VISUALIZING: The "Twin Paradox"

In Einstein's ground-breaking 1905 paper, On the Electrodynamics of Moving Bodies, he provides the basis for the well-known "Twin Paradox" (where one twin takes a space journey at high speeds, and finds, upon returning home, that he or she has AGED less than the stay-at-home sibling):

If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be [1 - α] seconds slow*.

To VISUALIZE Einstein's thought experiment, let "A" be a location on Earth, where the stay-at-home twin resides, and the "closed curve" be the path followed by the traveling twin, moving at 87% of the speed of light (v/c = β = 0.8660) where α = 0.5 as depicted below.

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

As depicted, Blair and Aden are 20 years old when Aden takes off on a long space journey at ultra-high speed while Blair remains home. Aden's journey, at an average speed of 87% the speed of light, extends out to the vicinity of a Neutron Star (or a Black Hole) where Aden's spaceship "slingshots" and speeds back to Earth.

When Aden is at the half-way point, Blair has AGED 30 Earth-years and is 50 years old. However Aden, due to being in a state of high Kinetic Energy with respect to Blair, has AGED only 15 years and is only 30 years old.

By the time Aden returns from the journey, Blair has AGED an additional 30 Earth-years and has reached the ripe old age of 80. However Aden has only AGED an additional 15 years, and returns home a sprightly 50 year-old!

WHAT DOES THIS VISUALIZATION TELL US?

First of all, this is only a "thought experiment" and there are many practical limitations that make it unrealistic. None of our current spacecraft are capable of even 1% of the speed of light, much less the 87% imagined for Aden. Furthermore, even if we had such a spacecraft, and even if it carried only a clock and not a fragile human being, considering the G-forces involved,  it would take a number of years to accelerate up to 87% of the speed of light, perform the "slingshot", and decelerate to land safely on Earth.

A more realistic depiction would include those years of acceleration and deceleration and would require some portions of the journey to be faster than 87% of the speed of light so as to average 87%.

Note that the Einstein quote is from Einstein's 1905 SPECIAL RELATIVITY paper and he (wisely) specifies that the "closed curve" be at "constant velocity". It would take an additional ten years, and Einstein's 1915 GENERAL RELATIVITY paper to account for the Relativistic Effects of the acceleration and deceleration required for a practical journey. It turns out that the acceleration and deceleration of the traveling twin in the spacecraft would actually increase the difference in AGING somewhat. However, 60 years of Earth gravity, to which the stay-at-home twin would be exposed, would actually decrease the difference in AGING a bit.

On the other hand, some of the explanations of the "Twin Paradox" I found on the Internet expose what I think are misinterpretations of inertial reference frames and simultaneity.

CONFUSION IN EXPLANATIONS OF THE "TWIN PARADOX"

Symmetry and Simultaneity Run Riot !

In my explanation above, I state that, at the half-way point, Aden, on the spaceship, has aged 15 years while Blair, on Earth, has aged 30 years. Well, some would complain, if Aden has not yet done the "slingshot" turn-around, it is improper to state anything about difference in aging between Blair and Aden since both are still in their original frames of reference! They claim a symmetry where each twin sees the other as moving and the other as having a slower clock. They claim there is no such thing as simultaneity.

Different Inertial Frames

For example, right near the top of the Wikipedia explanation:

... each twin sees the other twin as moving, and so, according to an incorrect naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged more slowly. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the two twins. [My emphasis]

Well, prior to the turn-around, each twin had one and only one inertial frame, not "two different inertial frames". So, does this explanation mean to say they both aged more slowly, or neither aged more slowly? Does it mean to say that, during the turn-around, the traveling twin suddenly got younger or the stay-at-home twin suddenly got older?

Sadly for me (an old engineer who cannot understand the meaning of minus two people - see the Physist/Engineer joke in my prevous posting ) YES, they do seem to think that the ages of the twins can suddenly change, based on how they do their calculations!

Gravitational Time Dilation

Further confusion in the Wikipedia explanation:

.... Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration.

According to this Wikipedia quote, the Einstein/Born explanations invoke "gravitational time dilation to explain the aging as a direct effect of acceleration."  Well, the traveling twin certainly had to be accelerated and decelerated during launch and recovery and during the turn-around, and we learn from General Relativity that Relativistic Effects of gravity are equivalent to high-speed effects at certain levels of acceleration and speed. However, the amount of reduction in aging is proportional to the total length of time the traveling twin is at ultra-high speed, and the thought experiment could be lengthened to hundreds or millions of years, such that the acceleration/deceleration periods are an insignificant fraction of the travelling twin's journey.

Age Jump Instantly At the Turn-around

Yet further confusion in the Wikipedia explanation:

... For a moment-by-moment understanding of how the time difference between the two twins unfolds, one must understand that in special relativity there is no concept of absolute present. ...For different inertial frames there are different sets of events that are simultaneous in that frame. This relativity of simultaneity means that switching from one inertial frame to another requires an adjustment in what slice through spacetime counts as the "present". ...

... Just before turnaround, the traveling twin calculates the age of the Earth-based twin ... [but] ... Just after turnaround, if he recalculates, ... there is a jump discontinuity in the age of the Earth-based twin. ... [If the twins] regularly update each other on the status of their clocks by way of sending radio signals (which travel at light speed), then all parties will note an incremental buildup of asymmetry in time-keeping, beginning at the "turn around" point. Prior to the "turn around", each party regards the other party's clock to be recording time differently from his own, but the noted difference is symmetrical between the two parties. After the "turn around", the noted differences are not symmetrical, and the asymmetry grows incrementally until the two parties are reunited. Upon finally reuniting, this asymmetry can be seen in the actual difference showing on the two reunited clocks. [My emphasis]

OK, the twins are far apart for much of this thought experiment so radio signals between them will take years to reach their destinations. Therefore, even if the turn-around plans have been settled and the Relativistic Effects calculated before the launch, the stay-at-home twin will not know for sure whether or not they have been successful. The spacecraft may have blown up or gone off the planned course. Similarly, the traveling twin will not know the status of the stay-at-home. The Earth may have been destroyed by a meteor, etc.

But, it blows my mind that some physicists can imagine an instantaneous jump in age by any human being (much less a clock) due to a spacecraft turning around, or a calculation based on a delayed radio message.

MY (SIMPLE) EXPLANATION OF THE "TWIN PARADOX"

Yes, if two spaceships pass in the night, all they can measure is relative speed (even if one happens to be Spaceship Earth). According to all that is currently known, observers on each spaceship will measure the other as being shorter in the direction of travel than it really is (length contraction) and that the other's clock is running slow (time dilation). I got that.

As one Internet source noted, when two cars pass on a highway and each driver looks in their rear-view mirror, the other car appears to be getting smaller.  Of course, in the case of the cars, we know that neither is really getting smaller.

So, what is different in the case of the twins?

Well, for one thing, the spacecraft was loaded with fuel and the stay-at-home twin watched it blast off and accelerate. The traveling twin felt the acceleration to ultra-high speed. Due to that expenditure of fuel, the spacecraft was raised to a higher level of Kinetic Energy than it had when it was sitting on the launch pad.

Throughout its journey, the spacecraft continued at high speed relative to the Earth (assuming that any frictional losses of energy were made up by further expenditure of fuel).

I maintain that the Relativistic Effects of a slowdown of aging (clock rate) for the travelling twin compared to the stay-at-home twin is due to a relatively higher level of Kinetic Energy. (When we get to General Relativity later in this Blog series, we will learn that high levels of Potential Energy due to the acceleration of gravity have similar Relativistic Effects.)

Ira Glickstein

* Note: I've substituted "1 - α" for the equivalent, but more complex equation in Einstein's original paper, where α is the Square Root portion of the Lorentz Transformation ( ) as described in my previous Blog posting.

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

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