# How The Lorentz Contraction Results From Time Dilation - A Detailed Calculation

The following calculations show that a ** direct consequence** of the

*relativistic time dilation phenomenon*

**the**

*IS**relativistic distance contraction phenomenon*, also known as the

*Lorentz Contraction*. This is just another

*analogous*method of visualizing the distance contraction phenomenon. And in doing so, we revisit the example of the muons, which are created at the very outskirts of our atmosphere, and blaze their way towards a collision with the earth at the very high relative speed of 0.998

*c*. Moreover, we will see (at the end of our calculations) that these two phenomena, namely, time dilation and distance contraction, impose

*‘equivalent’*effects on time and space, respectively. The value of the speed of light

*c*is, of course, the universally constant value of 300,000

*km/second*, or 9.843 x 10

^{8}

*ft/second*.

* [ 1 ]* The muons are moving at a speed of

*v*

*=*0.998

*c*, relative to

**. Thus, calculating this relative speed in units of**

*us**feet per second*gives:

*v* = 0.998 x (9.843 x 10^{8}* ft/second*) = 9.823 x 10^{8}* ft/second*.

* [ 2 ]* Due to

*relativistic time dilation*,

**will observe that**

*we***clocks (if the muons were somehow able to carry clocks with them!) are ticking at a**

*their***rate than**

*slower***clocks by the factor**

*our*which is, as we know, the *relativistic time dilation factor.* Plugging in the numbers, then, we find that ** we** observe that the

**clocks are ticking at a rate that is:**

*muons’*Hence, as observed from ** our** point of view (standing here on earth), the rate at which

**passes for the**

*time itself***is 15.82 times**

*muons***than the**

*slower***rate at which time passes for**

*normal***!**

*us** [ 3 ]* Furthermore,

*also observe that the*

**we****traverse the**

*muons***34,184**

*entire**feet*of

**distance – that is to say, the**

*normal***distance of our atmosphere and earth that is measured by**

*normal***using**

__us__**– before finally disintegrating (because the muons actually penetrate the earth’s crust).**

__our__rulers* [ 4 ]* Now, since

*TRAVEL TIME = DISTANCE TRAVELED / SPEED OF TRAVEL*,

then the ** amount of time** that it takes the muons – as observed from

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

*our***using**

__us__**– to travel this**

__our__clocks**distance thus equals:**

*normal*(34,184* ft*) / *v* {where *v* is, of course, the ** relative speed** of the muons}, which then equals:

(34,184* ft*) / (9.823 x 10^{8}* ft/second*), or 34.8 x 10^{-6}* seconds*, which is 34.8* microseconds*.

Therefore, as observed from ** our** point of view, it takes the muons 34.8

*microseconds*of

*our***time to travel**

*normal***measured**

*our***distance of 34,184 feet.**

*normal** [ 5 ]* But, due to the

*time dilation phenomenon*,

**observe a corresponding**

*we***amount of time of only**

*lesser*elapse on the ** muons’** clocks, during this journey! We will call this amount of time Δ

*t*. And so, carrying this calculation through gives us:

_{MUONS}We note here that the value for the amount of time Δ*t _{MUONS}* could have just as easily been calculated by multiplying the 34.8

*microseconds*that elapse on

**clocks (during the muons’ journey) by the quantity 1 / 15.82 (which is, as we know, the**

*our**inverse*of the

*time dilation factor*), because we have already calculated that

**will observe the**

*we***clocks ticking**

*muons’***15.82**times

**than**

*slower***clocks. That is to say, we have already calculated the**

*our**time dilation factor*to be equal to

**15.82**. Hence, carrying this calculation through gives us:

To recap, from **our** point of view,

**observe a time interval of 34.8**

*we**microseconds*elapse on

**clocks during the muons’ 34,184**

*our**ft*journey, but

**observe a corresponding**

*we***time interval of only 2.2**

*lesser**microseconds*elapse on the

**clocks during this journey! That is, from**

*muons’***point of view, the**

__our__**clocks are ticking in**

*muons’**slow motion*, specifically:

(34.8 *microseconds*) / (2.2 *microseconds*), or **15.82** times *slower*

than ** our** clocks, which are, of course, ticking at the

**rate of time passage!**

*normal** [ 6 ]* Now, since

*DISTANCE TRAVELED = SPEED OF TRAVEL x TRAVEL TIME*,

then, from the ** muons’ **point of view, since it only takes Δ

*t*, namely, 2.2

_{MUONS}*microseconds*, of

**for the earth (which is, of course, moving at the relative speed of 0.998**

*normal time**c*) to reach

*them*,

*will ‘measure’ a travel distance of only*

**they**(9.823 x 10^{8}*ft/second*) x (2.2 x 10^{-6}*seconds*), or only 2,161 *ft!*

Hence, the ** muons** will ‘see’ the

*full*34,184

*ft*distance to be

**, or**

*contracted**compressed,*into a distance of only 2,161

*ft*

**Consequently, the**

*along the direction of relative motion!**muons*will ‘see’ a travel distance that is

(34,184* ft*) / (2,161 *ft*), or **15.82** times *shorter*

than the travel distance *we* see!

* *

Therefore, the fact that *time is dilated* by a factor of 15.82 for the (relatively) moving muons directly results in the muons “seeing” a *travel distance* that is *contracted by a factor of *1 / 15.82 *along the direction of relative motion*. That is to say, the *distance contraction* phenomenon is a ** direct result** of the

*time dilation*phenomenon. The following diagram will help to illustrate this fact (we note here that due to the

*symmetry*of special relativity,

**will observe the muon itself - that is, its diameter - to be also contracted along the direction of motion!).**

*we*And finally, we can summarize this example using a purely *algebraic* form, as follows:

* [ 1 ]* First,

**observe a muon moving at the speed**

*we**v*, relative to

**.**

*us***[ 2 ]**** We** observe that the

**clock is running**

*muon’s***than**

*slower***clock by a factor of**

*our** [ 3 ]* Furthermore,

*also observe that the*

**we****traverses the**

*muon*

*entire***distance – that is to say, the**

*normal***distance of our atmosphere and earth that is measured by**

*normal***using**

*us***– before finally disintegrating (because the muon actually penetrates the earth’s crust). We will call this normal distance**

*our rulers**L*.

_{0}* [ 4 ]* Now, since

then, from ** our** point of view, i.e., as measured by

**clock, the amount of time that it takes the muon to travel the**

__our__**distance,**

*normal**L*, is equal to

_{0}** [ 5 ]** But, due to the

*time dilation phenomenon*,

**observe a corresponding**

*we**amount of time of only*

**lesser**elapse on the ** muon’s** clock, during this trip.

* [ 6 ]* Now, since

then, from the ** muon’s** point of view, i.e., as “measured” by the

**using**

__muon__**, the**

__its__rulers*therefore “sees” a*

**muon****travel distance,**

*contracted**L*, which is given by the expression:

_{C}To summarize what this equation is saying, then: From the ** muon’s** perspective,

**“sees” the**

*it**full*

**distance,**

*normal**L*, to be

_{0}**, or**

*contracted**, into a*

**compressed****distance,**

*shorter**L*, by a factor of

_{C}## Go Back To "The Lorentz Contraction, or, How Motion Affects Space (Part 1)

- The Lorentz Contraction, or, How Motion Affects Space (Part 1)

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

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