Bayesian Inference Adviser

Bayes Theorem has practical applications. Use it to make real world decisions.

Bayes Theorem is not just an obscure artifact of the statistics of probability handed down to us from centuries ago. You can use it now to make decisions that will affect your financial well-being.

A relatively simple Excel-based tool helps you choose the right course of action in the face of uncertain probabilities and inexact test results. It is available for FREE.

What can the Bayesian Inference Adviser Tool do for You?

I know you are probably not in the oil business (and neither am I), but the best way to understand an abstract concept is to go through a practical example. The example below has to do with oil exploration and drilling, but the Bayesian Inference Adviser can handle any problem where you know:

1) the probability of success if you take action without doing further testing,
2) the cost of further testing and the probability the test results will be reliable,
3) the cost of taking some action,
4) the financial benefit to you if the action is successful, and
5) the compensation you expect for taking the risk.

You can download the Bayesian Inference Adviser at:
https://sites.google.com/site/bigira/stuff-ira-knows/BayesianInferenceAdvisorV3.xls?attredirects=0&d=1

The Bayesian Inference Adviser will compute the most likely financial implications of your actions. What if you proceed without further testing? What if you get a positive test result and proceed? What if you get a negative test result and proceed? Several other examples in very different domains are included later in this posting, but for now, put on your hard hat and let us imagine we are in the oil business!

What is TIME? (In the context of Relativity)

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

My entry for Alan Alda's 2013 Flame Challenge was submitted way back in February 2011. 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.]

Ira Glickstein

VISUALIZING - Using My FREE Excel Spreadsheet for Special and General Relativity

Nowadays it is common to use computer models 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 so on. I've created an interactive Excel Spreadsheet for Special and General Relativity.

User selects a Star, Planet, or Set Angle Option. 

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.

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.

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.

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.

Ira Glickstein

VISUALIZING My "Insight" into Lorentz "Gamma" and SpaceTime

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 lower-case 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 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). Quoting Einstein:

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*.

* 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.

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.

Relative ENERGY (Kinetic and Gravitational Potential Energy) Determines Relativistic Effects

LEFT: Jumpy Jill is in a friction-free tunnel and cycles forever. She is "at rest" at the top and bottom of the hole and therefore has zero Kinetic Energy (KE). She does have positive Gravitational Potential Energy (GPE) with respect to the center of the tunnel. Her maximum speed is achieved at the center. 

RIGHT: Far-Out Fred is launched into space at a hair less than Escape Velocity. He travels a very long distance over nearly infinite time before he runs out of KE and is momentarily "at rest". He falls back towards the planet, enters the friction-free hole at a hair less than Escape Velocity, reaches maximum speed at center, decelerates, exits hole at a hair less than Escape Velocity, heads into space, and cycles forever. 

Note: All speeds and relative energy are with respect to an observer "at rest" at the very center of the planet.

Relativistic Effects are Proportional to Kinetic Energy and Gravitational Potential Energy

Does relative TOTAL ENERGY determine the magnitude of Time Dilation and Length Contraction? Here are some equations, arguments and examples that support the idea that these Relativistic Effects are due to Total Energy with respect to a reference observer. I will appreciate comments and corrections.

Time Dilation is a relativistic effect that slows the rate of a clock that is moving and/or accelerating with respect to an “at rest” reference clock. 

"Coordinate" Time Interval between two events, Δt, ("delta tee") measured by a clock "at rest" in a reference frame (zero speed, zero gravity). 

"Proper" Time Interval between the same two events, Δτ, ("delta tau") measured by a clock that is moving and/or accelerating with respect to the reference frame (due to relative gravity and/or speed).

Δτ is equal to or greater than Δt due to Time Dilation. 

Lorentz Factor, γ = Δt/Δτ, equal to or greater than 1.0. (γ  is lower-case "gamma")

Lorentz Factor due to relative speed, γ_K = 1/SQRT(1-v²/c²) where v is the relative velocity between the "at rest" clock and the moving clock, and c is the speed of light in a vacuum. 

Lorentz Factor due to relative gravity, γ_G = 1/SQRT(1-2GM/rc²), where G is the gravitational constant, M is the mass of the spherical planet (or massive object) that is the source of the gravitational field, and r (which must be equal to or greater than the radius of the planet) is the distance of the clock from the center of the planet.

Time Dilation,TD = Δt - Δτ, is the "coordinate" time interval minus the "proper" time interval. 

Time Dilation,TD = (γ - 1)Δτ, an equivalent way to express Time Dilation. Please note that the (γ - 1) factor is common to the equation for Time Dilation as well as the expressions for KE and GPE.

Kinetic Energy of the clock, KE = (γ_K  – 1)m₀c², where m₀ is the "rest mass" of the clock.

Gravitational Potential Energy of the clock with respect to infinity, -GPE = (γ_G  – 1)m₀c² .

Escape Velocity, v_E, from a non-rotating, no atmosphere planet, P, is the speed, perpendicular to the surface, where KE is exactly equal to -GPE. (The minus sign is due to GPE being defined as zero at infinite distance from P, increasing in magnitude to its largest NEGATIVE value at the center, and then increasing asymptotically to zero an infinite distance from P.) A rocket launched at v_E will just break free of the gravity of P and continue into space forever, with no need for further propulsion.  v_E = SQRT(2GM/r), where G is the gravitational constant, M is the mass of P, and r (which must be equal to or greater than the radius of P) is the distance of the object from the center of P.

A clock moving at v_E (the Escape velocity associated with the surface of P), but far from any massive object (zero gravity), will have exactly the same TD as a clock “at rest” (zero speed) on the surface of P. Note that this occurs when the KE of the former is exactly equal to the -GPE with respect to infinity of the latter.

As an Engineer (not a Physicist) these facts scream out that relative ENERGY (KINETIC and/or -GRAVITATIONAL POTENTIAL) causes TD. I find it surprising that these facts are not mentioned at all in the Wikipedia entries for Escape_velocity nor Time_dilation, nor by most internet resources for relativistic effects. In the few instances where mentioned, the exact relationship of ENERGY to RELATIVISTIC EFFECTS is treated as an unimportant coincidence. It seems to me that RELATIVE ENERGY IS THE PRIME CAUSATIVE FACTOR FOR RELATIVISTIC EFFECTS, as confirmed by simple examination of the applicable equations.

Summary

A change in kinetic energy causes a linear change in TD.  A change in speed will cause a change in TD, but not in a way that is linearly related to the Lorentz factor. For example, a 1% increase in speed, from 0.098c to 0.099c will cause a 0.01% increase in TD, while a 1% increase from 0.98c to 0.99c will cause a  41% increase!

A change in GPE causes a linear change in Time dilation. A change in gravity will cause a change in TD, but not in a way that is linearly related to the Lorentz factor. (Indeed, the relationship between gravity and Time dilation is not even monotonic! When a clock is within a massive object, gravity decreases while Time dilation increases! [Thanks to John Rennie at the physics stack exchange for kindly (and with extraordinary patience) corrected my misconceptions about gravitational potential energy, see here.]

The equations for TD, E_K , and -E_G. share a common element: (γ - 1)

TD =  (γ  –  1)Δτ.

 E_K = (γ_K  – 1)m₀c² 
-E_G = (γ_G  – 1)m₀c²

When E_K = -E_G  then (γ_K  – 1)  = (γ_G  – 1) = (γ –  1) 

Therefore, for a clock with unit mass (1 kg), the time dilation per second of "proper" time, is (γ_K  – 1)/m₀c², due to relative speed, and (γ_G  – 1)/m₀c², due to relative gravity.

It should be "obvious" that TD experienced by a clock "at rest" on the surface of P is exactly equal to the TD experienced by a clock in deep space moving at the Escape velocity, v_E, associated with the surface of P.

LINEAR and MONOTONIC Relationship between ENERGY and Relativistic Effects

The Excel graphs below compare ENERGY, Velocity, and Gravity with the Time Dilation.

The analysis is based on the Kinetic Energy resulting from relative velocities from zero to 295,599,350 meters per second (0 to 98.6% of c, the speed of light) in deep space -or- the Gravitational Potential Energy resulting from being "at rest" within the gravitational attraction of either an Earth-like massive sphere or a compact Neutron Star-like massive sphere (with a mass of two Suns and a radius of only 13 km, experienced from an infinite distance down to the center.

Time Dilation is LINEAR with Respect to Kinetic Energy and NON-Linear with Respect to Velocity

The above graphic shows the relationship between Time Dilation, Velocity, and Kinetic Energy. Note that, for all speeds from zero to the speed of light, TD is exactly proportional to KE, and TD is not proportional to Velocity.

Time Dilation is LINEAR with Respect to Gravitational Potential Energy and NON-Linear and NON-Monotonic with Respect to Gravity (Example for Earth-Like Spherical Massive Object)

The above graphic shows the relationship between Time Dilation, Gravity, and Gravitational Potential Energy. Note that, for this example, using a spherical mass with the same mass and radius of the Earth, TD is exactly proportional to GPE. Note also that the relationship between TD and Gravity is not even monotonic!
From Infinity to the Surface, as Gravity INCREASES in (negative) magnitude, TD increases, but non-linearly.
From the Surface to the Center, as Gravity DECREASES in (negative) magnitude, TD increases, also non-linearly.

Time Dilation is LINEAR with Respect to Gravitational Potential Energy and NON-Linear and NON-Monotonic with Respect to Gravity (Example for Neutron Star-Like Spherical Massive Object)

The above graphic shows the relationship between Time Dilation, Gravity, and Gravitational Potential Energy. Note that, for this example, using a spherical mass with the same mass and radius of a Neutron Star, which is much, much more massive than the Earth, and very much smaller, so its density is far greater than the Earth, TD is exactly proportional to GPE. Note also that the relationship between TD and Gravity is not even monotonic!
From Infinity to the Surface, as Gravity INCREASES in (negative) magnitude, TD increases, but non-linearly.
From the Surface to the Center, as Gravity DECREASES in (negative) magnitude, TD increases, also non-linearly.

Conclusion

TOTAL ENERGY with respect to the center of P determines the magnitude of Time Dilation and Length Contraction for the entire journeys of both Fred and Jill. These journeys involve movement at various speeds, including momentary instants of being “at rest”, and various levels of gravitational attraction, including momentary instants of zero gravity.

The equations for TD, E_K , and -E_G. share a common element: (γ - 1)

TD =  (γ  –  1)Δτ. where γ = Δt/Δτ,

 E_K = (γ_K  – 1)m₀c², where   γ_K = 1/SQRT(1-v²/c²)  
-E_G = (γ_G  – 1)m₀c², where   γ_G = 1/SQRT(1-2GM/rc²)
When E_K = -E_G  then (γ_K  – 1)  = (γ_G  – 1) = (γ –  1) 

Therefore, for a clock with unit mass (1 kg), the time dilation per second of "proper" time, is (γ_K  – 1)/m₀c², due to relative speed, and (γ_G  – 1)/m₀c², due to relative gravity.

When E_K = -E_G  then (γ_K  – 1)  = (γ_G  – 1) = (γ  –  1) 

My Excel analysis confirms, for a wide range of relative velocities and gravitational situations, ENERGY remains linearly related to Time Dilation and Length Contraction This is definitely NOT the case for Velocity nor Gravity, each of which are asymptotically limited.

Therefore it is reasonable to conclude that relative TOTAL ENERGY determines the Relativistic Effects of any combination of speed and gravity.

[Edited 24 July 2021. I fixed a couple of typos and changed "ϒ" to "γ".]

Ira Glickstein