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Some Results of the  Lorentz Transformation Equations

 

I. Result 1 - clock rates:

From the above equation for "S-time interval" 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 S' . Why? 

Furthermore, since whenever

 

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

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

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 S'  to an outside observer in stationary system S.

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 S 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 S and frame system S'  moving away at relative velocity v, we have

and for a rigid rod fixed at

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

Now for a moving observer in S'  at some arbitrary time, t, we therefore have

In both cases or rather in both frame systems, S and S', the rigid rod will appear to shrink in the longitudinal x ( x' ) - axis direction by the inverse of the Lorentz Factor

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

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 S'  being made from S.

Conversely this will also be true for the inverse

where observations of system S are being made from S'.

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.

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