How Does Time Dilation Affect GPS?

Yep, you’ve read that title correctly. In this article we will be going through the phenomena describing the slowing down of time itself: time dilation! Impossible you say? Nope. Hard to get your mind around or even accept it exists? Definitely. This one will be a little less practical than other “Pilots who ask why” articles. However, if you can stick around, it will change your outlook on time itself forever. Let’s see if we can avoid an existential crisis along the way!

You might ask how this could possibly be relevant to pilots and other aviation focussed professions. Well, without the groundbreaking discoveries that describe the slowing down and (speeding up) of time itself, we would not have any functional Global Navigation Satellite Systems. This includes GPS, GLONASS, GALILEO and others. All these systems rely on satellites. Satellites with an average orbital speed of around 8700 mph, and some with a distance away from earth of around 22300 miles! 8700 mph is just a “little bit” faster than most of our helicopters and fixed wing aircraft. Even the legendary SR-71 Blackbird “only” achieved Mach 3 (around 2300 mph).

Each satellite carries an extremely accurate atomic clock that allows for the crucial time readings that are required for GPS to work. Here is the kick in the face though: None of these systems would work well enough for longer than a few seconds if it wasn’t for this groundbreaking discovery that time itself is not constant or ‘objective’, but varies with the observer’s speed (and experienced amount of gravity), weird huh?

Based on these very accurate clocks, any person with the right access to the system can get a position readout that has a 5-10 meter accuracy (down to only a few inches if you have more advanced receivers), but only if we correct for the time dilation effect that happens due to the satellites velocity as well as the smaller effect of gravity at 12550 miles away from earth. If we choose to be stubborn here (like so many to this day still!) and refuse to accept that time itself is relative, our position readout would be completely useless after about 2 minutes of usage, after which your position error would increase by about 10 km per day! Are you ready yet to accept that time is not a constant but relative to everyone individually? Good, let’s dive in, but first let’s go over how GPS works exactly.

GPS: THE FUNDAMENTALS

Before we can talk about how time dilation affects anything, let’s start from the beginning. GPS utilises 24 satellites, each with their own extremely accurate atomic clock that have a nominal accuracy of 1 nano second, that is 1 billionth of a second! The way this system works is actually quite ‘simple’. There are 3 components in the system:

  • The satellites
  • The ground stations
  • The user / receiver

The system uses a technique called trilateration (not to be confused with triangulation for all the pilots here!) to determine your position. Let’s break it down to the absolute root and keep it simple (compensation for this article’s topic). It all starts with someone like you or me wanting to know what our position is on earth. The question is often raised about how many satellites are required for GPS to work. The answer given is often 3, but it is actually 4. Let’s see what 3 satellites will give you. First, our receiver will start measuring the distances between itself and any satellite it can get its grubby hands on, by measuring the time it takes for the satellite’s signal to arrive at the receiver. While talking to the first satellite, we can now know we must be along the red line (below) anywhere on earth, so that does not really narrow it down.

Using a 2nd satellite, we now have 2 points on earth where those 2 range circles intersect, but which one is the correct one?

That is where the 3rd satellite comes in. As you can see below, as soon as our receiver works out the range of the 3rd satellite, we can detect which of the 2 possible location is our location, tadaa!

This technique called trilateration is different from triangulation in that it uses ranges from different points (satellites), rather than angles. VHF direction finding, as we are used to in aviation, is a form of triangulation, and is NOT comparable to what is going on with GPS as discussed above.

THE TIME PROBLEM

So now what? We know our position, happy days right? If only! This is where all the real problems start. Hopefully most of the beloved ‘Pilots who ask why’ audience will know that speed equals distance divided by time. This is all very straight forward if we want to measure the speed of 2 signals with the same frame of reference for all 3 variables. Let’s see, same frame of reference for distance? Check. What about for speed? “You think so?” Well, if time is not the same for each satellite, neither will the measurements for the signal’s velocity. How the hell are we supposed to counteract this madness? Here is how: we need a 4th satellite that can correct for the time dilation experienced by the satellites and compare it’s time reading with others and the ground station so the system can apply a correction! Let’s zoom in on what problem we are trying to solve here, as it can get confusing rather quickly!

Please keep in mind though, that just because we are able to describe its effects does not mean we as humans fully understand yet what is going on exactly. This is proven by the fact there are many different theories on how time dilation works exactly, but we are going to stick with the most prominent / commonly accepted: Einstein’s theory of special relativity. Let’s kick it off by going first of two main variables that affect time: velocity! Gravity (the 2nd variable) will be discussed in the future.

Imagine I throw a ball at 10 mph. What is our frame of reference here to be able to say the ball’s velocity is 10 mph? In this case it is the surface of the earth, which as we know rotates as well, while the earth orbits the sun etc. For now just keep it as ‘surface’.

Next, I climb on a driving truck (health and safety goes out the window here), and throw the same ball again, with the same speed. The truck is driving at 25 mph. If I now ask: what is the ball’s velocity? What would you say?

You would not really be able to answer this question before asking me ‘compared to what reference’? You see, if we say the ‘reference’ is the surface, then yes, the ball’s velocity would be 25 + 10 + 35 mph. However, compared to the truck, the ball is still going at 10 mph!

Now here is where things get super messy. Einstein discovered with his work on special relativity, that the speed of light is still the speed of light regardless of any reference. So if we take the example above, but use it on a plane with a landing light. Initially the plane is stationary on the ground. The light coming out of the landing light is, well, the speed of light, or C (299 792 458 m/s to be exact).

But now we are going to let the plane take off, let it fly at 500 mph, and measure again what the velocity of the light is. Have a guess! If we apply our previous logic, it should be C + 500 mph right? Unfortunately this logic does not work anymore when it comes to the speed of light. The speed of light is always the speed of light, so wether we measure those landing lights from the ground, the stationary plane or from the flying plane’s perspective, it will ALWAYS be EXACTLY 299792458 m/s, or C, regardless of who measures it! I know, confusing!

Is this nature’s way of making fun of how puny our piddly brains are as humans? Kind of, yea. Einstein discovered that in order for the speed of light to be constant for any observer, observers with a higher speed will have time moving slower for them, or seconds taking ‘longer’. The observer should get a lower reading of the light’s speed if we compare it to me standing on that truck. If I would measure that ball’s velocity, I would get 10 from the truck, not 35 remember?

The way this works out is because the the pilots inside the plane travelling at 500 mph, will experience longer seconds than observers on the ground, which makes the speed of light coming out of the landing light still exactly the same as C for either observer. Why? Fine, we will go one final layer deeper into this mess, but don’t blame me for not being able to sleep tonight.

Let’s have a quick look at a photon clock. It sounds fancier than it is. Just imagine a little photon (light particle, travelling with the speed of light) bouncing between 2 mirrors. Every time the photon hits 1 mirror, 1 second passes.

We can say that the distance travelled equals to 1 second. But now, imagine we move the clock, and we observe the bouncing photon from a ‘stationary’ surface. What we would see is something like this:

That’s right, the distance travelled for this little photon is longer now, as it is travelling along a longer path between the 2 mirrors. This orange line in the picture is what someone would see from a stationary surface. The weird thing is, the seconds for someone travelling with the clock are still the same seconds, and if that person would look at the photon, it would look like in the previous picture. A second is still a second. It only becomes a problem when we compare that perspective with someone that is not travelling along with it! Conclusion? Seconds take LONGER when a clock is moving, i.e time is slowing down! The question now is, how much? If we want GNSS to be of any use, we will need to know exactly how much time dilation we are being exposed to, and then correct for it.

So let’s try to get a grasp of how much time can actually slow down for us by using the time dilation equation. Don’t worry, we will break it down completely. Imagine I am standing on earth and you hop in a rocket that goes 0.8 times the speed of light (very fast, and not yet possible unfortunately, this is just to explain what is happening). I ask you to fly around for 60 seconds based on your watch inside the rocket, while I measure on earth how long that takes for me. How many seconds will actually have passed for me? We know that time will go slower for you and faster for me, hence the time that will have passed for me should be MORE than 60 seconds. Let’s have a look at it, the variables we have are:

  • The seconds on my watch, on earth
  • The seconds on your watch, inside the rocket
  • The rocket’s velocity
  • The speed of light (C)

Let’s fill them in using 0.8 times the speed of light, and 60 seconds on your watch inside the rocket:

There we go, 100 seconds! As you can now see, the difference is 40 seconds. That means that by going 0.8 times the speed of light for 1 minute, when you come back to earth, the people around you are already 40 seconds older than you. Imagine what a year’s trip or so would do to all that (Interstellar rings a bell?). Luckily, satellites are travelling not even close to speed of light, so the effect is much less severe. As you can see in the equation, dividing a small velocity by C usually gives a microscopic figure. However, as you get closer to the speed of light, this effect becomes much more prominent.

IS THAT ALL?

Well not really. The code inside the GPS satellites use the equation, plus some other variables, to calculate their individual amount of experienced time dilation. The satellites experience around 7 microseconds of ‘delay’ every 24 hours due to their speed. The code compensates for this at all times.

The other effect though that is not discussed yet in this article (we don’t want anyone’s head exploding today) is the effect of gravity on time, which is also affecting GPS. This however, due to the satellite’s distance away from earth and reduction in gravity, actually cause the clocks to tick faster by 45 microseconds every 24 hrs. The overall time difference therefore is 45-7=38 microseconds. But while the speed is dilating time, the lower amount of gravity is speeding it up, so the satellite’s clocks are actually ticking 38 microseconds faster than clocks on earth!

So why does gravity affect time as well? Good question, and this will be discussed in an article in the future, as it is a lot more complex (spacetime effects) and we want to keep things tidy! This topic can be incredibly confusing as it is talking about time changes. Which is like pointing out water to a fish! While this topic is a lot less practical than some of the other past and future topics, it is something that affects all of us. Next topic will be more practical though, not to worry!

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2 thoughts on “How Does Time Dilation Affect GPS?

  1. Jop

    Really enjoyed that article. However the 3 circles all intercepting in one place is not entirely due to the time dilation effect. It’s really due to not knowing the time at all. The big assumption you made with the 3 circles intercepting at one point is that you know what time it is.

    Unless your clock is synched with the satellite (very closely) your distance will be in error. But all 3 satellites will have the same time error and instead of a point you have a tri-corn hat shape (each has a pseudorange). The fourth satellite allows you to resolve what the error is until you have a very small tri-corn shape. And of course you’ve now got a clock perfectly synched with the satellite (hence GPS is used to synch wristwatches). THEN you start correcting for time dilation and atmospheric effects.

    Like

    1. George! Always love your input, thank you! You’re completely right. The first circles assume a “correction” on time already been applied. It’s an oversimplification so I fully agree with your comment. I have seen your GPS explanation before as well, which is interesting as the literature seems to be divided on this. I’ll do some further digging!

      Like

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