*quantifies*these Relativistic Effects, apparently unaware that Hendrick Lorentz had earlier come up with the same equation. The "Lorentz Transform" or "Lorentz Gamma" (equation near the top of the graphic below) solves for

*(Greek letter Gamma) given knowledge of the relative velocity of a body (*

**ϒ***) divided by the speed of light (*

**v***).*

**c**Simple enough, but, in my (perhaps overly anal :^) Engineering Mind it bothered me that I could not "picture" it in

*physical*terms.

VISUALIZING Relativity - PowerPoint Show

VISUALIZING Relativity - Excel Spreadsheet

VISUALIZING for Science and Technology - Blog Posting

VISUALIZING Einstein's "Miracle Year" - Blog Posting

VISUALIZING My Insight Into Lorentz Gamma - Blog Posting

VISUALIZING the "Twin Paradox" - Blog Posting

**AN OLD JOKE**

This situation reminds me of the old joke about the Historian, the Physicist, and the Engineer who happened to be waiting for a bus outside an office building. They noticed

*three*people (a man and two women) enter the building, and, some time later,

*five*emerge (a woman and four men).

Making conversation, the Historian asked, "How many people are in that building?"

The Physicist immediately answered, "

*Three*went in and

*five*came out, so there are

*minus two*people in that building!"

The Engineer shook his head. "Mathematically correct," he noted, "But, what in hell does '

*minus two*people' mean?"

"Do you have a better answer?" asked the Historian.

The Engineer thought for a while and replied. "Well, if we assume that is the only entrance and exit for that building, we can deduce that, prior to our arriving here, there were

*at least three men*in that building, and now there is

*at least one woman*in there."

**BACK TO LORENTZ**

Well, a couple of years ago, I ran the Lorentz Transform for several different values of

*and was startled to find some*

**v/c***familiar*numbers come up, among them

*.*

**0.5000, 0.7071, and 0.8660**Early in my engineering career I memorized the sines and cosines of

*. Those were the familiar numbers that popped up in my results for the Square Root of*

**30⁰, 45⁰, and 60⁰****. (The fact that I still remember those numbers, half a century later, confirms how anal my Engineering Mind really is. :^)**

*1-(v/c)²*For example, if you pick the simple case of half the speed of light (i.e.,

*), the Square Root term turns out to be*

**v/c = 0.5000***, which is the*

**0.8660***of*

**Cosine***. As the graphic above illustrates, if you plot Time vs Space with commensurate scales (i.e., Time in*

**30⁰***nanoseconds*and Space in

*feet*, since, as I also memorized those many years ago, light travels about

*one foot*in

*one nanosecond*), a unit long

*SpaceTime*vector, tipped

*from the Time axis, has its point at*

**30⁰***along the Space axis and*

**0.5000***along the Time axis!*

**0.8660****SOME EXAMPLES**

Let us call the Square Root part of the Lorentz Transform term

**(Greek letter Alpha) from here on, and notice that**

*α*

*α = 1/**. Furthermore, let us call the*

**ϒ***term*

**v/c***(Greek letter Beta), and the angle between the Time axis and the unit long*

**β***SpaceTime*vector

*(Greek letter Theta).*

**Θ**For

**Θ =**

**0***:*

**⁰**

*α = 1.0000 = Cos(*

**0**

**⁰****and**

*)*

**β****=**

*0.0000*=*Sin*

*(*

**0**

**⁰**

*)*For

**Θ =**

**30***:*

**⁰**

*α = 0.8660 = Cos(*

**30**

**⁰****and**

*)*

**β****=**

*0.5000*=*Sin*

*(*

**30**

**⁰**

*)*For

**Θ =**

**45***:*

**⁰**

*α = 0.7071 = Cos(*

**45**

**⁰****and**

*)*

**β****=**

*0.7071*=*Sin*

*(*

**45**

**⁰**

*)*For

**Θ =**

**60***:*

**⁰**

*α = 0.5000 = Cos(*

**60**

**⁰****and**

*)*

**β****=**

*0.8660*=*Sin*

*(6*

**0**

**⁰**

*)*For

**Θ =**

**90***:*

**⁰**

*α = 0.0000 = Cos(*

**90**

**⁰****and**

*)*

**β****=**

*1.0000*=*Sin*

*(*

**90**

**⁰**

*)*So, now it all makes sense (at least to an old engineer like me :^)! All the Square Root part of the Lorentz Transform is telling us is that if we pick a value for

*that is equal to the Sine of some angle,*

**v/c***, we'll get a value for the Square Root part that is the Cosine of that*

**Θ***same*angle,

*.*

**Θ**The simple VISUALIZATION is a unit vector in

*SpaceTime*tipped

**Θ***from the Time axis, and it works for any*

**⁰***between*

**Θ**

**0***and*

**⁰**

**90***.*

**⁰****OK, BUT WHAT IS THIS VISUALIZATION TELLING US?**

In the above graphic, the Time axis extends up to

*, but the projection of the unit long*

**1.0***SpaceTime*vector onto the Time axis reaches only to

*. So, what does*

**0.8660****tell us?**

*α = 0.8660*I used to have the impression that Relativistic Effects "slowed down time", and I believe quite a few of you who are reading this Blog accept that idea. However, the well known "Twin Paradox" (to be discussed in more detail the next Blog Posting in this VISUALIZING Series) tells us, IMHO, that it is not Time, per se, that "slows down" but rather AGING. For every year the stay-at-home Twin ages, the travelling twin ages only

**years. So, if**

*α***, and the stay-at-home twin ages**

*α = 0.8660**years, the traveling twin will age only*

**10***years.*

**8.66**What do

*I*mean by AGING? Well, it is simply the number read from a good-quality clock at some final Event, assuming the clock was set to zero at some initial Event, and that the clock was present at both Events. The clock may be a mechanical or electronic device, or a chemical or radioactive reaction, or a biological life form, such as bacteria, plants, or animals.

In the Twin Paradox example, both siblings are present at the separation Event and the reunion Event, each usually denoted by numerical values for

**and**

*t, x, y,**, for Time and the three dimensions of Space. Thus, when reunited, they are both at the exact same Time (and Space), the only difference is how much each of them has AGED.*

**z**

*Ira Glickstein*
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