-axis. Let the fees be fixed in the laboratory. A an unfavorable point charge
, who mass is
, is put at a point y that the
-axis close to the origin (so that
) and also is constrained to relocate only follow me the
-axis1 (Figure 1).

You are watching: Compared to clocks in a stationary reference frame, clocks in a moving reference frame run

The electrical field developed by the charges

in ~ the place of the charge
is

(7)

where

is the permittivity of space and
is a unit vector in the
direction. Since
(as we have the right to see from figure 1), we have the right to neglect
in the denominator the Equation (7). The pressure experienced by
,
, is climate essentially

(8)

This is a linear restoring force. Therefore our device of the three charges constitutes a an easy harmonic oscillator, and also the charge

oscillates through the period

(9)

Clearly, this system of the 3 charges may be considered to constitute a clock and also can be used for measure time in regards to the duration of oscillations

.

Let us currently assume the the 3 charges are put in a reference structure moving follow me the

-axis with consistent velocity
family member to the labo- ratory. The force
acting top top
as observed in the moving reference framework (where the charges
space at remainder now) is the exact same as that provided by Equation (8), except that y and also
are currently written v primes, that is

(10)

Figure 1. A negative point charge

oscillates along the
-axis under the action of two addressed positive allude charges
. This mechanism constitutes an primary school electro- magnetic clock.

To find the force acting ~ above

in the relocating clock together observed native the laboratory, we use Equations (2), (5) and (6) combined with Equation (10). Let united state assume the the velocity
of the moving reference structure is much bigger than the maximum oscillatory velocity of
. The velocity
of
loved one to the activities is then equal to
, and also its
component,
, is then simply
. Therefore Equation (2) becomes

(11)

from which us obtain

(12)

Substituting Equation (10) into Equation (12) and also taking into account that, by Equations (5) and (6)2,

and
, we attain for the force acting ~ above
in the relocating clock together observed native the laboratory

(13)

Taking right into account that the massive of the relocating charge

(actually the transverse mass, See, for example, H. Goldstein <3> ) is

(14)

and utilizing Equation (9) through

in it replaced by
, we obtain for the period of oscillations that the charge
in our moving clock #1 together observed from the laboratory

(15)

Comparing Equation (15) with Equation (9), we discover that

(16)

so the the rate of our moving clock #1 is longer by the factor

보다 its rate as soon as it is stationary.

Clock #2. This clock is the very same as clock #1 except that the field-producing fees

are now put along the
-axis at the point out
of the axis (Figure 2). The suggest charge
is again on the y axis close to the origin, therefore that
.

Clearly, when clock #2 is at remainder in the laboratory referral frame, the duration of oscillations the

is the exact same as that given by Equation (9) for clock #1, the is

(17)

Figure 2. A an adverse point charge

oscillates along the
axis under the action of two fixed positive suggest charges
. This mechanism constitutes an primary school electro- magnetic clock.

Let us currently assume the clock #2 is put in a reference frame moving along the

axis through velocity
loved one to the laboratory. As observed in that reference structure (where the charges
space at remainder now), the pressure
exhilaration on
is the very same as that offered by Equation (10), that is

(18)

To uncover the pressure acting top top the moving

as observed native the laboratory, we usage Equations (2), (5) and also (4) an unified with Equation (18). Together before, us assume that the velocity v of the relocating reference framework is much bigger than the preferably oscillatory velocity of
. The velocity
the
loved one to the laboratory is then again same to v, and also its
component,
, is simply
. Therefore Equation (18), just like Equation (10), becomes

(19)

from which us again obtain

(20)

Substituting Equation (18) right into Equation (20) us have

(21)

where us still need to transform

into
and
into
. This we perform with the aid of Equations (5) and (4).

First we notification that, by Equation (5),

(22)

To uncover the relation between

and
, we proceed as follows. Let united state designate the location of the 2
charges it was observed from the laboratory together
and
, wherein
describes the charge located to the best of the origin and also
refers to the charge located to the left of the origin. The distance between the 2 charges it was observed from the activities is then

(23)

Let the moment of observation in the activities be

(arbitrary time that observation). Making use of Equation (4), we have for the location of the two
fees measured in the moving reference framework in regards to their location ob- offered from the laboratory

(24)

and

(25)

The distance in between the two charges measured in the moving reference structure is then, by Equations (24), (25) and also (23),

(26)

and hence

(27)

Substituting

and also
in Equation (21), we because of this obtain because that the pressure acting ~ above the charge
in the relocating clock together observed native the laboratory

(28)

Taking into account the the mass of the moving charge

is

(29)

and utilizing Equation (17) through

in it replaced by
, we obtain for the period of oscillations the the fee
in our moving clock #2 together observed from the laboratory

(30)

Comparing Equation (30) with Equation (17), we uncover that for clock #2

(31)

so that, in comparison to Clock #1, the rate of the moving Clock #2 is dilated by the aspect

rather than by the factor
.

3. Discussion

The calculations simply presented expose two nature of the rate of moving clocks: 1) moving clocks run slower than the very same stationary clocks; 2) differently created (or oriented) synchronous stationary clocks may run at various rates and also therefore may become asynchronous when inserted into a relocating reference frame. Quantitatively, our very first clock behaves in accordance through the idea that the kinematic (relativistic) time dilation (the moving clock runs

time slower 보다 the stationary clock), but our second clock behaves in stark disagreement v the idea of the kinematic time dilation (the moving clock runs
times slower than the stationary clock). And yet, the price of the second clock is in accord through the fundamental relativistic change equations used in our calculations. To understand the actions of the 2nd clock, permit us take into consideration the electromagnetic forces responsible because that the work of the two clocks.

As is known, the electrical field developed by a moving suggest charge concen- trates itself around the aircraft perpendicular come the direction of motion and decreases along the heat of motion (see, for example <4> ). Therefore, as watched by a stationary observer, the electric force exhilaration on the fee

due to the electric field of the dues
in the moving clock #1 is various from the electric force exhilaration on the charge
due to the electric field of the dues
in the relocating clock #2. Likewise, the magnetic pressure acting top top the charge
(due to the magnetic field produced by the fees
in the moving clocks) is various in the two relocating clocks. Thus, conversely, the charge
in the stationary clocks #1 and also #2 is subjected to the same force, and the clocks are thus synchronous, in the relocating clock #1 the charge
is based on forces different from the pressures acting upon it in the relocating clock #2, and therefore the two moving clocks room no much longer synchronous (for a in-depth discussion and calculations that the electromagnetic forces acting in the in different way oriented and differently built clocks see <5> ). Because that this type of clocks us would like to denominate together Jefimenko’s non-Einsteinian clocks)3.

It may at very first appear that the principle of relativity is violated through the fact that synchronous stationary clocks may end up being asynchronous when moving. This is not so. Follow to the principle of relativity, it is difficult to tell whether a particular inertial reference structure is “actually” moving or is stationary. The truth that synchronous clocks may end up being asynchronous when inserted in a “moving” reference structure does no tell us which that the 2 frames is really moving, because the impact is reciprocal. It is straightforward to watch that if we started with synchro- nous clocks resting in the “moving” reference frame, then inserted them into the “stationary” recommendation frame and also used transformation equations express unprimed force components and also coordinates in regards to primed quantities, us would uncover that the clocks put in the “stationary” reference framework would no be synchronous as soon as viewed indigenous the “moving” recommendation frame. Hence placing synchronous clocks right into a various reference frame and also finding the the clocks come to be then asynchronous provide no details on even if it is or no one or the various other of these recommendation frames is “actually” moving or is stationary, i m sorry is in finish agreement with the rule of relativity.

There is, however, an essential implication an outcome as much as the proofs of the existence of the kinematic (relativistic) time dilation space concerned. The is currently generally believed that the existence of kinematic time dilation has been prove by actual experiment <6> <7> <8> . As has actually been currently pointed out by Jefimenko <9> , and as is clean from the calculations presented in this paper, the experiments the are taken as proofs of the fact of kinematic time dilation may have a straightforward alternative translate in terms of velocity- dependence forces present in the solution under consideration. Of course, we do not know what pressures are responsible for the decay of elementary school particles and also we know tiny about the pressures responsible because that the work of atom clocks. But there can be no doubt that the decay of elementary school particles, as well as the action of atom clocks, is controlled by some kind of forces. In the light of the calculations presented in this record and of similar calculations gift else- wherein <5> <9> , it is much much more natural and prudent to analyze the experiment allegedly proving the truth of kinematic time dilation together manifestations of the visibility of velocity-dependent forces and also interactions in the solution under consideration.

Moreover, it is currently clear the to prove the truth of kinematic time dilation that is crucial not just to demonstrate that relocating clocks run slower by the element

보다 the same stationary clocks, however it is additionally necessary come prove the the rate of the relocating clocks is not influenced by your mechanism, structure, orientation or by any other particular physical or geometrical residential property of the clocks.

In conclusion, in link with the above, us would favor to adduce here (or suggest to consider) a curious paradox:

Imagine the in the system

, i beg your pardon moves through a consistent velocity
loved one to the stationary system
, are mounted two clocks:
-clock (
-clock) Equation (16) and “
”-clock (
-clock) Equation (31). The clocks are mounted on a right line perpendicular come the right line
moves in the direction that the stationary system
, the clocks space equidistant from the line
and also let these clocks are synchronized in
At some distance native the clocks in the device
are mounted some “gimmick” with two sensors installed just like the clocks in
on a directly line perpendicular to the straight line
. Allow these two clocks at the same time at part arbitrary moment of time (in the mechanism
send light pulses in the direction of the equivalent sensors the “gimmick”. Allow a cat is put in this device. The sensors are created so the if they all at once (in the system
)4 receive the irradiate pulses, it motivated an automatic equipment which kills the cat in the “gimmick”. If the light pulses will not got by the sensors in ~ the same minute of time the the system
, the gear will not work and also the cat will remain alive.

It is apparent that

-clock and
-clock as soon as synchronized in the device
will remain synchronized in this mechanism constantly. Thus, from the point of view of the system
pulses emitted through
-clock and
-clock at the same time at some arbitrary minute of time in the device
to the sensors the the “gimmick” reach the corresponding sensors simultaneously! and thus the poor cat will be killed. However, native the approach of the system
-clock and also
-clock go with various rate. It means that the pulses are emitted in ~ the different moments the time in the mechanism
and also from different distances indigenous the device
and also will no reach the sensors simultaneously! Thus, indigenous the ideology of the device
the cat will remain alive! Let us name this cat together “Jefimenko’s cat” (by analogy through the well known Shroedinger’s cat).

As because that the “Jefimenko’s cat” yet it is not a “point the view”. According to our calculations, the non-synchronicity of clocks is a purely objective phenomenon. Its essence is the stationary systems and also moving systems are different physical systems, and so the physical phenomena occur in them in different ways.

NOTES

1The charge have to be constrained to stay on the axis since otherwise the is unstable with respect to a lateral displacement.

2Observe that

, being the
name: coordinates of
, transforms follow to Equation (6).