GPS & Relativity

Posted by in Random, Technology, Uncategorized

Relativity? Really?

Einstein

 

This topic is really relevant these days. Why? Because everything these days are going fast! mobile phones, computers, cars & even wrist watches are getting smart & fast! As our world become faster and faster, seconds – even micro seconds & nano seconds becomes relevant. There comes the relevance of Relativity! It has started to appear even in the mainstream media often these days. One of the more recent & stunning example is the movie Interstellar, by the celebrity director Christoper Nolan who is very well known for making people puke rainbows! He hasn’t failed this time also I believe. No matter what the plot is, the guy never ceases to amaze his audience. Even if they don’t get anything from it, they’ll all be talking huge about it. (p.s: I’m not a Nolanite, but I like his movies). The movie is about the future of Earth, alternate worlds, time dilation & sh*t! This post isn’t about the movie, I don’t really wanna be a movie critic!

Interstellar Meme

When the word “Relativity” is heard, people often don’t care because ‘it is arcane &  of no real world use’  Well, they’re all wrong! Relativity is something of serious importance.

Consider GPS for example. We all have GPS in our mobile phones. Some even have it on their cars. Its a nifty service which can guide you to your destination or lets you find your pace accurately. Is it a simple technology? Absolutely not! But its easy to use & that’s why its there on your mobile phone (conspiracy theorists may say that the NSA wants to spy the entire population of the world and that’s why they’re giving it for free 😛 ). GPS has 24 satellites orbiting around the Earth.  The satellite orbits are distributed in such a way that at least 4 satellites are always visible from any point on the Earth at any given instant.  All these satellites have an atomic clock built in, having an accuracy of 1 nano second (1 nano second = 1 billionth of a second!). A GPS receiver in your mobile phone or car determines its current position and heading by comparing the time signals it receives from a number of the GPS satellites (minimum 4 required) and Trilaterating on the known positions of each satellite. GPS doesn’t use Triangulation!

Trilateration

Trilateration

Its a popular misconception we got from the movies. Triangulation involves the measurement of angles to locate while Trilateration involves determining absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. Anyways, the accuracy of the results obtained are quite astounding & that’s why you end up being at your destination rather than being at some random place (If you’re using Apple Maps, I feel sorry for you mate 😛 ). 5 to 10 meter accuracy is often obtained & its quite remarkable knowing that you’re on the surface of the Earth & what you’re getting are absolute values. A GPS receiver in a car can even give you accurate readings of position, speed, and heading in real-time!

To achieve this level of precision, the clocks in the GPS satellites must have an accuracy of 20-30 nanoseconds. However, because these satellites are constantly moving relative to the observers on the surface of Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy.

1) Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we will see their clocks ticking more slowly. Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion.

2) Another prediction of General Relativity is that, clocks closer to a massive object will seem to tick more slowly than those located further away. The GPS satellites are in orbits high above the Earth, where the curvature of spacetime (only spacetime has an absolute reality independent of the observer) due to the Earth’s mass is less than it is at the Earth’s surface. Therefore, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. Calculations based on this has predicted that the clocks on GPS satellites should get ahead of the clocks on the ground by about 45 microseconds per day.

The combination of these two relativistic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38). This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day & the whole system would be useless for navigation in a very short time!

The people who designed the GPS system included these relativistic effects when they designed and deployed the system. To counteract the General Relativistic effect once on orbit, they slowed down the ticking frequency of the atomic clocks before they were launched so that once they were in their proper orbit stations, their clocks would appear to tick at the correct rate as compared to the reference atomic clocks on the ground. Also,  each GPS receiver has a built in computer that (among other things) performs the necessary relativistic calculations.

 

Relativity, some food for thought! 🙂