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the  Lorentz - FitzGerald Transformation Equations


Some Results of the  Lorentz Transformation Equations


midpoint flash

S-time interval 

I. Result 1 - clock rates:

From the above equation for  "time clocks - time interval" ( that is, time as observed in system  time clocks  ), 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  time clocks. Why? 

time clocks

Furthermore, since whenever

time clocks 

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

Lorentz length contraction

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

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  the  Lorentz - FitzGerald Transformation Equations  runs slower than the nearer clock 1 at  time clocks in relatively moving frame system  time clocks to an outside observer in stationary system  time clocks.

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

failure of time simultaneity

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

time clocks,

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

time clocks

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  length contraction  and frame system  length contraction moving away at relative velocity length contraction, we have

time clocks

and for a rigid rod fixed at

rigid rod

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

time clocks

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

Lorentz length contraction

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

Lorentz Factor

That is, for an observer in  Lorentz Contraction Factor,  a rigid rod in "moving away" frame system  Lorentz Contraction Factor will appear to shrink by an amount given by the Lorentz Factor, and equally for a relatively "moving away" system  Lorentz Contraction Factor  for a stationary observer in system  Lorentz Length Contraction Factor, 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.

Lorentz Length Contraction Factor

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!

the  Lorentz - FitzGerald Transformation Equations

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

In this case, let there be just one clock at, say, Lorentz Length Contraction Factor,  hence

Lorentz Length Contraction Factor

and assume time

Lorentz length contraction

then

Lorentz Length Contraction Factor

reduces down to

Lorentz dilated time

for observations of  the  Lorentz - FitzGerald Transformation Equations being made from  Lorentz dilated time.

Conversely this will also be true for the inverse

inverse time dilation

where observations of system  Lorentz dilated time  are being made from  Lorentz length contraction. 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

Lorentz dilated time

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

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 Science Calculator Mac application

An unknown particle, the  Lorentz - FitzGerald Transformation Equations , 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 0.99Lorentz dilated time together in a beam of ( other ) known particles.

(i). Determine the proper lifetime for the  Lorentz - FitzGerald Transformation Equations:

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,  Lorentz dilated time system.

Therefore by relativistic time dilation,

Lorentz dilated timeAlbert Einstein published his "On The Electrodynamics of Moving Bodies" in 1905 wherein he incorporated the earlier mathematics of Irish George FitzGerald and Dutch Hendrik Lorentz

(ii). Determine distance travelled for  Lorentz dilated time:

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

Lorentz Inverse Transformation Equations

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

Lorentz length contraction

since because we are imagining as moving coincident ( actually imagining as being stationary ) with  the  Lorentz - FitzGerald Transformation Equations,  therefore we're ( relatively ) stationary with  Lorentz dilated time and  Lorentz dilated time,  hence  Lorentz dilated time.

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

Lorentz dilated time

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

Lorentz dilated time

Finally,

Lorentz length contraction

Notice that

Lorentz length contraction

which means that any ( atomic ) clock "attached" to  Lorentz length contraction  will move slower as seen ( i.e., measured ) by an observer in stationary system  Lorentz length contraction 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

Lorentz length contraction

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

Lorentz length contraction

That is,

Lorentz length contraction


the  Lorentz - FitzGerald Transformation Equations

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