GPS, brought to you by General Relativity

If you wish to make Google Maps, you must first unify space and time.

It’s easy to forget how miraculous modern technology is. Consider, for example, that wherever I am, whenever I want, I can take out my phone and pinpoint my location to within a few metres. Even the people who invented the Global Positioning System (GPS) would find that incredible (they thought the best we could do was a few hundred metres). As a theoretical physicist, what I find even more incredible is that, were it not for the curious minds of the twentieth century wrestling with the concepts of time and space, satellite navigation technology would be essentially unusable. Forget a few hundred metres. Unless you understand the laws of special and general relativity, you’ll be lucky to get within 10 kilometres, as I shall demonstrate shortly.

Before we get into time dilation, let’s consider the basics of satellite navigation. The answer to the question “Where am I?” is always given relative to some point of reference. So, step one is to establish a reference point, preferably several. These are, you guessed it, the satellites in orbit. You need at least one for every dimension you want to measure. Therefore, to accurately establish an object’s latitude, longitude and altitude, no matter where it is, we need every point on Earth to be in the field of view of at least three satellites at any given time. Four, actually, because the clock in your phone isn’t accurate enough to do the distance calculation with just three. GPS uses 31 satellites in its constellation, though it could manage with 24.

The next problem to solve is how receivers in planes, ships, or your phone determine their distance from satellites orbiting Earth at an altitude of 20,180 kilometres and travelling at 3.9 kilometres per second [citation needed]. Part of the solution is the incredibly accurate atomic clock on board every navigation satellite. Included in the radio signal your phone receives from them is the time the satellite sent it. Your phone has a record of the current time and can thus figure out how long it took the signal to travel from space. To convert that time to a distance, we need to know how quickly the radio waves travelled. Perhaps you remember the following equation,\[Speed = \frac{Distance}{Time}\:.\]

Luckily for us, radio waves (indeed, all forms of light) travel at a constant speed in a vacuum. That speed is \(c\), or 299,792,458 metres per second. Not only does this speed never change, but every observer (moving with uniform velocity) will always agree on the speed at which light travels in a vacuum. Here we see that theoretical physics has already become useful, for this is one of the fundamental principles of special relativity. It’s also more confusing than it sounds. (Note: nowhere in these considerations will we deal with a perfect vacuum. Light moves ever so slightly slower in the atmosphere, so chuck a 0.03% error on some of my numbers.)

Imagine you’re sitting in a train travelling at 200 kilometres per hour. Of course, you don’t feel like you’re moving that fast. You feel like you’re stationary, along with everyone around you. When you look outside, it’s as if everything else is moving backwards at 200 kilometres per hour. Now imagine a rather fast person sprinting along the platform as you whizz past. If they are travelling in the same direction as the train, they will still appear to be moving backwards, but slower than everyone else on the platform just standing there: 170 kilometres per hour instead of 200. If they were running in the opposite direction, they would seem to pass you much faster than everyone else, at 230 kilometres per hour. So far, so straightforward.

Let’s add a contrived setup to this thought experiment: in the train are also two vertically stacked mirrors separated by a distance \(x\). Below is a ray box that shoots light straight upward, causing it to bounce between the two mirrors (the bottom panel is a one-way mirror). Equipped with a stopwatch and inhuman reflexes, you measure the time it takes the light to bounce from one mirror to the other as \(t = x/c\). Now, to someone standing still outside the train, the light ray will appear to take a diagonal path between the mirrors. This path is clearly longer than the distance \(x\). However, remember that it is an experimentally verified fact that every observer (moving with constant velocity) will observe light moving at the same speed, \(c\). Therefore, it must be the case that the two observers disagree on the time taken for the light ray to reach the other mirror. In a different time \(t'\), the stationary observer sees the light ray move along the hypotenuse, length \(ct'\), of a right-angled triangle whose base measures \(vt'\), where \(v\) is the speed of the train, and whose height is \(x\). Of course you remember Pythagoras’ theorem (right?), so you’ll be able to work out that \((ct')^2 = (vt')^2 + x^2\). With some algebra, you can then deduce the relationship between the two time measurements:\[t’ = \frac{ct}{\sqrt{c^2 - v^2}}\:.\]In other words, moving clocks appear to tick more slowly than stationary ones.

This effect is called time dilation. What’s even more screwed up is that observers don’t even agree on whose clock is ticking slower. To you, it is the person on the platform moving, and therefore whose clock is ticking more slowly. Since stuff on Earth doesn’t move that fast (compared to light), we don’t notice these minute time differences. However, satellites move at 3,900 metres per second. Thus, from our perspective on Earth, they appear to lose about 7 microseconds each day. That’s almost 100 times the smallest time difference that atomic clocks onboard GPS satellites can distinguish! If we had never discovered special relativity but decided to do satellite navigation anyway, I’m sure we’d be awfully confused as to why our super-accurate atomic clocks are out by 2 kilometres every day.

Oh, but it gets worse. It turns out special relativity is the easy one, still doable with the maths you learn before you go to university. If you want to incorporate gravity into this framework, you need an entirely different kind of maths called differential geometry. Luckily for Einstein, mathematicians like Gauss, Riemann and Lobachevsky had done a great deal of important work in developing this field. After years of intense thought, discussion and collaboration, with many missteps and dead ends along the way, he cracked it. The theory of general relativity is a thing of beauty. Perhaps you’ve heard of a few of its implications, like black holes, wormholes, and the fact that gravity isn’t a force in the traditional sense, but due to the warping of spacetime. Turns out it messes with time more, too, and by cheating a bit you don’t need university-level maths to show it.

The orbital velocity of a GPS satellite is entirely due to the gravitational force that keeps it in orbit. We can demonstrate this by linking the kinetic energy of the satellite to its gravitational potential energy,\[\frac{1}{2}mv^2 = \frac{GmM}{r}\:,\]where \(m\) is the mass of the satellite, \(M\) is the mass of Earth, and \(r\) is the distance from the centre of Earth to the satellite. The gravitational constant \(G = 6.6743 \times 10^{-11}\) (in SI units) quantifies the strength of gravity. A quick rearrangement of the formula above gives us the orbital velocity purely in terms of \(G\), \(M\) and \(r\),\[v^2 = \frac{2GM}{r}\:.\]Substitute this expression into our formula for time dilation, and we get\[t’ = \frac{ct}{\sqrt{c^2 - \frac{2GM}{r}}}\:.\]

The average radius of Earth is about 6,371 kilometres, and its mass is 5.97219 × 10²⁴ kilograms. Therefore, due to gravitational time dilation, clocks on the ground lose about 46 microseconds. Notice that I didn’t say “appear to” here. Unlike time dilation in special relativity, clocks in the vicinity of a stronger gravitational field absolutely tick slower than those in weaker fields. Put differently, we on Earth and the astronauts onboard the ISS would both agree than the terrestrial clocks tick slower. The total relativistic correction is therefore about 39 microseconds. Translated to a distance, that is over 10 kilometres of error generated by the warping of spacetime and the universality of the speed of light.

I’m sure, because it happens to me constantly, that at the time of the discoveries of early twentieth-century physics, plenty of practical-minded folk said something like, “Sounds cool, but what are the practical applications?” To which the answer, at the time, would have been “Who knows?”. That is simply the nature of curiosity-based research. We don’t do it with a specific application in mind; we do it because we have a question about how the natural world works and want to find an answer. A fortunate result is that it often leads to immensely useful applications years down the line. Of course, sometimes those applications are not so great. Other times, however, they become so fundamental to how the world works that the average person would simply be unable to function without them. Satellite navigation is but one of many examples. Stay tuned for more!