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

VISUALIZING: Einstein's "Miracle Year"

In 1905 an obscure, 26-year old Assistant Examiner in the Swiss Patent Office revolutionized the world of Physics forever. Albert Einstein published four ground-breaking papers in a single year!

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

1- PHOTOELECTRIC EFFECT

This paper may be considered the foundation of Quantum Theory. Einstein theorizes that Energy is not continuous, but rather comes in DISCRETE QUANTA.

2- BROWNIAN MOTION

Statistical physics.

3- (SPECIAL) THEORY OF RELATIVITY (INVARIABILITY)

Perhaps his most important paper, On the Electrodynamics of Moving Bodies, begins with a VISUALIZATION of a magnet and a coil of wire. He notes that, when the wire is held still and the magnet is moved, an electrical voltage is produced in the wire. The opposite is also true, he notes, when the magnet is held still and the wire is moved.

Einstein's critical INSIGHT was that it was the RELATIVE movement of the wire and the magnet that induced the electrical voltage.

He knew about electrical technology because his father owned a company that manufactured electrical equipment. However, by the 1880's, when Einstein was a child, many other people had this knowledge, yet, it was not until 1905, with this paper, that the critical implications of this simple VISUALIZATION were recognized.

Of course, by the time he had the insight and wrote this paper, Einstein had a diploma in Physics and Mathematics from the Zurich Polytechnic. That enabled him to extend his insight lightyears [pun intended] beyond mere electrical technology and clarify Maxwell's electrodynamics.

Einstein proposes two startling postulates:

1) "The same laws of electrodynamics and optics will be valid for all frames of reference." and
2) "Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."

These two postulates summarize the essence of Einstein's 1905 theory, which has come to be called his SPECIAL Theory of Relativity, to distinguish it from his later 1915 GENERAL Theory of Relativity. [Special Relativity applies only to so-called Inertial frames of reference moving at a constant speed, while General Relativity extends the ideas to frames of reference that are subject to acceleration, including the effects of gravity in the vicinity of a massive body. Some commentators including me (and even Einstein at one time :^) think these theories would have been better named the Special and General Theories of Invariability.]

The first postulate of Special Relativity establishes that ALL frames of reference are equal with respect to the applicability of the Laws of Nature. Some social commentators have, IMHO, over-interpreted the applicability of this idea, claiming that everything is relative,  including basic concepts of ethics. The actual meaning, IMHO, is the Invariability of the Laws of Nature, being the same to all observers, at all times and places.

The second postulate does away with the need for a "luminiferous ether" the then prevalent idea that light and other electromagnetic waves require a medium in which to propagate, the way sound waves require air, water or some other medium. Again, IMHO, some scientists have over-generalized that postulate to deny any concept of Absolute Space or Absolute Time. All Einstein says is that all attempts to measure the Speed of Light will yield the same value, which is another example of Invariability.

Quoting Einstein directly:

... the unsuccessful attempts to discover any motion of the earth relatively to the 'light medium,' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. 

and

...The introduction of a 'luminiferous ether' will prove to be superfluous inasmuch as the view here to be developed will not require an 'absolutely stationary space' provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

Thus, while sound waves require a medium for propagation, light and other electromagnetic waves do not. For example, assume I am some distance from a wall and I make a loud, sharp sound (such as a gunshot), and then measure the return-trip time between the original sound and the echo, I will measure a shorter time in still air than I will if there is a wind blowing the air towards or away from me.

The reason for this is that, for still air, the outbound and return trip will take exactly the same amount of time, while, for moving air, it will take longer for the sound going against-the-wind, and shorter for the sound going with-the-wind.

You might think the longer and shorter changes would cancel out, but that is not so, The against-the-wind slowdown will be greater than the with-the-wind speedup. To VISUALIZE that, imagine the extreme case where the wind is blowing at the speed of sound. The with-the-wind sound will get there at twice the speed of sound, but, the against-the-wind sound will never get there!

Not so with light and other electromagnetic waves. The 1887 Michelson-Morley experiment demonstrated that there was no difference in the return trip time for light signals aligned with the 67,000 Miles per Hour motion of the Earth around the Sun and those at right angles to that motion.

Thus, there is no "ether wind", but there are differences between Kinetic (and Potential) Energy levels experienced by different Observers. 

IS THERE ANY SUCH THING AS ABSOLUTE SPACE AND TIME?

Consider the so-called "Twin Paradox", where the stay-at-home twin will have aged more than the rapidly-moving travelling twin when they are reunited. This has been explained by the fact the travelling twin changes frames of reference when he turns around to come back home. True, but, IMHO, the real difference is that the travelling twin has spent his life at a higher speed and thus a higher level of Kinetic Energy than his stay-at-home sibling, and thus aged more slowly.

Alternatively, imagine a stay-at-home twin who spends her life on the surface of a massive body, while her travelling sibling goes on a slow-speed journey to far-away empty space, living most of his life far from any massive body, and then slowly returning. Which will age more slowly? Well, in this case it is the stay-at-home twin! IMHO, the real difference is that the stay-at-home twin has spent her life at higher gravitational acceleration, and thus a higher level of Potential Energy than her travelling sibling, and thus aged more slowly.

OOPS! If you bought the above explanation you have to ask: "Higher Kinetic or Potential Energy with respect to what?" Well, at least with respect to his or her sibling! Higher Kinetic Energy is based on being at a speed that is greater than the speed of some reference point, is it not? Similarly, higher Potential Energy is based on being under the influence of gravitational acceleration that is greater than the acceleration of some reference point, is it not?

Thus there is some point (such as the Center of Mass of the Universe?) or points (such as Lagrange points L4 and L5?) that have lower Kinetic and/or Potential Energy levels than others, and are thus special in that residents will age more rapidly there. Perhaps there is a point (or points) in the Universe where aging is more rapid than ANYWHERE ELSE. At that (perhaps imaginary) point or points, residents experience Absolute Time (and/or Space) and are truly "At Rest" because they are at some Absolute Zero Kinetic and/or Potential Energy level.

[Note: Lagrange points, L1 to L5, are where the combined gravitational pull of two large masses provides precisely the centripetal force required to orbit with them. L4 and L5 are stable.]

4- EQUIVALENCE OF MATTER AND ENERGY
This paper is merely the source of the World's Best-Known Equation, E=Mc2  

CONCLUSION

Four published papers. Pretty good work for a 26-year old in a single year - or a lifetime! 

Ira Glickstein 

VISUALIZING: for Science and Technology

Computer Model Visualization of Crash-Dummy

Nowadays it is common to use computer models, such as the crash-dummy in the adjacent image, to help us VISUALIZE and better understand complex situations and systems. Prior to the advent of computer models, we had to use mental models in our "mind's eye", along with physical aids such as paper maps and diagrams, modelling clay, and other means.

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

Albert Einstein was a great physicist, with all the requisite mathematical tools. However, he rejected purely mathematical abstraction and resorted to physical analogy for his most basic insights. For example, as part of the thought process that resulted in his theories of Special Relativity (1905) and General Relativity (1915) he imagined himself riding along a beam of light; or as an observer standing along the tracks as a train zipped by at near-light-speed; or as a scientist sealed in a closed box and not able to tell if the box was stationary on the surface of the Earth, subject to gravity, or in deep space, far from massive objects, but subject to acceleration due to being dragged by a rocket at ever-increasing speeds.

VISUALIZING the Solution
Using Math and Graphics

Of course, Einstein and virtually all scientists and technologists use mathematical abstractions to quantify the meaning in our visualization models. We change the initial conditions and run these models to simulate what may or may not happen in different situations.

COMPUTER MODELS FOR VISUALIZATION

As personal computers and the Internet have become endemic, manual typewriters, paper maps, physical books, and so much else has been displaced by automated versions. Similarly, computer visualizations and models have displaced older methods - except for that old reliable "mind's eye" which remains as important as ever.

During my career as a Senior System Engineer at IBM and Lockheed-Martin I made extensive use of computer models and visualizations and have continued to do so since retirement.

In particular, I have created visualizations for the Atmospheric "Greenhouse" Effect and Einstein's Special and General Relativity. 

VISUALIZING THE ATMOSPHERIC "GREENHOUSE" EFFECT

As a Guest Contributor to the World's most popular Climate site, I authored a four-part series on Visualizing the Atmospheric "Greenhouse" Effect that attracted over 65,000 page views and over 2000 comments (see:  Physical Analogy,  Atmospheric Windows,  Emission Spectra, and Molecules and Photons,) The following graphics are some of the animated visualizations I created for that series.   

Physical Analogy

Model of a Physical Greenhouse

Model of the Atmospheric "Greenhouse"Effect

Modeled Down to Photons and Air Molecules

VISUALIZING EINSTEIN'S SPECIAL AND GENERAL RELATIVITY

Perhaps the most well-known equation in the world is E = mc2, recognized by virtually every person. But, what does it really mean?

And, many people know about the so-called "twin paradox", where one twin goes off on long mission at high speeds into space, and comes back younger. But why does this happen and exactly what causes it?

If "everything is relative" why isn't the stay-at-home twin also also younger? So, everything is not relative, and perhaps Einstein's original name for his theories "Invariance" is more apt -for the fact all observers, including those moving at different speeds, measure the same speed for light.

If the traveling twin is younger due to experiencing high speed and acceleration, then it is aging that has slowed down, not time, per se.

Furthermore, what, precisely, is TIME? And how is TIME united with SPACE to form SpaceTime?

When you Google any of this stuff you are quickly buried in equations and tensor mathematics that no one (even an engineer like me) can really understand!

Well, all this bothered me for most of my life until, back in 2012, I decided to answer Alan Alda's Flame Challenge "What is Time?" and produce a short video. In the research process for that project, I think I had a critical insight into TIME, SPACE, and RELATIVITY that may help you VISUALIZE this important scientific theory.

Time - the fourth dimension (2013 Flame Challenge) from Ira V Glickstein on Vimeo.

Since that time, I've continued to delve into Relativity and I've come up with what I think is a unique way to visualize and ... perhaps ... even understand it. The following images are screenshots from an Excel spreadsheet I created to provide myself (and you :^) a "hands-on" experience with the relativistic effects of high speed (kinetic energy) and high acceleration (potential energy), including time dilation, length contraction, and the curvature of SPACE and TIME. It is available free.

Free Excel Spreadsheet for VISUALIZATION of Einstein's Special and General Relativity. 

Image is of the Main Panel where user selects a Star, Planet, or Set Angle Option. In the case illustrated, the SpaceTime angle is set to 30º, where velocity is half the speed of light. This causes clocks to slow down by 13.4%, which corresponds to 49 days per year or 482 seconds per hour. Right side shows Special Relativity Effects due to the Kinetic Energy of moving at half the speed of light in empty Space. Left side shows equivalent General Relativity Effects, where Time "curves" due to the Potential Energy of being "at rest" close to a Black Hole.

Free Excel Spreadsheet for VISUALIZATION of Einstein's Special and General Relativity. 

Image is of the SpaceTime view of the right side of the Main Panel (where the vector sum of TRAVEL + AGING = 1) plus the Minkowski-Like SpaceTime view (where the simple sum of TRAVEL + AGING =1).  

Free Excel Spreadsheet for VISUALIZATION of Einstein's Special and General Relativity. 

Image is of the Minkowski-Like view (described above) compared to a Planck view, where both Space and Time are assumed to be discrete, and Each tiny cell is 1 Planck Time (tP ) by 1 Planck Length (ℓP).

THE MAP IS NOT THE TERRITORY!

As my principal PhD advisor, Howard Pattee, taught me, "The MAP is NOT the TERRITORY". That sage statement means that no model is exactly the same as the thing being modeled (else it would be the real thing.)

We make models because the real thing is too complex and difficult for us to visualize, or -like the Global Climate- is not readily available for us to experiment upon.

The MAP is NOT the TERRITORY !

Many a General (or football coach) has moved symbols around on a map of the field of battle, convincing himself and his staff of inevitable victory, only to find his opponent also had a model, perhaps a better one plus superior forces to carry it to victory. 

We generally model only the most important or critical parts of the situation or complex system we are trying to visualize. We consider the model to have been successful if the results match actuality to some level of fidelity, at least for those significant portions. If subsequent testing reveals that the model does not comport with reality, we must improve or discard it.

CONCLUSION

This is the first of what I believe will turn into a series detailing my VISUALIZATION of Einstein's Relativity. Please stay tuned!

Ira Glickstein

What is Time? Alan Alda's 2013 "Flame Challenge"

Time - the fourth dimension (2013 Flame Challenge) from Ira Glickstein on Vimeo.

My entry for Alan Alda's 2013 Flame Challenge was submitted last week. It is in the form of a short video  answering the deceptively simple question "What is Time?" (click above to view the video).

Alan Alda is on a mission to help youngsters become interested in science. In conjuction with the Center for Communicating Science at SUNY Stony Brook, he started the Flame Challenge in 2012 with the question "What is Flame?" They received some 800 entries.

I expect they will get even more this year with the question "What is Time?"

ABOUT MY ENTRY

I think I've come up with a unique way of viewing "Time - the fourth dimension".  Due to a strict limit on the length of the video, and the fact that it is aimed at 11-year old students, I have had to greatly simplify the material. This Blog posting includes additional material that will be useful to adult readers and science teachers who wish to know more about my way of viewing Time.

There are three big ideas here:

1. TIME is NOT a clock (any more than Space is a ruler or Heat is a thermometer), nor is it rotation of the Earth or motion or the order of events, etc.  
2. TIME is the fourth dimension, plain and simple. It appears different to us because the whole Universe is speeding along the Time axis at the speed of light.  
3. TIME slows down when we move in Space because nothing can move faster than the speed of light, so any motion in Space must take away from the speed in Time such that the vector sum of the Space and Time velocities exactly equals the speed of light.

WHAT TIME IS NOT

Time is not the tick, tick, tock of a click, click, clock, any more than Space is a ruler or Heat is  a thermometer!

 

Nor is Time the rotation of the Earth on its axis that gives us day and night divided into 24 hours. Nor is it the movement of the tilted Earth in orbit around the Sun that gives us the seasons, nor any other kind of motion. Nor is it the spontaneous decay of certain atoms that give radioactive materials a half-life. Nor is it simply the ordering of events.

WHAT TIME IS

Time, plain and simple, is the fourth dimension, very much like the first three dimensions of Space.

 

The Time dimension appears different to us because you and I and the whole Universe are hurtling along it at very nearly the speed of light as a consequence of the “Big Bang” expansion some 13.7 billion years ago, in which our Universe, along with the dimensions of Space and Time, originated.

Since Time itself originated with the "Big Bang" it may not be meaningful to even ask the question "What happened before the Big Bang?" In any case, we may never know what caused it.

 

The Universe originated as an incredibly energetic and dense point of Energy/Matter that suddenly expanded. During the initial moments of the expansion, it is not clear if there was anything like the sub-atomic and atomic particles of Matter or the radiation of Energy with which we are familiar today. However, when Matter and Energy, as we know it, formed, all particles with mass were expelled along the Time axis, or at very tiny angles with respect to that axis. You and I, along with everyone and everything else, are still moving along or near that dimension at very close to the speed of light, c, which is as fast as anything can go.

 

We do not notice our ultra-rapid travel along the Time dimension as motion because the whole Universe is moving along with us. Therefore, we notice only relative motion between ourselves and other people and between ourselves and other things.

 

For example, people on the equator are happily unaware that they are moving Eastward at about 1,000 miles per hour due to the rotation of the Earth on its axis. Unless you live in one of the polar regions, you are moving Eastward at hundreds of miles per hour. Even if you are on an airplane, travelling  "Westward" from New York to Chicago or Los Angeles at 500 miles per hour, your net velocity is Eastward, due to the rotation of the Earth! We are equally unaware that the whole Earth is speeding along at over 67,000 miles per hour on its orbit around the Sun!
 

WHY TIME CAN BE SLOWED A BIT
 

We live in four-dimensional SpaceTime where everything must move at the speed of light, c, either along the Time axis, along a Space axis, or in a combination of Time and Space at an angle, Θ, to the Time axis. If movement is totally aligned with the Time axis, Θ = 0 and we are said to be “at rest” in Space, and we move along the Time axis at the normal rate (c, about one foot per nanosecond).

 

If we are not "at rest" in Space, Θ > 0, and we move through SpaceTime in a combination of Space and Time such that the vector sum of our Space and Time velocities is exactly c. Since nothing can go faster than c, any movement in Space must slow down our movement in Time. This was recognized over 100 years ago by Lorentz, Minkowski, and Einstein, who use the terms "Dilation of Time" and "Contraction of Space". This is usually expressed in terms of the Lorentz factor:

where c is the velocity of light and v is the velocity of an object in Space.

As an engineer, I found that way of expressing relativistic effects of travel at significant fractions of the speed of light not to be "understandable" from my physical (and perhaps anal) point of view.

After knocking my head against the wall over an inordinate amount of Time, I finally realized that I could get an exactly equivalent Lorentz factor by considering the angle Θ, between the Time axis and the velocity vector of an object through SpaceTime.

[above image modified 12 April 2013]

It turns out that v (the velocity of the object in Space) divided by c is equal to the Sin Θ, and that 1/ϒ, the Lorenz factor, is equal to the Cos Θ.  

WHAT ARE DIMENSIONS?

 

This may sound like a simple question, and the answer is pretty simple, but, just to be sure we are all on the same page (see figure below):

 

0 - A POINT has ZERO dimensions

1 - Drag the point along the FIRST dimension ("x" of Space) and you get a LINE, that has ONE dimension.

2 - Drag the line along the SECOND dimension ("y" of Space) and you get a SQUARE (or rectangle) that has TWO dimensions.

3 - Drag the square along the THIRD dimenson ("z" of Space) and you get a CUBE (or rectangular solid) that has THREE dimensions.

4 - Drag the cube along the FOURTH dimension ("t" of Time) and you get a HYPER-CUBE (or hyper-rectangular solid) that has FOUR dimensions.



 

SUMMARY

When movement is a combination of Time and Space, and the velocity in Space is v, an object is moving through SpaceTime at an angle Θ, such that: v/c = SinΘ, and 1/ϒ (the Lorentz factor) = Cos Θ.

The figure below shows the situation for seven different values for the angle of travel through SpaceTime, from Θ = 0⁰ to Θ = 90⁰ .

Θ = 0⁰  [Sin Θ = 0.0000,  Cos Θ = 1.0000]   AT REST IN SPACE
For an object that is "at rest" in Space, Θ = 0⁰. Even when an object is not moving along the Space axis, it is moving along the Time axis. Since everything in SpaceTime must have a speed of c, an object "at rest" in Space must be moving at speed c in Time. Note that for this condition, v/c = 0 and the Lorentz factor ϒ = 1. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ.

Even the fastest rockets and satellites developed so far go only a tiny, tiny fraction of c. Therefore, for all practical purposes, the angle, Θ, is ≈ 0⁰ (approximately equal to ZERO degrees). For example, the Earth is travelling around the Sun at a speed of 67,000 miles per hour, faster than any rocket, but that is only 0.001 % of the speed of light. At 67,000 miles per hour, v/c =  0.00001 and  Θ = 0.0000017⁰.

Θ = 15⁰ [Sin Θ = 0.2588, Cos Θ = 0.9659]    MOVING 26% OF c IN SPACE
An object is moving through SpaceTime at an angle of Θ = 15⁰. It moves through Space at 26% of c and through Time at 97% of c. Note that for this condition, v/c = 0.2588 and the Lorentz factor ϒ = 0.9659. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ.

Θ = 30⁰ [Sin Θ = 0.5000, Cos Θ = 0.8660]   MOVING 50% OF c IN SPACE
An object is moving through SpaceTime at an angle of Θ = 30⁰. It moves through Space at 50% of c and through Time at 87% of c. Note that for this condition, v/c = 0.5000 and the Lorentz factor ϒ = 0.8660. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ.

Θ = 45⁰ [Sin Θ = 0.7071, Cos Θ = 0.7071] MOVING 71% OF c IN SPACE
An object is moving through SpaceTime at an angle of Θ =45⁰. It moves through Space at 71% of c and through Time at 71% of c. Note that for this condition, v/c = 0.7071 and the Lorentz factor ϒ = 0.7071. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ.

Θ = 60⁰ [Sin Θ = 0.8660, Cos Θ = 0.5000] MOVING 87% OF c IN SPACE
An object is moving through SpaceTime at an angle of Θ = 60⁰. It moves through Space at 87% of c and through Time at 50% of c. Note that for this condition, v/c = 0.8660 and the Lorentz factor ϒ = 0.5000. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ.

Θ = 75⁰ [Sin Θ = 0.9659, Cos Θ = 0.2558]   MOVING 97% OF c IN SPACE
An object is moving through SpaceTime at an angle of Θ = 75⁰. It moves through Space at 97% of c and through Time at 26% of c. Note that for this condition, v/c = 0.9659 and the Lorentz factor ϒ = 0.2558. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ.

Θ = 90⁰ [Sin Θ = 1.0000, Cos Θ = 0.0000]      TIME STANDS STILL 
Light (and other forms of electro-magnetic radiation) move through SpaceTime at an angle of Θ = 90⁰. Light moves through Space at 100% of c and, therefore, since nothing can go faster than c, Time stands still. Note that for this condition, v/c = 1.0000 and the Lorentz factor ϒ = 0.0000. Note also that, for this case Sin Θ is equal to v/c and Cos Θ is equal to 1/ϒ. Anything with mass cannot achieve this speed in Space because it would take an infinite amount of energy to get it up to this speed in Space.
 
[ADDED 11 March 2013] In response to some skepticism about my contention that the whole known Universe is speeding along the Time dimension at nearly the speed of light, I did more research and found support from Brian Greene, Professor of Physics and Mathematics at Columbia U, who has been featured on the PBS Nova series. He writes:

“Special relativity declares a similar law for all motion: the combined speed of any object’s motion through space and its motion through time is always precisely equal to the speed of light” [Excerpt From: Greene, Brian. “The Fabric of the Cosmos.” Vintage Books, 2007. See http://www.pbs.org/wgbh/nova/physics/fabric-of-cosmos.htm for his PBS series.]

I have provided more detail in the first comment below.

Ira Glickstein

Time, the flexible dimension

[From Joel] I was lying in bed this morning noting that the clock somehow had not sprung forward. I lay there thinking about time and in particular about waiting. At the moment, I'm waiting for some oral surgery to heal and anticipating the moment when I can eat some serious solid food. There's nothing like waiting for a wound to heal to turn your attention to how much of our lives are spent in anticipation of a future event.

Generally speaking, I try to avoid waiting. If there's a long line to go to a movie or participate in a buffet, I'll forgo the film or go hungry before I'll wait on line. But, waiting to heal is not something voluntary and sometimes the discomfort doesn't let you do other things that would normally let you ignore the wait. There seem to be at least two approaches to such a situation. Some people "bitch and moan" to pass the time while others just curl into a ball and make the world go away.

We have various interesting expressions concerning waiting. We sometimes call a hobby or a sport like golf a "pastime" from the French "passe temps," literally "pass the time way." Obviously, pastimes are diversions (another French word) which distract the mind from focusing on what seems like the interminable wait until we achieve our end state. The trouble is that sometimes the "diversion" is just that. It relieves the pain of waiting but also postpones or even prevents the achievement of the goal. Beer parties keep one's mind off waiting to complete one's education, but graduation never arrives for many who overindulge. We use the expressions "time marches on" and "time and tide wait for no man." Humans, like most animals and plants, have biological rhythms, known as circadian rhythms, which are controlled by a biological clock and work on a daily time scale. These affect body temperature, alertness, appetite, hormone secretion etc. as well as sleep timing. Our illusion of time slowing down or speeding up in emergencies is at conflict with our biological clock. Or is it?

Human evolution

[by JohnS]

My my, what did I start with my rebuttal, see A rebuttal to “Who was Cain’s wife?”. How did physics get into the discussion? How did time enter the discussion? There are at least three aspects of time, probably more. There is relativistic, Einsteinian, time, quantum time and classical Newtonian time. There are also two forms of evolution, physical evolution, the evolution of the universe and biological evolution, the evolution of flora and fauna on earth.

When we discuss human evolution, our discussion should be limited to classical time and biology. I’ll redefine my preposition. With the rise of mankind a third form of evolution arose within man, intellectual evolution. Humans are still subject to biological evolution however, intellectual evolution moves mankind at a much faster pace and allows humans to rise above and dominate nature. It may even allow humans to circumvent nature and biological evolution. It is improbable that any other earthly species will rise to compete with or replace humans. UNLESS humans disappear as the dinosaurs did.

Is it possible that any natural catastrophe can cause mankind to disappear? Can mankind cause his own downfall? Man is a very adaptable animal and I find it hard to conceive of his disappearance. Therefore, man will continue to dominate the earth until physical evolution makes the earth uninhabitable and by that time mankind may have gone into space.