motion, in the contrary to these finite differences of age, can amount to arbitrary big value in IFR of any of the twins. And therefore, they still will lead to mutually contradictory information about the age of the twins. The reveal of the physical essence of the imaginary twin paradox is the aim of this article.

2.   The initial causes of the twins paradox

      As it is shown in [1], GR actually admits the possibility of existence of separate FR - that is the FR of the physical vacuum (PV), in which relict radiation frequency is isotropic. We will consider motion of objects in this PVFR, space and time of which according to Newton [2] are absolute. However, of cause, according to the principle of relativity we could as well take any IFR as basic FR. WL of uniform straight-line motions of two objects in the absolute space are shown in the picture. The first one is moving at the absolute velocity . The second one is withdrawing from the first one at the relative velocity of , and then drawing closer to it at the relative velocity of .

      and here are the absolute velocities of motion of the second object correspondingly in direct and reverse direction. At this, for simplification of mathematical expressions it is assumed that distances and spatial coordinates are measured in light units of length. Therefore, the eigenvalue of the velocity of light is: .

  Fig.:  1 — WL of the first object; 2 — WL of the second object at the time of its withdrawing from the first one; 3 — WL of the second object at the time of its drawing closer to the first one; 4 — WL of the light.

     Let in the PVFR the first object comes to the point simultaneously with coming of the second object to the point. And the proper (standard) time of motion of the second object from the point to the  point is . Then an interval of absolute time, corresponding to this proper time and read in the PVFR from the moment of coming of the first object to point and the second object to the point will be the following: . The point coordinate, read from the point, will be determined at this by the following dependency: , where .

     Intervals of absolute time between events in the point on the first object and in the point on the second object, which are simultaneous in the IFR of the second object, depend on the value of velocity of the second object in the point:.

here is a coordinate of the first object, observed in the IFR of the second object.

     Since , then:

,   (1)

and:

.   (2)

    Therefore, depending on the value of absolute velocity of the second object in the point, events corresponding to various positions of the first object in the absolute space will be simultaneous with the event in the point of the FR of the second object. So, correspondingly, at :

;

at :

,        ;

at :

  ,

      Let the modules of relative velocities of motion of the objects in the process of their withdrawing and drawing closer are equal to each other . Then changing of position of the first object by the second twin will not be observed at the moment of changing of the direction of motion by the second object. However a transition from simultaneity in the second twin FR with the moment of change of its motion of some events to simultaneity of other events on the second object corresponding to other position of the latter in the absolute space will happen at this: . That is, at a transition of the second object from the motion at the velocity to the motion at the velocity a change of positions of the first object considered as simultaneous with the position of the second object in the point realizes. In that way, a drop of coordinate time (which is observed in second twin FR), corresponding to the events on the first object, occurs:

,     (3)

where:

.

      And consequently, an exception from the consideration of a part of path-like proper time of the first object, determining age of the first twin, takes place. Therefore, the second twin comes to a wrong conclusion about decreasing of total time that has run out on the first object from the moment of separation to the moment of meeting of the twins. This determines the physical essence of imaginary twin paradox (paralogism).

3.   Results of direct observations

Considering the drop of coordinate time, full path-like proper time of the first object, observed by the second twin, will be the same as in the FR of the first object:

,      (4)

here is time duration of the motion of the second object in the reverse direction by its proper clock, and: . Presence of the drop of proper time of the first object ("observed" by the second twin mediately through its two IFRs) does not at all mean that information about events, which have occurred on the first object between the and points, does not come to the second object. At the moment of changing the direction of the second object information about an event, which has happened on the first object at the moment of time when it was in the point at some distance from the point, arrives to it:

,      (5)

    Immediately after changing the direction of the second object a displacement of radiation spectrum of the first object, observed by the second twin, will also change. This can lead to a false conclusion of the second twin that the first object was withdrawing from it only during the time:

,     (6)

and is approaching it during time. Therefore, the full time of objects’ drawing closer will be evaluated by it this way:

,    (7)

       Considering this, proper time intervals of the first object corresponding to mutual drawing closer and withdrawing of the objects will be regarded by the second twin with the following values:

,      (8)

,           (9)

       This, of cause, does not correspond to the values, observed in the FR of the first object. However, this disagreement is explainable by incorrectness of the definition (made from a false premise about the change of direction of motion by not the first but by the second object) by the second twin of the moment of stoppage of withdrawing and starting drawing closer of the objects by the first objectТs clock. In spite of this, the total value of proper (standard) time of the first object, evaluated by the second twin, will be the same as it is observed in the FR of the first object:

.

     And consequently, information about all events occurred on the first object arrives on the second object.

      Because of the motion of the second object in the direct and reverse direction at different absolute velocities, shrinkage of distances to the objects before and after the change of its motion will be observed as dissimilar by the second twin. At change of the distance to the point leads to mutual pseudo-superposition of time intervals and by the clock of the second twin counting standard [3] (path-like) time. This mutual pseudo-superposition of time intervals is caused by the withdrawing of the first object from the position with the coordinate to the position with the coordinate at the velocity more than the velocity of light at the point of observation. "Flow of time back" concerned to the transition of the second object from one IFR to another IFR will take place at such an "observation" (mediately through the two IFRs) no matter how smoothly the transition from to will realize. Direct observation, as it was shown earlier, does not find out this. The given pseudo-effect is concerned to the calculation of and values on the basis of supposition about similarity of improper (coordinate) values of the velocity of light in all intrinsic space of the second object, moving noninertially in the process of transition from to . In fact this is not right. Improper values of the velocity of light in the points of presence of the first object in the process of its transition from the distance to the distance can not be less than the velocities of displacement of the first object in the FR of the second object. And these velocities noticeably exceed the velocity of light in the point of observation of radiation spectrum displacement because of the fast change of relativistic shrinkage of the distance to the first object in the FR of the second object.

     Considering the change of improper value of the velocity of light in the intrinsic space of the second object in the process of its noninertial motion, time superposition in the intrinsic FR of the second object will not be observed. Standard time, determined in this FR from the quantity of wavetrains having come from the source of standard radiation of the first object will concur with its value, determined by a clock, motionless relatively to the first object.

4.  Conclusions

     The physical essence of imaginary twin paradox (clock paralogism) lies in neglect of necessity of re-calculation of time coordinates of events at a transition from one IFR to another. To avoid similar paralogisms it is necessary to consider that improper (coordinate) values of the velocity of light [3] in FR of accelerating objects can arbitrarily exceed the eigenvalue of the velocity of light, which is a gaugeЦinvariant quantity [1].

Reference list

[1].   Danylchenko P., Gauge foundations of special relativity, in: Gauge-Evolutional Interpretation of Special and General Relativities, Vinnitsa, O. Vlasuk, 2004, p.15а

(read, http://pavlo-danylchenko.narod.ru/docs/Foundations_Eng.html

download http://pavlo-danylchenko.narod.ru/docs/Foundations_Eng.rar)

[2].   Newton, I., Philosophiae Naturalis Principia Mathematica, London, 1686, revised by A. Cajori, Sir Isaac NewtonТs Mathematical Principles of Natural Philosophy and His System of the World, Univ. of California Press, Berkeley and Los Angeles, 1934, paperback, 1962

[3].   Möller C., The Theory of Relativity, Oxford: Clarendon Press Oxford, 1972

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