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Thank you Albert Einstein for GPS

I’m a huge fan of Albert Einstein. As many physicists will agree his theories are the most pure expression of genius in the history of humankind. So as an engineer I’m excited that I get the opportunity to design products which use a technology dependent on two of his greatest theories. This is the case with the Global Positioning System (GPS).

GPS was developed by the US military at the height of the cold war in the 1960’s. It was primarily used by the Navy to allow ships to track their location while at sea. With this early version a vessel could only get an update on its location once per hour. Throughout the next few decades the U.S. continued to make improvements to the system with the launch of more satellites. In the 1980’s the Reagan administration made the decision to release GPS technology for civilian purposes. The decision was ignited in 1983 by the USSR shooting down a Korean Airlines passenger jet that accidently flew over Soviet territory while in route from Alaska to Korea.

The GPS system consists of 24 satellites in high orbit around the Earth. These satellites travel at about 9,000 miles per hour and orbit at an altitude of about 12,000 miles (for comparison the International Space Station orbits at around 250 miles). Each satellite carries an atomic clock with an accuracy of around 1 nanosecond. The satellites are distributed so that from any point on the Earth’s surface there is a line of sight to at least 4 satellites. A GPS receiver on the ground (or on an airplane in flight) receives timing and location data from these satellites on a 1.575GHz radio carrier signal.

Without Albert Einstein’s universe altering theories of relativity there would be no GPS. Crazy ideas like time slowing down with velocity, or mass bending the curvature of a four dimensional fabric called space-time are critical for GPS to work. Instead of an accuracy measured in feet (or inches for the military) it would be measured in miles without Einstein’s theories being taken into account. Not exactly accurate enough for driving directions! In fact, within a very short amount of time the errors would accumulate to the point of making the entire system useless.

A GPS receiver is able to calculate the distance to each satellite by timing how long it takes for the radio waves, which travel at light speed, to arrive at the receiver. By knowing the exact distance and location (encoded in the signal) of three satellites the receiver’s microprocessor is able to trilaterate its location. Trilateration is a more complex form of triangulation that uses intersecting spheres to determine a location based on distances to three other points in 3-dimensional space. A fourth satellite is required as a redundant check and to provide timing corrections to the receiver which don’t have the luxury of a built in atomic clock.

Measuring the travel time of radio waves traveling over such relatively short distances requires extreme precision. To achieve the necessary accuracy required by GPS the radio wave timing needs to be accurate to within a few nanoseconds. For this to happen you must invoke Einstein’s theories of Special and General relativity. Special relativity deals with objects moving at high speed, whereas general relativity is a theory of gravity.

Einstein published his theory of Special Relativity in 1905 and forever changed our fundamental understanding of the universe. In it he proved that, contrary to Newton and common sense, the speed of light is truly the universal constant not time or distance. His theory showed that as an object approaches the speed of light time slows down in relation to a “stationary” observer on Earth. I put quotes around stationary because nothing is really stationary and at this moment you are moving on a spinning Earth as it travels around the sun in a rotating galaxy in an expanding universe.

According to special relativity if your twin brother were to fly away in a spaceship at nearly the speed of light, his time slows down relative to your time. He may return after only what was an hour for him to find you are 50 years older. The time difference all depends on how close to the speed of light he travels. This effect is known as time dilation and is shown by the following Lorentz transformation equation:

where t is the time in one inertial frame (a frame of reference moving at a constant velocity – for example the GPS receiver on Earth), t’ is the time in another inertial frame (for example the orbiting satellite), v is the relative velocity between the two frames of reference, and c is the speed of light.

When the relative velocity is low compared to the speed of light then the above equation reduces to simply t’ = t meaning there is no time dilation. This is our everyday world. But as the relative velocity approaches the speed of light then t’ approaches infinity (meaning time slows down, or even stops at the speed of light). For the case of GPS satellites which are traveling at 9,000 mph, time flows about 7,000 nanoseconds slower per day as compared to a receiver on Earth.

The reason special relativity was called special is because it was a limited theory that was only for one special case – when acceleration equals zero. Einstein’s special relativity equations only worked in cases where an object was traveling at a constant speed in a straight line so as to have no acceleration.

Einstein eventually realized that acceleration and gravity are equivalent (called the Equivalence Principle). In 1915, after nearly a decade struggling with the math behind his new theory, he published his masterpiece the General Theory of Relativity. The math behind general relativity (known as tensor analysis) is many times more complex than that required for special relativity. After over 200 years of rule, Newton’s theory of gravity was overthrown by Einstein’s new theory on gravity.

Einstein’s general theory showed that gravity from mass warps the fabric of space-time. Like velocity in special relativity, he showed that gravity also causes time to slow down. In the case of GPS, the satellites in space experience slightly less gravity than the receiver on the ground. Because of this difference in gravity, time travels about 45,000 nanoseconds faster per day for the satellites compared to the ground based receiver.

Special relativity says that time slows down for the fast moving satellites, whereas general relativity predicts that time runs faster for the satellites since they experience less gravity. The effects combine to make time flow about 38,000 nanoseconds (45,000 – 7,000 nanoseconds) per day faster for the satellites relative to the receiver on Earth. The radio waves coming from the satellites travel about 1 foot per nanosecond. That means without relativistic effects included, the distance measurements to each satellite would be off by 38,000 feet (or about 7 miles) per day!

Whether providing directions to a restaurant, or allowing pilots to track their location anywhere in the world, or helping you find a lost pet, GPS impacts us all in so many ways. Just remember that were it not for a man over a hundred years ago with a burning curiosity to understand nature none of this would be possible. “I have no special talents. I am just passionately curious.” claimed Albert Einstein.

8 comments on “Thank you Albert Einstein for GPS

  1. eafpres
    May 27, 2015

    Hi John.  I enjoyed the article.  If I take the inhabitants of the US as a target, and define a “understanding cross section”, calculating the cross section for understanding both how important is GPS and that Relativity Theory is essential for a practical system, I get a number very close to zero.

    Relativity has always fascinated me.  An interesting thing about the Lorentz transform is that you can derive it using a simple geometric argument and the assumption that c (speed of light) is a true constant in a given medium.  

  2. teelengineering
    May 27, 2015

    Thanks for the positive feedback!  Having some understanding of relativity really gives you a new perspective on our universe.

    Yeah, one beauty of special relativity is that mathematically its very simple once you make the right assumptions (that light speed in constant, not time or distance).  General relativity on the otherhand is mathematically very complicated.  At one point I spent months teaching myself the math behind general relativity (tensor analysis) and you can definitely see why it took Einstein a decade to formulate the math behind it.  Very complex mathematically and it requires math that most engineers have never seen.  When he released this theory I think only a couple of people in the world really understood the math behind it.

     

  3. David Ashton
    May 29, 2015

    John, thanks, a very interesting article.  While I don't have anything like your understanding of relativity, I know a bit about it (I cut my teeth on an excellent explanation in one of the old Time-Life science books :-).  I have a habit of looking for commonplace demonstrations of Physics – for example the doppler effect when a train goes past you, or how your hands under a hot-air dryer suddenly get a lot hotter once all the water has evaporated.  But I had never thought of any comonplace applications that demonstrate relativity.  I guess you can't see the proof of this (you can't switch the relativity compensation off!) but it's interesting to know it is there.

  4. teelengineering
    May 29, 2015

    Thanks for the comments David.  I too love seeing examples of physics in our every day lives.  GPS has been one of my favorite examples for a long time so it was fun writing this article.  Relativity isn't exactly a theory of our everyday world so seeing how it affects a technology we all use everyday is quite unique.

  5. netrick
    May 31, 2015

    Thank first Pierre-Louis Moreau de Maupertuis and his colleagues: La Condamine, Celsius,

    Clairaut, etc…who made exact measurements of the earth's shape in …1735-40 in

    Lappland and Ecuador. 

    Originally, GPS was called “Transit” and it was used to reset the position of the inertial

    platforms in the “Polaris” nuclear subs, and correct the drift of their integrating loops.

    The Transit satellites looking like little windmills were more “polar” than the present

    GPS ( 65 °). A Transit receiver system consisted in : 

    * One upside-down double conoic antenna ( 5 feet high )

    * One low noise cooled preamp box ( alike half a desk ) 

    * One power-supplies cabinet

    * One “Digitizing” cabinet where each ADC was a full 19 inch rack

    * One control panel cabinet with a lot of manual switches and “Nixie” tubes

    * One computer cabinet, with a “militarized” DEC PDP-11

    * One data storage cabinet with IBM style drums

    I/O was done with a militarized TTY ( about 200 pounds ) connected with 

    ASCII-2 / RS-232.

    * A synchronization with Omega hyperbolic positioning system was also provided.

    I was working for ITT and we received the rights to sell the system for civilian

    applications in 1968. But only in NATO countries. Customers were oil companies

    for positioning of off-shore platforms, super-tankers ( after Torrey-Cannyon

    disaster ), coast-guards, meteo frigates, etc…

    Competitor was Magnavox with a lower performance system.

    Standard unit price was …1.8 million $ ….of that time !!!!!

     

     

  6. RichQ
    June 3, 2015

    The mention of GPS being developed in the 60's is a bit off. Those first navigation satellites were the Transit series, and they worked on a slightly different principle. The receiving station used the doppler shift of the received signal to determine the range to the satellite rather than the signal's propagation delay, because in those day they couldn't outfit the satellites with precise, synchronized clocks. But they could build sufficiently precise frequency sources. They also used a lower orbit, about 600 miles.

    The GPS satellites we use today were first launched in 1978 but the 24-satellite constellation wasn't fully populated until 1995, according to the offical Air Force GPS site – af.mil/AboutUs/FactSheets/Display/tabid/224/Article/104610/global-positioning-system.aspx

     

  7. teelengineering
    June 3, 2015

    Thanks for the comment Rich.  I knew the early Transit system lacked precise atomic clocks but didn't realize they instead used Doppler shift.  Thanks for sharing.

  8. RichQ
    June 3, 2015

    I started my engineering career at JHU/APL, the creators of Transit. I remember doing a comparative study of navigation systems there, back when there were only a few GPS satellites in orbit and Transit was still in use. An amusing tidbit I learned then: when Transit first became operational the Navy compared the positional fixes from conventional surveying against the Transit results and determined they were pretty good when it came to most continental positions. Islands, however, were not as good. Hawaii was about a mile off and Sydney was so far off a nuclear missile strike would completely miss, so its true position was made classified information for the next decade. So I was told, anyway.

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