Physical essence of
twins paradox*
Danylchenko P.
SPE УGeoSystemФ, Vinnitsa, Ukraine
Contact with author: pavlo@vingeo.com
The initial cause of
unambiguously younger age of the second twin than age of the first twin at the
moment of their meeting is pointed. This initial cause is not the accelerated
motion of the second twin, but the fact of changing of its direction or just
the velocity of its motion in space itself and, consequently, the fact of its
transition from one inertial reference system of spatial coordinates and time
(IFR) to another. This is concerned to changing in new IFR of spatial as well
as of time coordinates of events, which have realized before, including events,
information about which have not come to the twin by the moment of its
transition to new IFR. It is shown that imaginary
twin paradox (clock paralogism) takes place in general relativity (GR) only because of the impossibility of mutual
distinguishing of standard time (path-like
proper time of moving object) and coordinate-like internal time of the IFR (or
any other FR) and because of neglect of the necessity
of re-calculation of events time coordinates as a result.
1.аааааа 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 Уthought experimentsФ 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.аааааа
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 
. 
аand 
аhere 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: 
.
ааааааааа Fig.:аа 1 Ц WL of the first object; 2 Ц WL of the
second object at the time of its withdrawing from the first one;ааааа 3 Ц WL of the second object at the time
of its drawing closer to the first one; 4- WL of the light.
ааа
Let
in the PVFR the first object comes to the 
аpoint simultaneously
with coming of the second object to the 
аpoint. And the proper
(standard) time of motion of the second object from the 
аpoint to the 
аpoint 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 
аpoint and the second
object to the 
аpoint will be the
following: 
. The 
аpoint coordinate,
read from the 
аpoint, will be
determined at this by the following dependency: 
, where 
.
Intervals of absolute time
between events in the 
аpoint on the first
object and in the 
аpoint on the second
object, which are simultaneous 
аin the IFR of the
second object, depend on the value of velocity 
аof the second object
in the 
аpoint: 
. 
аhere is a coordinate
of the first object, observed in the IFR of the second object.
Since 
, then:
,ааа (1)
and:
.аааааааа (2)
аааа Therefore, depending on the value of absolute velocity of the second
object in the 
аpoint, events
corresponding to various 
аpositions of the first
object in the absolute space will be simultaneous with the event in the 
аpoint of the FR of
the second object. So, correspondingly, at 
:
;
at 
:
,аа 
;
at 
:
а,
ааа 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 
аat 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 
аvelocity to the
motion at the 
аvelocity a change of
positions of the first object considered as simultaneous with the position of
the second object in the 
аpoint 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:
,ааааа (3)
where:
.
ааа 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). аааааааа
3.аааааа 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:
, аа (4)
аhere 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
(УobservedФ 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 
аand 
аpoints, 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
аpoint at some
distance from the 
аpoint, arrives to it:
,аааааааааааа (5)
ааа 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:
,аааааааааааааа (6)
and is approaching it during 
аtime. Therefore, the full time of
objectsТ drawing closer will be evaluated by it this way:
,аа аааа (7)
ааа 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:
,аааааааааааааааааааа (8)
,аааааааааааааааааааааааа (9)
аThis, 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.
ааа 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 
аchange of the
distance to the 
аpoint 
аleads to mutual
pseudo-superposition of time intervals 
аand 
аby 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 
аcoordinate to the
position with the 
аcoordinate at the
velocity more than the velocity of light at the point of observation. УFlow of
time backФ concerned to the transition of the second object from one IFR to
another IFR will take place at such an УobservationФ (mediately through the two
IFRs) no matter how smoothly the transition from 
аto 
аwill realize. Direct
observation, as it was shown earlier, does not find out this. The given
pseudo-effect is concerned to the calculation of 
аand 
аvalues on the basis
of supposition about similarity of improper (coordinate) values of the velocity
of light 
аin all intrinsic
space of the second object, moving noninertially in the process of transition
from 
аto 
. 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 
аdistance to the 
аdistance 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.
ааа 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.аааааа 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]. а 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]. а 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]. а Möller C.,
The Theory of Relativity, Oxford:
Clarendon Press Oxford, 1972
*а
Supplemented and completed version of the article from the collection of
articles УGauge-Evolutional Interpretation of Special and General RelativitiesФ,
Vinnitsa, O. Vlasuk, 2004