Some Results of the  Lorentz Transformation Equations

 

I. Result 1 - clock rates:

From the above equation for  " - time interval" ( that is, time as observed in system    ), it is obvious that the units of time for clock 2 will be greater as compared to units of time for clock 1 in moving system  . Why? 

Furthermore, since whenever

 

clock 2 is much farther away from clock 1 in moving frame system   -  the slower clock will be the clock at than at to an outside stationary observer in frame  , and hence an

will also be greater signifying a slower rate of time passing in frame system   as seen by an observer in frame system   !! 

Ia. Corollary - space-time:

The greater the distance separating clocks 1 and 2, the slower will be the rate of time passing to an outside stationary observer! 

This phenomenon has already been demonstrated as for when the further distant clock 2 at    runs slower than the nearer clock 1 at   in relatively moving frame system   to an outside observer in stationary system  .

The conclusion is therefore inescapable: time is dependent on space as these Special Relativity equations demonstrate!!

II. Result 2 - "The Failure of Simultaneity at great distances":

Whenever

And for ever greater separating distances for clocks 1 and 2,

,

there will be ever greater disparities in time of light received respectively at clocks 1 and 2 for a stationary observer in frame system as shown by

.

In other words, in physical reality there is "no simultaneity of clock events" when either great distances or great velocities of clocks are involved relative to a stationary observer!!

III. Result 3 - Length Contraction: 

Again, as between stationary system    and frame system   moving away at relative velocity , we have

and for a rigid rod fixed at

in the "moving away" frame system  , we have length

Now for a moving observer in   at some arbitrary time, , we therefore have

Therefore in stationary frame systems, , the rigid rod will appear to shrink in the longitudinal - axis direction by the inverse of the Lorentz Factor

That is, for an observer in  ,  a rigid rod in "moving away" frame system   will appear to shrink by an amount given by the Lorentz Factor, and equally for a relatively "moving away" system    for a stationary observer in system  , this same rod will also appear to be contracted!! It's all relative! And it's called reciprocal length contraction.

This contraction effect is called the Lorentz Contraction Effect.

IV. Result 4 - Time Dilation ( time interval increase ):

In this case, let there be just one clock at, say, ,  hence

and assume time

then

reduces down to

for observations of   being made from  .

Conversely this will also be true for the inverse

where observations of system    are being made from  . This type of time dilation for non - accelerating, inertial system motion is mutually reciprocal which precludes "The Twin Clock Paradox" construct.

Lorentz Transformation Rules Summary

Rule 1:  Every clock will appear to go at its fastest rate when it is at rest relative to the observer; hence, any motion relative to an observer slows the apparent rate of any clock.

Rule 2:  Every rigid rod will appear to be at its greatest longitudinal extent when it is at rest relative to the observer, whereas transverse or perpendicular extants relative

to the direction of motion are always uneffected. Therefore any longitudinal motion relative to an observer shrinks any rigid rod in the direction of motion by an amount

given by the Lorentz factor.

Lorentz Inverse Transformation Equations Example^∗

^∗ note : this example is used in the future upcoming Relativity Calculator Mac application

An unknown particle, , appears and then disappears with a "lifetime" of 1.80 x 10^{- 8} sec in a particle accelerator such as CERN's LHC ( Large Hadron Collider ) and is observed during it's "lifetime" to have a concurrent velocity of 0.99 together in a beam of ( other ) known particles.

(i). Determine the proper lifetime for :

The proper lifetime ( proper time ) is the lifetime of the particle measured by an observer moving coincident with the particle in the particle's own frame of reference,   system.

Therefore by relativistic time dilation,

(ii). Determine distance travelled for  :

The particle accelerator resides in ( actually comprises ) stationary system  ,  therefore we use

Since we are trying to determine distance travelled in our particle accelerator, or stationary    system, we must place ourselves as "observers" exactly within the particle's own frame of reference (  that is, relatively moving system   ),  so as to recreate the coordinates for the appearance and disappearance of the     particle in CERN's LHC ( Large Hadron Collider ):

since because we are imagining as moving coincident ( actually imagining as being stationary ) with  ,  therefore we're ( relatively ) stationary with   and  ,  hence  .

Going back to our stationary system  , the particle accelerator, from which we are making these "outside" ( external ) observations, Lorentz Inverse Transformation Equations give

Hence relative to stationary    or particle accelerator, our Lorentz Inverse Transformation Equations become

Finally,

Notice that

which means that any ( atomic ) clock "attached" to    will move slower as seen ( i.e., measured ) by an observer in stationary system   as compared to an observer of time "attached" to this particle;  this is the meaning of time dilation or time interval expansion.

(iii). If time dilation did not exist in nature:

This means that hypothetically the speed of light is infinite or instantaneous, in which case

Finally and continuing with the hypothetical that time dilation is not a reality in nature, also assume that    possesses a velocity considerably less than c, speed of light, or  ,  then

.

That is,

.

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