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  spacetime:
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 of Time 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.
And in order, therefore, to maintain a universal constant speed of light in any light sphere in any direction by Einstein's special relativity proposition, longitudinal length contraction must be invoked!
More simply, length contraction is imputed in order to maintain a universal constant speed of light when determining time dilation in both Einstein's Special and General Relativity equations!
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.
Video: Time Dilation Experiment
source: "Time Dilation  An Experiment with Mu  Mesons", ©1962, presented by The Science Teaching Center of the Massachusetts Institute of Technology with the support of the National Science Foundation, demonstrated by Profs. David H. Frisch, M.I.T. and James H. Smith, University of Illinois. Video source: https://www.youtube.com/watch?v=2e9ltbbOwtc
Lorentz Transformation Rules Summary
Lorentz Inverse Transformation Equations Example^{∗}
^{∗} note : this example is used in the future upcoming Relativity Science 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 :
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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