Tag: Special relativity

What is Time

Image above by Kevin from The Beginning at Last

Many and strange are the universes that drift like bubbles in the foam upon the River of Time—first sentence of Wall of Darkness by Arthur C. Clarke, 1949.

What is Time? What is the arrow of time? Does entropy define an arrow of time? Is there a past and a future? Has time always existed as, for example, in the eternal inflation cosmological model in which universes continuously pop into existence via Big Bangs like the bubbles on the River of Time in the Arthur C. Clarke quote above. Does time have a beginning as a recent Big Bang model by Stephen Hawking imply. If time came into existence with the universe itself then there is no place for creation, just as in an infinitely existing universe. There cannot be anything north of the north pole, and there cannot be anything before time itself.

In the theories of relativity time is relative. The speed of clocks as well as the order of events differ from observer to observer. The presence of energy or mass will bend time. There is no universal flow of time. There is no universal time.

The goal of this blog is to create a list of what I call super facts. Important facts that we know to be true and yet they are surprising, shocking or disputed among non-experts. It is a type of myth busting. However, I also create posts that are not super facts but that feature other interesting information, such as this book review and book recommendation.

In the picture there are objects from the past as well as futuristic elements and several clocks | What is Time
This picture reminded me of the mysteries of time. This is a submission for Kevin’s No Theme Thursday

About Time: Einstein’s Unfinished Revolution by Paul Davies

The book I am about to present focuses a lot on relativity. This is not strange since the theories of relativity shed a lot of light on time. Yes, pun intended. I’ve come across a lot of people who make various claims about time, and even consider themselves philosophers of time, and yet they do not even understand the special theory of relativity.

Because of this they say a lot of profound sounding things about time that ultimately are nonsensical. There is no way around it. If you want to say profound things about time and expand human knowledge on this topic you need to first have some understanding of relativity as well as other related physics such as thermodynamics and entropy. This book will provide that.

I should say that the book is somewhat outdated, being written in 1995 and 1996. However, most of it is still relevant and I know of no other book that approaches the topic in such an honest and rigorous way. The book comes in two formats hardcover and paperback. I bought the paperback version.

  • Hardcover –  Publisher – Simon & Schuster; First Edition (March 13, 1995), ISBN-10 : 0671799649, ISBN-13 : 978-0671799649, 316 pages, item weight : 1.3 pounds, dimensions : ‎ 6.25 x 1.25 x 9.5 inches, it costs $18.09 on US Amazon. Click here to order it from Amazon.com.
  • Paperback –  Publisher – Simon & Schuster; First Edition (April 9, 1996), ISBN-10 : 0684818221, ISBN-13 : 978-0684818221, 316 pages, item weight : 0.704 ounces according to Amazon but I think that should be 0.704 pounds, dimensions : ‎ 9.21 x 6.14 x 0.8 inches, it costs $12.28 on US Amazon. Click here to order it from Amazon.com.
Front cover of hardback version of the book About Time: Einstein's Unfinished Revolution by Paul Davies
Front cover of hardback version of the book About Time: Einstein’s Unfinished Revolution by Paul Davies. Click on the image to go to the Amazon page for the hardcover version of the book.

Amazon’s Description of About Time: Einstein’s Unfinished Revolution

An elegant, witty, and engaging exploration of the riddle of time, which examines the consequences of Einstein’s theory of relativity and offers startling suggestions about what recent research may reveal.

The eternal questions of science and religion were profoundly recast by Einstein’s theory of relativity and its implications that time can be warped by motion and gravitation, and that it cannot be meaningfully divided into past, present, and future.

In About Time, Paul Davies discusses the big bang theory, chaos theory, and the recent discovery that the universe appears to be younger than some of the objects in it, concluding that Einstein’s theory provides only an incomplete understanding of the nature of time. Davies explores unanswered questions such as:

* Does the universe have a beginning and an end?

* Is the passage of time merely an illusion?

* Is it possible to travel backward — or forward — in time?

About Time weaves physics and metaphysics in a provocative contemplation of time and the universe.

My four-star review for About Time: Einstein’s Unfinished Revolution

The link above will take you to my original review for About Time: Einstein’s Unfinished Revolution. Below is a slightly modified version of the review.

Our Evolving Concept of Time

Science, particularly physics, has revealed to us some profound insights about time, and yet it remains a mysterious topic. In the first chapter of the book “A Very Brief History of Time” the author describes how the concept of time has evolved throughout human history. The ancient philosophers had a very fuzzy notion of time, and they often dismissed both motion and time as illusions. The author tells us about “cyclic time”, and the “linear time” concept from Judeo-Christian thinking, and how it was incorporated into science as a “time variable”. This made it possible to solve the paradoxes created by such Greek thinkers as Zeno, who based on apparent paradoxes concluded that motion and time cannot really exist. Those “false” paradoxes were solved with the concept of linear physical time.

Physical time also made Calculus and Physics possible, and it became clear that there was a difference between physical time, which is a measurable entity used to sequence events, and experienced time, which is the subjective human feeling of a past, present, and a future.

Most of the rest of the book is focused on the Special Theory of Relativity and the General Theory of Relativity. The theories of relativity revolutionized how we view time, and they enabled a much deeper understanding of time. Special relativity refuted the concept of universal absolute time. Not only do clocks in different systems run at different speeds, but there is no universal “Now”.

However, the theories of relativity are not all there is to know about time, and we are not even done drawing conclusions about time based on the theories of relativity. This is why the book is called “About Time Einstein’s Unfinished Revolution”. The book also discusses quantum time, the arrow of time, worm holes, neuroscience and modern psychological experiments.

The book contains a lot of information and yet it scratches only on the surface. Time is a quite complex concept when you start digging into it. Take, for example, the discussion on the arrow of time. The conclusion of the books seems to be that there really is an arrow of time. Time has direction. The author is presenting both “sides” of the issue and to really get a grasp of the issue you need to clarify and look at the physics. Time certainly seems to have a direction if you look at the concept of entropy. The Cosmos is filled with irreversible processes. Our memory, which is essential for our experience of a self, is another example.

However, time has a direction even on the most fundamental level in subatomic particles. Kaon, a type of meson, decays in a way that is not time symmetric. The same may be true for the neutron. These particles “know” the difference between the past and the future. One could easily imagine how this chapter could be turned into a several thousand pages long book by just digging deeper into the related physics.

“About Time Einstein’s Unfinished Revolution” covers a lot of interesting topics related to time and it is often quite informative. The book is also well written, well organized, it has a good pace, and it is very interesting. It strikes a good balance between depth and breadth, and it does not overwhelm its readers. However, I do have some critique of the book, which prompted me to give the book four stars instead of five stars.

To understand some of the discussions in the book you should have a decent understanding of the theories of relativity, in my opinion. The author is explaining the theories of relativity to some degree, but the explanations did not seem to be that good. I did not have a problem with understanding these explanations since I already studied these topics, but I don’t think I would have understood what I needed to understand had I encountered the theories of relativity for the first time while reading this book.

Another criticism I have of the book is that he introduces ugh Everett’s many-worlds interpretation of quantum mechanics and in my opinion promotes it. This interpretation is in my opinion both strange and implausible and is not necessary for the discussions in the book.

In short, it proposes that every quantum measurement creates new universes, each corresponding to a different possible outcome of that measurement. In essence, our Universe is just one of many, and each quantum event splits the universe into multiple versions, where each version experiences a different outcome.

The problem with this interpretation is not so much that you get multiverses, but that you an infinite number of new fully evolved universes every nanosecond, which in turn create an infinite number of universes, which in turn create an infinite number of universes, and so on infinitely, and there is no way to empirically verify their existence. It is a so-called solution to a problem that is fictitious in my opinion and is derived from an old naïve interpretation of quantum waves. It is “hip” and “cool” interpretation, but I don’t see how it is useful. I should add that I have no problem with, for example, the multiverse idea stemming from string theory.

Anyway, except for my two complaints I think this is a pretty good book that I can recommend.

Front cover of paperback version of the book About Time: Einstein's Unfinished Revolution by Paul Davies.
Front cover of paperback version of the book About Time: Einstein’s Unfinished Revolution by Paul Davies. Click on the image to go to the Amazon page for the paperback version of the book.

About Paul Davis

Paul Davies is an internationally acclaimed physicist, cosmologist, and astrobiologist at Arizona State University, where he runs the pioneering Beyond Center for Fundamental Concepts in Science. He also chairs the Search for Extraterrestrial Intelligence Post-Detection Taskgroup, so that if SETI succeeds in finding intelligent life, he will be among the first to know.

The asteroid 1992OG was officially renamed Paul Davies in his honor. In addition to his many scientific awards, Davies is the recipient of the 1995 Templeton Prize–the world’s largest annual prize–for his work on science and religion. He is the author of more than twenty books, including The Mind of God, About Time, How to Build a Time Machine, and The Goldilocks Enigma. He lives in Tempe, Arizona.

Other Relativity Related Posts

To see the Super Facts click here

The Pole-Barn Paradox and Solution

Super fact 39 : Relativistic length contraction goes both ways. If two observers are moving compared to each other both will observe the length of the objects in the other’s system to be shorter in the direction of motion. The first observer will think that a yard stick in the second observer’s frame will be shorter whilst the second observer will think that the yard stick in the first observer’s frame is the shorter one.

Assume a pole and a barn are of equal length when both objects are stationary. If the pole is moving (at a high speed) compared to the barn, then the pole will be shorter than the barn from the barn’s perspective but longer than the barn from the pole’s perspective. Does the pole fit inside the barn or not? This is referred to as the pole-barn paradox, or the barn-door paradox, or the ladder paradox (if a ladder is used instead of a pole).

I call this conundrum a super fact because whilst most people have heard of relativistic time dilation and perhaps length contraction, the fact that it goes both ways comes as a surprising head scratcher. The situation is analogous to my super fact post “Time Dilation Goes Both Ways” where I state:

Super fact 38 : If two observers are moving compared to each other both will observe the other’s time as being slower. In other words, both observers will observe the other’s clocks as ticking slower. Time slowing down is referred to as Time Dilation. And this post is about how time dilation goes both ways.

Both the time dilation paradox and the pole-barn paradox are solved by the non-simultaneity in relativity. However, the pole-barn paradox is more concrete and perhaps more in your face. You can easily imagine the problematic paradox.

A picture of a girl, Amy who is speeding past a man, Alan and his barn. Amy has a pole. The pole is contracted along the direction of motion from Alan’s perspective and the barn is shorter along the direction of motion from Amy’s perspective | The Pole-Barn Paradox and Solution
Amy is speeding past Alan and his barn at a high speed. Amy has a pole. Because of the high-speed Amy’s pole appears shortened and will easily fit in Alan’s barn. However, to Amy it is Alan’s barn that is contracted, and her pole has the normal length and will therefore not fit in Alan’s barn.

Postulates of Special Relativity

The two postulates of special relativity are:

  • The laws of physics are the same in all inertial frames of reference. An inertial frame is a system that moves at a constant velocity.
  • The speed of light in a vacuum is constant for all observers, regardless of the motion of the light source.

The first postulate is called the principle of relativity and goes all the way back to Galileo Galilei. It means that no experiment can determine whether you are at rest or moving at a constant velocity. The reciprocity of length contraction follows from this postulate. If the length of the pole in the example above is half as long as the barn in both the barn frame and the pole frame then you could tell who was standing still and who was moving from that fact, and that violates the first postulate. The first postulate demands that if the pole is half as long in the barn frame and that the barn is half as long in the pole frame.

The second postulate is the more shocking one and is special to relativity. It was discovered experimentally at the end of the 19th century but was too difficult for scientists to accept at first so various ad hoc explanations were put forth to explain it away, until the theories of relativity were created. I designated this postulate as my super fact #4 and you can read about it here.

Length Contraction

Time dilation means that a time interval between two events in a certain frame is longer by a factor B in a frame moving relative to the first frame (see picture below). Let’s imagine Amy moving at the speed v compared to Alan and his barn. Amy passes the left side of the barn at a certain time and soon after the right side. The time difference from Alan’s perspective is T and the width of the barn is L, so L = vT. From Amy’s perspective the time difference is T’ and width of the barn L’ and L’ = vT’. We denote Amy’s measurements with a prime. Note the velocity must be the same in both systems. However, Amy’s clock ticks slower (from Alan’s perspective) so T’ = BT or T = T’/B (time dilation). So, L’ = vT’ = vT/B = L/B.

If the derivation of the formulas above is confusing to you, ignore the math, and just remember that Alan measures a shorter time for the passing of the pole (because Amy’s clock is slower) from his perspective and therefore the pole must be shorter as measured from his system. If Alan measures two seconds for the passing of the pole than Amy measures maybe four seconds. It is Amy’s pole, so her longer measurement corresponds to the proper length of the pole whilst Alan’s measurement is the contracted length. Note the length contraction can only happen along the direction of motion, not perpendicular to it. To read more about length contraction click here.

This picture shows the formula for time dilation, the expression for the beta factor, and the formula for length contraction | The Pole-Barn Paradox and Solution
The beta factor used in the formula for time dilation as well as length contraction.

Solution to the Pole-Barn Paradox

So, Amy’s pole cannot fit in Alan’s barn. The pole is moving fast so it must move in and out of the barn. Now let’s create the paradox. Imagine the barn having doors on each side that open for the moving pole and then close for a moment to entrap the pole and then they open as the pole leaves the barn. Here is the paradox, if they open and close at the same time, than the pole can be inside the barn (entrapped) from Alan’s perspective but not from Amy’s perspective. From Amy’s perspective the pole does not fit.

However, the solution to the paradox lies in “open and close at the same time”. If the doors open and close at the same time from Alan’s perspective, then they don’t open and close at the same time from Amy’s perspective.

From Amy’s perspective the door on the left side will open first and let the pole in and then after that the right door will open. After the pole has fully entered the barn and some of it is sticking out on the right-hand side then the left door will close but the door on the right will remain open  until the pole is entirely outside. Relativistic non-simultaneity solves the paradox.

A picture of a girl, Amy who is speeding past a man, Alan and his barn. Amy has a pole. The pole is contracted along the direction of motion from Alan’s perspective and the barn is shorter along the direction of motion from Amy’s perspective. There are two doors on each side of the barn. In Amy’s world the left door is open letting the pole into the barn, whilst the right door is closed. In Alan’s world both doors are close thus enclosing his shorter pole.
In Alan’s frame the doors can be closed at the same time and enclose Amy’s pole. In Amy’s frame the doors open and close to let the pole through but they don’t open and close at the same time.

Finally, below is a YouTube video that explains and solves the pole-barn / barn-door / ladder paradox simply and efficiently in a little over two minutes.

Book Recommendations on Relativity

To see the other Super Facts click here

Time Dilation Goes Both Ways

Super fact 38 : If two observers are moving compared to each other both will observe the other’s time as being slower. In other words, both observers will observe the other’s clocks as ticking slower. Time slowing down is referred to as Time Dilation. And this post is about how time dilation goes both ways.

A lot of people know that if someone moves very fast his clocks will run slower. That’s relativity. If someone speeds through space in a rocket ship, close to the speed of light his time will slow down. When one hour passes on earth only half an hour may pass in the rocket. What comes as a shock to many people is when they find out that the converse is also true. When one hour passes in the rocket only half an hour will pass on earth.

Clearly that looks like a contradiction, but there is an explanation. I consider this a super fact because it is so strange and almost impossible for people to believe, and yet it is true.

The image shows two clocks side by side. On the left is a wall clock and on the right a wristwatch | Time Dilation Goes Both Ways
The guy on earth says my clock (left) is ticking double as fast as the rocket man’s clock (right). The rocket man say’s my clock (right) is ticking double as fast as the clock on earth (left). Who is right? Surprisingly both of them.

Postulates of Special Relativity

The two postulates of special relativity are:

  • The laws of physics are the same in all inertial frames of reference. An inertial frame is a system that moves at a constant velocity.
  • The speed of light in a vacuum is constant for all observers, regardless of the motion of the light source.

The first postulate is called the principle of relativity and goes all the way back to Galileo Galilei. It means that no experiment can determine whether you are at rest or moving at a constant velocity. The reciprocity of time dilation follows from this postulate. If the time for the rocket man in the example above was ticking at half the speed compared to the time for the guy on earth and they both agreed, then you could tell who was standing still and who was moving from that fact.

The first postulate demands that they disagree. The guy on earth thinks the rocket man’s clock is ticking at half the speed of his own clock, whilst the rocket man think it is earth man’s clock that is going slow. Therefore, you can’t tell who is standing still, which is what the first postulate requires.

The second postulate is the more shocking one and is special to relativity. It was discovered experimentally at the end of the 19th century but was too difficult for scientists to accept at first so various ad hoc explanations were put forth to explain it away, until the theories of relativity were created. I designated this postulate as my super fact #4 and you can read about it here.

The picture shows two people Alan and Amy. Alan is on the ground. Amy is flying by Alan in a rocket speeding left. Both Alan and Amy are pointing lasers to the left | Time Dilation Goes Both Ways
In this picture Amy is traveling past Alan in a rocket. Both have a laser. Both measure the speed of both laser beams to be c = 299,792,458 meters per second. The speed of light is a universal constant.

Time Dilation

In the pictures below I am showing two rocket systems in space, Amy’s rocket and Alan’s rocket. They are travelling at a high speed compared to each other. Each rocket has a light clock that consists of a light beam bouncing up and down between a mirror in the ceiling and a mirror on the floor. The two light clocks are identical, and each bounce corresponds to a microsecond.

Amy is passing Alan at a high speed, and therefore Alan will see Amy’s light clock running slower than his because Amy’s light beam must travel further. Remember, the speed of light is identical for both light clocks (light speed is a universal constant). For those interested I am also deriving the formula for time dilation.

The picture shows two systems, each with a clock consisting of light beams bouncing between mirrors. In this set up Alan is stationary compared to us and therefore his light beam only moves vertically.
Alan and Amy have identical light clocks. We call the time it takes for the light beam to go from the floor to the ceiling (one clock tick) Dt in Amy’s case and Dt’ (reference frame) for Alan. Amy is speeding past Alan towards the left. From Alan’s perspective Amy’s clock is running slower. Using Pythagoras theorem, it is possible to derive the formula for time dilation shown in the lower left corner.

Since Amy moving left is the same as Amy standing still and Alan moving right you can say that Alan is the one moving fast. In this case it is Alan’s light clock that is ticking slower because from this viewpoint it is his light beam that has to travel further. From Amy’s perspective it is Alan’s clock that is going slower.

The picture shows two systems, each with a clock consisting of light beams bouncing between mirrors. In this set up Amy is stationary compared to us and therefore her light beam only moves vertically | Time Dilation Goes Both Ways
It is equally correct to say that Amy is standing still and that it is Alan that is moving fast to the right. This time (pun not intended) it is Alan’s clock that is ticking slower. Dt corresponds to Alan’s clock ticks and Amy’s clock ticks are Dt’.

This seemingly contradictory situation is resolved by the fact that Amy’s and Alan’s perspectives will drift apart as they continue their journey. They will increasingly disagree on whether events are simultaneous or not, and they will disagree in which order events occur. This is another shocking fact, or as I refer to it, super fact. It is strange but it resolves the apparent contradiction of reciprocal time dilation. I am explaining this in greater detail in this post.

The Twin Paradox

But what happens if one of Amy or Alan decides to turn around so that they meet up again. If Amy’s clock runs slower from Alan’s perspective and Alan’s clock runs slower from Amy’s perspective, how can you reconcile that when they meet up again? It turns out that whoever is turning around or accelerating or decelerating to turn back is the one who will have the least time pass. If Amy is the one turning back, then she will age less than Alan. During her acceleration she will see Alan’s clock starting to run faster and faster until he is older her.

Let say Alan’s clock is running half the speed of Amy’s clock from Amy’s perspective and Amy’s clock is running half the speed of Alan’s clock from Alan’s perspective. Let’s also say that Amy traveled to the left for 10 years before turning around.

From Alan’s perspective she would have traveled 20 years before turning around. However, from Amy’s perspective 5 years would have passed on Alan’s clock. As she turns around Alan’s clock will run faster and catch up so that when they meet up again Amy will be aged 20 years, while Alan will be aged 40 years. That is 35 years of catching up for Alan’s clock from Amy’s perspective. Alan’s clock advanced 35 years from Amy’s perspective after Amy turned around. In the end Amy will be the younger one.

The picture shows Amy on the left turning around and Alan on the right. Text explains what happens | Time Dilation Goes Both Ways
Observe that the fast-forward advancement of Alan’s clock from Amy’s perspective happens only while Amy is in the process of turning around (accelerating / decelerating). Further, how fast the fast forward happens depends on the distance as well. Once Amy is traveling at a constant speed again (inertial frame) Alan’s clock will run slower again from Amy’s perspective.

A somewhat halting but OK analogy for the 35 years of catching up that happens on Alan’s clock from Amy’s perspective is when you turn a boat around on a wavy sea. As you are moving in the direction of the waves the waves will hit you much less often (if at all) but after you turn around and move against them the waves will hit your boat very frequently. Alan’s clock will run faster for Amy whilst she is turning around.

Book Recommendations on Relativity

To see the other Super Facts click here

The Special Theory of Relativity by David Bohm

The goal of this blog is to create a list of what I call super facts. Super facts are important and true facts that are nevertheless highly surprising to many, disputed among the public, or unnecessarily misunderstood. They are special facts that we all can learn something important from. However, I also make posts that are not super facts but feature other interesting information, such as this book review and book recommendation.

The Special Theory of Relativity by David Bohm

David Bohm a close colleague of Albert Einstein was one of the most interesting theoretical physicists of the 20th century. This book “The Special Theory of Relativity” is one of the most interesting and thorough introductory books on the Special Theory of Relativity that I have come across. It is a series of lectures on the topic. It features algebra, equations, and a little bit of differentials but not too much.

You may want to have some math, physics and some relativity under your belt before you tackle this book. In my estimate it is written for those who have taken high school AP physics and AP algebra, maybe calculus as well, or one or two college level physics classes and math classes (or more). It is probably too basic for professional physicists, but it is not written for laymen. I bought the paperback version.

  • Hardback –  Routledge; 1st edition (October 10, 1996), ISBN-10 : 0415148081, ISBN-13 : 978-0415148085, 256 pages, item weight : 13.6 ounces, dimensions : ‎ 5.75 x 0.75 x 9 inches. It is out of stock, and it costs $75.60 – $144.00 on US Amazon. Click here to order it from Amazon.com.
  • Paperback –  Publisher : Routledge; 1st edition (September 4, 2006), ASIN : 0415404258, ISBN-13 : 978-0415404259, 304 pages, item weight : 12 ounces, dimensions : 5.08 x 0.69 x 7.8 inches. It is out of stock, and it costs $12.36 – $17.21 on US Amazon. Click here to order it from Amazon.com.
  • Kindle –  Published : Routledge; 1st edition (September 29, 2015), ASIN : B009W3W6MG, 306 pages, it costs $10.10 – $13.77 on US Amazon. Click here to order it from Amazon.com.
The front cover of “The Special Theory of Relativity by David Bohm.”
Front cover of the paperback version of The Special Theory of Relativity by David Bohm. Click on the image to go to the Amazon page for the paperback version of the book.

Amazon’s Description of the Book

In these inspiring lectures David Bohm explores Albert Einstein’s celebrated Theory of Relativity that transformed forever the way we think about time and space. Yet for Bohm the implications of the theory were far more revolutionary both in scope and impact even than this. Stepping back from dense theoretical and scientific detail in this eye-opening work, Bohm describes how the notion of relativity strikes at the heart of our very conception of the universe, regardless of whether we are physicists or philosophers.

This is my five-star review for The Special Theory of Relativity

Note, I wrote this review in 2014, so it is relatively old, pun intended.

Lorentz Electrodynamics, Special Relativity, and our Perception of Reality

This book is a thorough and well written introduction to the “Special Theory of Relativity”. In addition to the basics of special relativity it covers the history of Special Relativity and it includes 60-pages of Lorentz Electrodynamics. The book also discusses Minkowski Diagrams, the Twin Paradox, relativistic Doppler effects, K-Calculus, and philosophy related to relativity. The book does not discuss General Relativity.

Bohm does not derive many formulas for electrodynamics, optics, quantum physics, thermodynamics, etc., and therefore this book does not resemble a textbook. Bohm’s focus is on a deeper understanding of the special theory of relativity itself, and on time and space.

He discusses perception of reality and includes discussions on child development, psychology and neurology related to perception, the meaning of the relational concepts in relativity, the structure of scientific revolutions (T.S. Kuhn), our perceptions of time and space, philosophy, and other related topics that cannot be classified as physics.

In K Calculus you draw the world lines of light pulses sent at constant intervals between different observers. Then you calculate what is essentially the Doppler factor K and uses it to explain what is going in relativity. In contrast the Lorentz transform is concerned with the space-time coordinates that you measure after taking into account that light have to travel a certain distance and that this takes time. Basically, the “actual time and space coordinates”.

K Calculus on the other hand is including both relativistic effects and the distance and time it takes for light to travel. K Calculus is thus not what you “measure” but what you “observe”. K Calculus makes it very easy to explain special relativity to an audience that is not strong in math and it also adds a new understanding to the special theory of relativity.

However, I see an inherent risk with K-Calculus in that it can end up fooling the student into believing that he understands relativity, for example, if a student incorrectly comes to believe that time dilation is some form of Doppler Effect.

Except for the fact that some of the mathematical derivations were unnecessarily complex I thought this book presented relativity lucidly, the philosophical discussions were insightful, and it added to my understanding of the topic. It should probably not be your first book on Special Relativity but it is a very good second book. I highly recommend it for those who want to think through the concepts of relativity a little deeper.

The back cover of "The Special Theory of Relativity by David Bohm"
Back cover of the paperback version of The Special Theory of Relativity by David Bohm. Click on the image to go to the Amazon page for the kindle version of the book.

To see the Super Facts click here

GPS uses relativity for accuracy

Superfact 23: GPS uses relativity for accuracy. Global Positioning Systems or GPS uses Special Relativity and General Relativity to guide you to your destination. In fact, GPS systems would be rendered useless without the Theories of Relativity.

Businessman finger pin for location points and search addresses on the world map application. Marking destination for travel or finding business places in GPS Satellite coordinates system online web | GPS uses relativity for accuracy
Stock Photo ID: 2502019165 by mayam_studio

Did you use Einstein’s Theories of Relativity to get to the grocery store today?

The theories of relativity may seem strange and impractical, something you only use for astrophysics, black holes, cosmology and extreme velocities. They feature strange concepts such as time dilation, the stretching and bending of space, events simultaneous to some are not to others, the universal constancy of the speed of light in vacuum, the energy and mass equivalency, etc. 

Therefore, it is a bit surprising that without the theories of relativity the GPS app on your phone would not be able to guide you to the grocery store. That’s why I call it a super fact that GPS uses relativity for accuracy.

Space satellite orbiting the Earth. 3D rendering
Stock Illustration ID: 1372134458 by Boris Rabtsevich

GPS and Time Dilation

GPS is a satellite-based  radio navigation system that provides location information and time anywhere on Earth. It is amazingly accurate. The basic GPS service provides users with approximately 7.0-meter accuracy, 95% of the time, anywhere on or near the surface of the earth.

The fact that the information is provided by satellites that orbit earth at high speeds and high above earth’s surface makes General Relativity and Special Relativity necessary. The GPS system needs to calculate precisely the time it takes for signals to travel from the satellites to a receiver on Earth for it to work. GPS satellites travel at high speeds causing a large enough time dilation that must be accounted for. In addition, they orbit earth high above earth’s surface where earth’s gravitational field is weaker than on earth’s surface. Clocks run faster in weaker gravitational fields due to gravitational time dilation, so you must correct that as well.

If you ignore relativity, you will accumulate a discrepancy of six miles in one day.  You are not going to find the grocery store that way, unless you use the old-fashioned method of reading a map. In a sense, if your GPS device finds the grocery store for you, you have proven Einstein right.

Below is a YouTube video animation visualizing the GPS system.

GPS Facts

  • The GPS project was started by the U.S. Department of Defense in 1973. It is also owned by the U.S. Department of Defense.
  • The GPS satellites were sent up by the United States Air Force (and not NASA).
  • The first NAVSTAR satellite, later called GPS, was launched in 1978.
  • There are 31 GPS satellites currently in orbit.
  • The system requires 24 GPS satellites.
  • The 24-satellite system became fully operational in 1993.
  • The Global Positioning System cost (the US government) $1.8 billion annually to operate and maintain.
  • The Global Positioning System is free to use for the public worldwide.
  • Making GPS free to civilians worldwide was a decision by President Ronald Reagan in 1983 after a Korean airliner was shot down for straying off course.
  • GPS satellites carry extremely accurate atomic clocks. As explained, GPS must account for relativity, special relativity as well as General Relativity.
  • Other satellite systems help improve GPS, including WAAS (in the U.S.), EGNOS (in Europe), and MSAS (in Japan).
  • GPS is not the only satellite navigation system. Other countries have their own satellite navigation systems. GLONASS (Russia), Galileo (EU), and BeiDou (China).
  • Ukraine is helped by both GPS and Galileo.
  • Russian forces have been actively jamming GPS signals in Ukraine.

Uses of GPS

  • Examples of consumer electronics that use GPS are smart phones, tablets, Smartwatches, Car navigation systems, Cameras (DSLR with GPS), some models of laptops, fitness trackers (Fitbit), and drones.
  • Examples of vehicles using GPS are cars, delivery vans, trucks, aircraft, trains, ships and boats.
  • Military uses of GPS include guided missiles, guided munitions, tactical radios, communication systems, soldier-worn devices for location tracking, military vehicles and military aircraft.
  • Additional examples of GPS use include construction equipment for site positioning and machine guidance,  tractors for precision farming and other agricultural machinery, surveying equipment, pipeline inspection drones, other inspection drones and rovers, emergency locator beacons, pet trackers, smart collars, livestock monitoring, personal trackers, and geocaching devices.

As you can see, GPS is extremely useful, and there are a lot of interesting facts about GPS.

To see the other Super Facts click here