# The Lorentz Contraction, or, How Motion Affects Space (Part 3)

In Part 2, we were able to deduce, using only special relativity's two fundamental postulates, that as observed from * our* point of view, Scotty's (relatively) moving Photon Clock H must indeed contract in length, that is to say, contract

*along the direction of relative motion*by a factor of

*sqrt(1 - v*, in order to keep its original synchronization with his Photon Clock V and thus uphold the

^{2}/ c^{2})*principle of relativity*. We will now discuss some consequences of this astonishing distance contraction phenomenon, formally known as the

*Lorentz Contraction*.

## The Lorentz Contraction: What Can We Conclude From All Of This Stuff?

** We** had initially believed that from

**point view, we would observe Scotty’s**

__our__*Clock H*

**with his**

*fall out of sync**Clock V*, because of

**that**

__our__erroneous prediction*t*>

_{H}*t*. But we need not be too harsh on ourselves, for this error was a

_{V}**of**

*direct result***(very rational, reasonable, and SANE)**

*our***that**

*assumption*

*Clock H’s length*__would__*, as observed from*

**remain unchanged****by the relative motion of Scotty’s train****perspective! But,**

__our__**to our**

*contrary***, we must instead face the**

*flawed assumption***that: From**

*reality***point of view standing at the train crossing,**

__our__**will observe Scotty’s**

*we**Photon Clock H*to be

**, or**

__contracted__**, along the horizontal direction, that is,**

__shortened__**. Thus, as his T.O. Special thunders through the crossing at the speed**

*along the*__direction__of his train’s__relative motion__*v*

*relative to*,

__us__**will observe that**

*we*where *h _{C}* is the

*contracted*, or

*shortened*length of his

*Photon Clock H*, as shown in

*Figure 1.3*, and

*h*is its

*normal*length, as shown in

*Figure 1.2*. And this

**amount of shortening of**

*necessary**Clock H’s*length will

**that**

*ensure**t*=

_{H}*t*and

_{V}

__NOT__*t*>

_{H}*t*, as measured by

_{V}**using**

__us__**watches, thus keeping**

__our__*Clock H*in synchronization with

*Clock V*as observed from

**point of view, for**

__our__

__ANY__*relative speed*

*v*, and hence upholding the

*first postulate*, that is, the

*principle of relativity*.

Stating it less formally, one final time: Because of his train’s motion *relative to us*, then as seen from

**perspective, Scotty’s**

__our__*Clock H*will become

**than its original, that is, its**

*shorter***length in the horizontal direction by the factor**

*normal*so that the *total distance* that its photon must travel to complete 1 tick is ** decreased** to the

*same**total distance*that

*Clock V’s*photon must travel to complete 1 tick. And this shortening of the length of

*Clock H*will therefore keep both clocks, as observed from

**point of view, in sync with each other, as they**

__our__**be.**

*must**Figures 1.1*,

*1.2*, and

*1.3*comprise a clear and concise summary of Einstein’s logic behind this truly astounding result.

** Moreover**, since there is absolutely

**special, magical, or supernaturally spooky about Scotty’s photon**

*nothing**Clock H*– heck, at the end of the day, it’s just two ordinary mirrors mounted onto a metal shaft; and besides,

*Clock H’s*mirrors can’t even tell the wicked stepmother just what a hideous, butt-ugly warthog she truly is – then we are forced to conclude that as observed

*from*,

**point of view**__our__**objects, living or inanimate, that are aboard the relatively moving train, and of course the train itself, must**

__all__**shorten by the**

*also***factor**

*same**sqrt(1 - v*along the direction of the train’s relative motion! Hence Scotty, if he is standing up, will appear to

^{2}/ c^{2})**, along with his**

*us**vertically*positioned photon

*Clock V*and all other ‘vertically’ positioned objects, to be

**than normal. On the other hand, those passengers who are lying down along the direction of the train’s relative motion and snoozing will appear to**

*thinner***, along with their bunk beds and all other objects positioned ‘horizontally’ along the direction of relative motion, to be**

*us***than normal! And the entire train itself must, of course, also contract and become**

*shorter**shorter*than its normal length!

So, our *first* formal conclusion is that: *An object that is in RELATIVE UNIFORM MOTION, that is to say, an object that is moving in a STRAIGHT LINE and at a CONSTANT SPEED relative to an observer, will appear, to the observer, to be CONTRACTED*,

*or SHORTENED from its*

__normal__length by the__factor__*sqrt(1 - v*

^{2}/ c^{2})

*ALONG THE DIRECTION OF RELATIVE MOTION, where**v*

**is the SPEED OF THE OBJECT RELATIVE TO THE OBSERVER, and**c**is the universally constant SPEED OF LIGHT. We shall call this factor,****namely,***sqrt(1 - v*

^{2}/ c^{2})**,**

*the “(relativistic) distance contraction factor”.*Therefore, if *L _{0}* is the

**length of a straight rod that is moving at a speed**

*normal**v*

**to an observer, then the observer will measure the rod to be**

*relative***, or**

*contracted***, to a length**

*shortened**L*, where

_{C}provided that the rod *lies*** along the direction of relative motion**, that is to say, provided that the rod

*is pointing***. If the observer, however, is instead**

*in the direction of relative motion***, so that the rod is**

*moving*__with__the rod**, then of course, the observer will**

__at rest relative to the observer__**see the rod to be contracted whatsoever, and will therefore measure the length of the rod to be just its**

*not***length,**

*normal**L*. Let’s take this scenario a small step further. If the rod and the observer are still

_{0}**, but are now**

*at rest relative to each other**both*moving at a speed

*v*

**, then we already know that**

__relative to us__**will measure the rod to be contracted to a length of**

*we**L*along the direction of relative motion (not to mention the observer, who will also appear contracted by the same factor). Nothing new, yet. If, however, we ‘watch’ the observer as

_{C}= L_{0}x sqrt(1 - v^{2}/ c^{2})

__she__again**, then we will ‘see’ that, from**

*measures the length of the rod for herself using*__her__ruler**point of view, she will**

*her***measure the length of the rod to be just its**

*still***length,**

*normal**L*,

_{0}**, as viewed from**

*BECAUSE***perspective,**

*our*

__her ruler__will__also__be contracted by the*factor**sqrt(1 - v*

^{2}/ c^{2}) !And let’s not forget our *second* formal conclusion, which states that: *The GREATER the relative speed, **v , of the moving object, the GREATER the amount of CONTRACTION of the object along the direction of relative motion*

**.**

And why is this the case? Because, *mathematically speaking*, as* v* becomes ** greater and greater** and approaches the speed of light,

*c*, the value of the quantity

*v*gets

^{2}/ c^{2}**to a value of**

*closer and closer***, and hence the value of the**

*one**relativistic distance contraction factor*,

*sqrt(1 - v*, becomes

^{2}/ c^{2})**, and approaches a value of**

*less and less***. Putting it into some**

*zero**symbols*to obtain a clear depiction of our logic, we get:

Consequently, distances get ** shorter and shorter** along the direction of motion, and in fact approach

*zero**as*

**length***v*approaches the speed of light

*c!*Stating things in a more ‘pictorial’ way, if Scotty’s train travels

*faster and faster,*then a

*GREATER AND GREATER**would*

__DIFFERENCE__

*BUILD UP between the***that**

__TOTAL DISTANCES__*Clock H’s*

**and**

*Clock V’s*photons would need to travel to complete their respective ticks,

*as ‘seen’ from the*view,

**point of**__observer’s__**Therefore, a**

*HAD CLOCK H*__NOT__SHORTENED TO BEGIN WITH!*shorter and shorter*

*Clock H*, as ‘seen’ from the

**perspective, can**

__observer’s__**, or**

__OFFSET__**this**

*“*__CANCEL OUT__”**, and in this manner,**

*ever increasing difference between the total distances**Clock H*can keep itself

**with**

*in sync**Clock V*at

**relative speeds, as**

*any and all***by the**

*required*

*first postulate!*^{[6]}

## The Lorentz Contraction: Some Examples

Let’s work out a few examples whose scenarios have become very familiar to us by now, to best illustrate this truly extraordinary phenomenon. First up is Scotty’s train, as it blazed through the crossing at the relative speed of 0.866*c*. From ** our** point of view, not only did we observe that his clocks were running 2

*times slower*than ours (i.e., 2 times slower than

*normal*), but we also observed that the length of his train was 2

*times shorter*than its

*normal*length, that is to say, its

*normal*length was contracted

*by the factor*0.5 in the direction of relative motion. So let’s put our formula to the test. If we let

*L*equal the train’s

_{0}**length and**

*normal**L*equal its

_{C}**length, then our formula tells us that:**

*contracted*** Specifically**, if we took our rulers and measured the length of Scotty’s locomotive while it was still at the train station, that is,

*while his locomotive was*

**still at rest**__relative to us__**,**so that there could be

**length contraction (**

*no**nor*time dilation) occurring, and found its length to be, say, 50 feet, then

**length would be the locomotive’s**

*this***length,**

*normal**L*. And later on, as Scotty and his train rumbled past us through the crossing at the relative speed of 0.866

_{0}*c*,

**would now observe his locomotive’s length, again, as measured by**

*we***rulers, to be**

*our**contracted*, or

*shortened*, to only 25 feet

*along the direction of its motion*, namely, shortened to the length

*L*. Reiterating what we had calculated above, then,

_{C}*L*= 0.5

_{C}*L*, and particularly, in this case, (25 feet)

_{0}*=*0.5(50 feet). Rumble, young man, rumble...

*Figure 2* shows that ** we** would observe Scotty’s locomotive to shorten more and more in the direction of motion, as its speed, relative to

**, continuously increased to near that of the speed of light.**

*us*Of course, from ** Scotty’s** point of view, the situation was

**due to the**

*reversed***of the principle of relativity. To**

*symmetry***, the lengths of his train and everything in it, including his two photon clocks, remained at their**

*him***lengths and were**

*normal***contracted whatsoever, while all of his clocks, including again, of course, his two photon clocks, were running at the**

*NOT***rate of time passage! To**

*NORMAL***, it was the**

*Scotty***that was moving at 0.866**

*outside world**c*instead, and therefore, he observed distances in the outside world to be contracted by the factor 0.5 along the direction of relative motion, and clocks in the outside world to be running 2 times slower than his clocks!

And indeed, just as the *astronauts* derived the exact ** same** equation as

**did for the**

*we***, but used**

*relativistic time dilation factor***point of view or frame of reference, which was their spaceship, to derive it instead, then so too would**

*their**Scotty*derive the exact

**equation as**

*same***did for the**

*we***from**

*relativistic distance contraction factor***point of view or frame of reference, which is his train! For instance, let’s say that**

*his***decide to set up the photon**

*we**Clocks V*and

*H*at

**so that**

*our train crossing**Clock V*is again in the vertical position, and

*Clock H*is again in the horizontal position (along the direction of the tracks, that is, along the direction of relative motion). Now, from

**point of view, we, our two clocks, and the outside world are all**

*our***. In fact, stating things more briefly, from**

*at rest with respect to each other***point of view standing at the train crossing, the**

*our***itself is**

*earth***. Hence, as observed from**

*at rest***perspective, there can be**

*our*

*NO**length contraction*(

*NOR**time dilation*) affecting our two photon clocks. Therefore, from

**point of view,**

*our**Clock V’s*photon and

*Clock H’s*photon

**travel**

*both**total distances of 2*

**equal***h*(where

*h*is, of course, the

**distance between the mirrors),**

*normal**at the*

*always**to complete their respective ticks.*

**constant**speed of light c,And as a direct result of this *fact*, our *Clocks V and H* **must therefore always remain in sync**, as observed from

**point of view. And why**

__our__**This must be so, because if our photon clocks, or, for that matter, if**

*must this be so?***,**

*ANY*__properly__working timepieces__at rest relative to the earth__*were to*

*ever fall out of sync with each other*, then

**as a result of**__merely being re-positioned__**could conclude,**

*we***, that,**

*without any doubt***. And with that conclusion, we would depth charge the**

*"Yes, the earth is*__definitely__moving!"

*and sink it into oblivion yet again!*

*principle of relativity*** But** what, then, will

*Scotty***about**

__predict__that he__should__observe**photon clocks as his train rumbles through the crossing at the relative speed**

*our**v?*Well, not to beleaguer the point, but from

**perspective, he and his train are**

__Scotty’s__**while it is we, our photon clocks, and everything else in the outside world, that is to say, it is the surface of the earth itself that is moving by him and his train at the relative speed**

*at rest**v*instead. First of all, then,

**will observe, just as we had observed about**

*he**his*

*Clock V*, that the photon in

*our Clock V*travels a

*double diagonal path*to complete 1 tick,

**. Secondly,**

*due to the*__earth’s__relative motion**will also**

*he***that he**

*predict***observe, just as we had predicted that we should observe about**

*should**his*

*Clock H*, that the photon in

*our Clock H*will travel a

*forward and backward path*to complete 1 tick, again,

*due to the earth’s relative motion*. And,

*since Scotty will*

__likewise__measure the speed of the photons to__always__equal the constant value*c*as per the

*second postulate*, then

**will therefore**

*he*

*ALSO*

**erroneously****, just as we had erroneously calculated, that as observed from**

*calculate***point of view, that is, as measured by**

__his__**using**

__him__**watches,**

__his__*t*>

_{H}*t*

_{V}*!*And hence, his inherently flawed calculation will

**lead**

*ALSO***to believe that**

*him***will observe**

*he*

*our Clock H*

*slow down and fall out of sync with our Clock V!*But Scotty knows the *principle of relativity* all too well by now. And ** he** therefore realizes that if, from

**point of view,**

__our__**were to observe**

__we__

*our**Clock H*to indeed lose sync with

*our**Clock V*for

*, then*

**no apparent reason****knows that**

*he***could conclude, without any doubt, that,**

__we__**, thereby chain-sawing the**

*“Yes, the earth is the one that is*__truly__,__absolutely__moving!”*principle of relativity*into nice, bite-sized pieces! And so, the lesson to be had in this symmetrical tale of two timepieces is that,

**to his prediction,**

*contrary***will**

*Scotty*

__unavoidably__**observe, just as we had observed about his**

*also**Clock H*, that

*our Clock H*is

**along its length in the horizontal direction by the factor**

__shortened__*sqrt(1 - v*, so that, as measured from

^{2}/ c^{2})**point of view,**

*his**t*=

_{H}*t*instead!

_{V}**And so, Scotty will likewise deduce that**

*In this manner, our Clocks V and H will be kept in sync, thereby*__upholding__the principle of relativity!where *h _{C}* is the

**length of our**

*shortened**Clock H*, as it appears to

**, and**

*Scotty**h*is its

**length, as it of course appears to**

*normal***. And since there is nothing extraordinary about our**

*us**Clock H*, and moreover, since, from

**point of view, it is the outside world and everything in it that is moving, then not only will Scotty observe our**

*Scotty’s**Clock H*to be shortened by the factor

*sqrt(1 - v*along the direction of relative motion, but in fact, he will observe

^{2}/ c^{2})*the outside world itself and everything in it*to be shortened by the factor

*sqrt(1 - v*along the direction of relative motion! And thus, just as the

^{2}/ c^{2})**demonstrated the**

__astronauts__**of special relativity’s**

*symmetry**time dilation phenomenon*by deriving

**, so too has**

*exactly the same formula for the time dilation factor as*__we__did**demonstrated the**

__Scotty__**of special relativity’s**

*symmetry**distance contraction phenomenon*by deriving

*exactly the same formula for the distance contraction factor as*__we__did!Next up in the batting order are those go-get-’em muons that tore through our atmosphere at the alarming relative speed of 0.998*c*, on their way to colliding with the earth. From the ** muons’** point of view, it was

**who were the ones**

*they***, while it was the atmosphere and the earth that were**

__at rest__**towards them at 0.998**

*hurtling**c*instead. And if we recall, this

**resulted in the**

*relative motion***‘seeing’ – along the direction of motion – a**

*muons***, or**

*contracted***travel distance**

*shortened***, that was 15.82**

*between the earth’s surface and themselves**than the*

**times shorter****distance. Here, we stress once more that the meaning of the expression**

*normal***is, in a manner of speaking,**

*‘normal distance’**“the muons’ travel distance, as measured by observers (such as*,

**), who had been**__us____at rest__with respect to the earth and the atmosphere in the first place*so that there could be*,

**NO**length contraction of this distance*.*

**as perceived from the**__observer’s__point of view*”*

We will now use the same nomenclature for the *length contraction formula* as before, ** but** we must remember that we are now observing things from the

*point of view, such that the atmosphere and the earth were the ones that were*

**muons’****. And so,**

*moving instead***, if we let**

*from the*__muons__’ perspective*L*equal the atmosphere’s and earth’s

_{0}

*normal**,*travel distance, and

**non-contracted***L*equal the

_{C}**travel distance, that is to say, the**

*contracted***travel distance that the**

*contracted***had to traverse in order to slam into the muons, then, the formula must again state that:**

*earth*And so, if **we** now took

**and**

__our__gigantic tape measure**the distance that the**

*measured***had to traverse in order to embed themselves beneath the earth’s crust (which is much easier said than done!), then, as we already know,**

*muons***would measure**

*we***distance to be 34,184**

*this**feet*. And moreover, since

*this**expanse of atmosphere and earth would (of course) be*

*, then*

**AT REST**__relative to us__**distance of 34,184**

*this**feet*would therefore be the

**,**

__normal__distance*L*. We know, however, that from the

_{0}**perspective, the atmosphere and the earth were**

__muons__’

*NOT AT REST**, but were in fact blazing past them at the relative speed of 0.998*

__relative to them__*c!*And therefore, the

**would have “observed” the**

*muons***, that is, the entire 34,184**

*normal distance**feet*, to be

**, or**

*contracted***to only 2,161**

*compressed**feet along the direction of motion*,

*as*

**(which, from the**

*measured by the*__their__‘rulers’**point of view, of course, would be rulers of**

*muons’***). Specifically, the**

*normal length**normal*distance

*L*would have been

_{0}*shortened*to the

*contracted*distance

*L*. Hence, reiterating what we had calculated above, from the

_{C}**perspective,**

*muons’**L*= 0.0632

_{C}*L*, namely, (2,161

_{0}*feet*)

*=*0.0632(34,184

*feet*).

Thus, it is also clear from our two examples that as the relative speed *increases*, so too does the amount of contraction *along the direction of motion*. Just as with time dilation, however, length contraction becomes perceptible only at speeds that are *significant fractions* of the speed of light. To appreciate just how *negligible* these relativistic phenomena are for objects moving at our “everyday” relative speeds, we consider a car being driven at 60 *mph* *relative to the road*. At this snail’s pace, the equations of *special relativity* which we have just derived tell us that we would “observe” a clock inside the car to be ticking at a rate that is 0.0000000000004 % less than the normal rate of time passage, and we would “observe” the length of the car to be contracted to 99.9999999999996 % of its normal length!

## Notes:

[6] We can also visualize the fact that the amount of distance contraction ** increases** as an object’s relative speed becomes

**– specifically, the fact that an object gets**

*greater and greater**shorter and shorter*along the direction of its relative motion as its relative speed becomes

*greater and greater*– by investigating the ‘behaviour’ of our three graphs shown below. We imagine that while we are standing at the train crossing, Scotty’s train zooms through it numerous times, each time at a

**, but**

*greater***, relative speed (e.g., the first time through the crossing, his train’s relative speed is 0.1**

*always*__constant__*c*; the second time through, it is 0.2

*c*; the third time, it is 0.3

*c*, and so on, in gradual increments). And for each time through the crossing, that is to say, for each value of the train’s relative speed,

**record the appropriate values, being either calculated or measured from**

*we***point of view, onto each of our three graphs. And so…**

*our*Our first graph is a plot of the ratio *d _{H} / d_{V}* versus the relative speed

*v*,

*given that the distance contraction phenomenon DID NOT EXIST AT ALL.*We recall that *d _{H}* equals

*the*. Correspondingly,

**predicted**total forward and backward**distance**that the photon in Scotty’s Clock H has to travel in order to complete 1 tick, as calculated from**our**point of view*d*equals

_{V}*the*. And

**actual**total double diagonal**distance**that the photon in Scotty’s Clock V has to travel in order to complete 1 tick, as observed from**our**point of view*v*is, of course,

*the speed of Scotty’s train relative to*Hence, our

**us.***first*graph is ‘saying’, in essence, that: If the

*relativistic distance contraction phenomenon*

**did not****, then as observed from**

*exist***point of view, as the relative speed**

*our**v*becomes

**, the**

*greater and greater***between these two distances, specifically,**

*ratio**d*, would

_{H}/ d_{V}**, as shown by the latter portion of our**

*increase at a*__faster and faster rate__*first*graph (we must note, at this point, that this

*“faster and faster rate”*is present throughout our

**graph as the relative speed increases, but becomes perceivable only in the latter portion, as the relative speeds approach the speed of light).**

*entire***, we would predict, exactly as we did earlier in our discussion, that as the relative speed**

*Hence**v*becomes

*greater and greater*, then the photon travel distance

*d*

_{H}**would**

__itself__**than the photon travel distance**

*become*__disproportionately__greater and greater*d*

_{V}**, as observed from**

__itself__**point of view, if relativistic distance contraction**

*our***. And since**

*did not exist**Clock H’s*and

*Clock V’s*photons must

*both*

**travel at the**

__always__*universally*speed of light

**constant***c*,

*then*>

**that d***our prediction*_{H}*d*would lead us to

_{V}**predict that**

*also***would observe**

*we**Clock H*fall

**with**

*further and further*__out__of synchronization*Clock V*as the relative speed of Scotty’s train becomes

*greater and greater*with each pass through the crossing!

We know, however, that this loss of synchronization **cannot** happen,

**of the fact that**

*because***is, by the**

*Scotty**principle of*

*relativity*(i.e., the

*first postulate*),

**in declaring that**

*completely justified***.**

*he and his train are*__at rest__**, from**

*Therefore***point of view, there can be**

*Scotty’s***distance contraction (**

*NO***time dilation) at work within the confines of his train!**

*nor***,**

*And therefore***must and will observe his**

__Scotty__*Clock H*and his

*Clock V*

__maintain__*.*

**their synchronization****,**

*Thus***must and will also observe his**

__we__*Clock H*and his

*Clock V*

**(although**

__maintain__their synchronization**will,**

*we***, observe**

*additionally**both of his clocks*ticking

**than**

*more and more slowly**our clocks*as his train’s relative speed

*increases*,

**).**

*because of*__time dilation__But now, how can *Mother Nature* ‘remedy’ our *first* graph so that it shows that *d _{H}* =

*d*and

_{V}

*NOT**d*, for

_{H}> d_{V}**values of the relative speed**

*any and all**v*, so that

**photons travel**

*both***distances in order to complete their respective ticks, thereby keeping**

*equal**Clock H*and

*Clock V*

**from**

*in sync***point of view**

*our***, as they**

*also***Well, we already know that the remedy is the**

*must be?**relativistic distance contraction phenomenon*, whose

**, which we’ll abbreviate as “DCF”, equals**

*distance contraction factor*Thus, we have our *second* graph, which is a plot of the DCF versus the relative speed *v*.

Let us take note of this graph’s important properties. First of all, when the relative speed of Scotty’s train equals 0, then the DCF has a value of 1. Surely, this must be the case, because it means that his train is **at rest relative to us** (and

**), and hence there can be**

*vice versa***distance contraction (**

*no***time dilation) at work from**

*nor***(or from**

*our***) point of view. Stating this mathematically,**

*Scotty’s*Secondly, as the relative speed of his train ** increases** with each pass through the crossing, the value of the DCF

**. But**

*decreases at a*__faster and faster rate__**does the DCF continue to**

*why***as the relative speed of Scotty’s train becomes**

*decrease*__disproportionately__*greater and greater?*It does so in order to

**the**

*COUNTERACT***value of the ratio**

__disproportionately__INCREASING*d*. Furthermore, it turns out that this

_{H}/ d_{V}*lopsided ratio*between

*d*and

_{H}*d*, which is plotted as a function of

_{V}*v*in our

*first*graph, is given by the equation

** But**, as we have discussed to great lengths by now,

**demands that Scotty’s**

*reality itself**Clock H*and

*Clock V*

**.**

*remain*__in____sync__**, the equation**

*And therefore*** must be incorrect**, because we already know that as observed from

**point of view, the photon travel distance**

*our**d*must

_{H}**, or**

*CONTRACT***in length so that it**

*SHORTEN*

__equals__*d*, in order for

_{V}*Clock H*to remain

**with**

__in sync__*Clock V*. Hence, this requirement that

*d*=

_{H}*d*immediately tells us that the

_{V}**equation for the ratio must simply be:**

__correct__So, how does Mommy Natural get to this latter equation? Enter the *relativistic Distance Contraction Factor*, or DCF, as we have called it, whose expression we know to be equal to

*sqrt(1 - v*. Now, let’s

^{2}/ c^{2})**the distance**

*SHORTEN**d*in our

_{H}*incorrect*equation by

*multiplying both sides of this equation by the DCF:*Hence, as observed from ** our** point of view, if

**, or**

*Clock H*__itself__contracts**(along the direction of the train’s relative motion) by a factor**

*shortens***, then**

*equal to the DCF**Clock H’s*photon travel distance

*d*will

_{H}**, such that it**

*also shorten by this*__same__factor**the photon travel distance**

*always*__equals__*d*for

_{V}**relative speeds, thereby keeping**

*any and all**Clock H*

**with**

__in sync__*Clock V*, as they must be. This result is clearly revealed by our

*third*graph.

In a manner of speaking, then: ** GRAPH 3 = GRAPH 1 x GRAPH 2, **for any and all relative speeds.

## Continue To Part 4

- The Lorentz Contraction, Or, How Motion Affects Space (Part 4 of 4)

The 2nd in a series of articles on Albert Einstein's Special Theory of Relativity

## Go Back To Part 2

- The Lorentz Contraction, or, How Motion Affects Space (Part 2)

The 2nd in a series of articles on Albert Einstein's Special Theory of Relativity

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