VISUALIZING: My Insight Into Lorentz Gamma

A key aspect of Einstein's Special Relativity is that, at high speeds, there is significant "Time Dilation" and "Length Contraction". In his 1905 Theory of Relativity paper, Einstein derives the equation that 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 γ (Greek lowercase letter gamma) given knowledge of the relative velocity of a body (v) divided by the speed of light (c).

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

LINKS TO RELATED POSTINGS AND RESOURCES
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 v/c and was startled to find some 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 30⁰, 45⁰, and 60⁰. Those were the familiar numbers that popped up in my results for the Square Root of 1-(v/c)². (The fact that I still remember those numbers, half a century later, confirms how anal my Engineering Mind really is. :^)

For example, if you pick the simple case of half the speed of light (i.e., v/c = 0.5000), the Square Root term turns out to be 0.8660, which is the Cosine of 30⁰. As the graphic above illustrates, if you plot Time vs Space with commensurate scales (i.e., Time in 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 30⁰  from the Time axis, has its point at 0.5000 along the Space axis and 0.8660 along the Time axis!

SOME EXAMPLES

Let us call the Square Root part of the Lorentz Transform term α (Greek letter Alpha) from here on, and notice that α = 1/ϒ. Furthermore, let us call the v/c term β (Greek letter Beta), and the angle between the Time axis and the unit long SpaceTime vector Θ (Greek letter Theta).

For Θ = 0⁰   :  α = 1.0000 = Cos(0⁰)   and β = 0.0000 = Sin(0⁰)
For Θ = 30⁰ :  α = 0.8660 = Cos(30⁰) and β = 0.5000 = Sin(30⁰)
For Θ = 45⁰ :  α = 0.7071 = Cos(45⁰) and β = 0.7071 = Sin(45⁰)
For Θ = 60⁰ :  α = 0.5000 = Cos(60⁰) and β = 0.8660 = Sin(60⁰)
For Θ = 90⁰ :  α = 0.0000 = Cos(90⁰) and β = 1.0000 = Sin(90⁰)

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 v/c that is equal to the Sine of some angle, Θ, we'll get a value for the Square Root part that is the Cosine of that same angle, Θ.

The simple VISUALIZATION is a unit vector in SpaceTime tipped Θ⁰ from the Time axis, and it works for any Θ between 0⁰  and 90⁰.

OK, BUT WHAT IS THIS VISUALIZATION TELLING US?

In the above graphic, the Time axis extends up to 1.0, but the projection of the unit long SpaceTime vector onto the Time axis reaches only to 0.8660. So, what does α = 0.8660 tell us?

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 α years. So, if α = 0.8660, and the stay-at-home twin ages 10 years, the traveling twin will age only 8.66 years.

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 t, x, y, and z, 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.

Ira Glickstein