Einstein's theory of relativity is one of the most significant achievements in the history of science. All the more so as it has been developed by only one person. Many of the conclusions drawn from this theory are literally striking and seem completely unbelievable at first glance. One of the conclusions of the theory of relativity is that time itself is by no means absolute. For millennia people considered time to be absolute. Even Isaac Newton, the founder of classical mechanics, believed that time would tick at the same speed everywhere in the universe. At the beginning of the 20th century, however, Einstein showed that this was a pure illusion and that different observers could not always agree on the passage of time. In the theory of relativity, it is known as **time dilation**.

Time dilation occurs when two clocks move relative to each other at some speed or these clocks are located at different points in the gravitational field. Let's say, for example, we have two observers - A and B. Observer A with his clock in standing on the ground, and Observer B is on board a fast-moving aircraft. As Observer B moves at high speeds relative to Observer A, it appears to Observer A, that Observer B's clock is ticking more slowly. Suppose now, that observer B with his clock is somewhere high in the mountains and observer A is at sea level. Since Observer A is now closer to the source of the gravitational field than Observer B, observer A`s clock moves more slowly than Observer B's clock.

**GPS** or Global Positioning System consists of 31 satellites, which orbit the earth and help us determine the exact location and time around the globe. Since they move at high speed relative to us and are located further away from the source of the gravitational field, they are inevitably affected by time dilation. Let's take a brief look now, at how GPS satellites help us pinpoint our location with such precision and why we must take these relativistic effects into account.

If we have a GPS signal receiver (such as a smartphone), then we have at least four GPS satellites at our disposal at all times. This means that no matter where we are on Earth, we have at least four GPS satellites above our heads. If we have four GPS satellites in sight, knowing the distances between the GPS signal receiver and the satellites will help us determine our location. This method is called **trilateration**. The image below helps to visualize the process in the plane, where the signal is received from the three GPS satellites.

If the time during which the signal from the satellite to the receiver reaches is known, the receiver can calculate the distance to the GPS satellite. The signal propagates at the speed of light, so the distance between the receiver and the satellite is equal to the product of the speed of light and time (S = c x t). Knowing the distances between the GPS satellite and the receiver, we can determine the receiver's location. Our location in this case is at the point where three circles intersect.

In reality, however, we need four satellites to determine our position, because we live in a three-dimensional world, not in a two-dimensional world and we need to know the altitude in addition to the latitude and longitude. So instead of a circle, we imagine a sphere, representing a spherically propagating signal from a GPS satellite.

Again, knowing the four distances between the receiver and the GPS satellites, we can determine our position in three-dimensional space. In this case. it is at the point where four spheres intersect.

The signal sent by the GPS satellite contains information about when the signal was sent, so the receiver can calculate how far the satellite is (S = c x t). But now relativistic effects must be taken into account. As satellites move at high speeds relative to us and far from the Earth, time dilation must take into account. GPS satellites have an atomic clock on board, which is extremely accurate, compared to, for example, the clock on our smartphone (you can read about how clocks measure time and what an atomic clock is from an earlier post "__Atomic wall clock__"). As the satellites fly at a high-speed relative to us, the time on board moves slower than the time on Earth.

We can use Lorentz transformations to find how much some time period in one system (measured on the ground) differs from a time period measured in another system (measured on a GPS satellite). One day is 24 hours, which makes 24 x 60 x 60 = 86 400 seconds. The speed of the GPS satellite is about 14 000 km/h = 3889 m/s and the speed of light is approximately 300 000 km/s = 3 000 000 00 m/s.

So, the clock in the fast-moving satellite falls behind 0,000007 seconds (7 microseconds) per day, compared to the clock on the ground.

Now we need to calculate the time dilation due to gravity. Since on the surface of the Earth, we are closer to the source of the gravitational field, the passage of time here slows down more compared to the clock on the GPS satellite. GPS receiver clock, located on the ground at the distance** r** from the Earth's center of gravity, experiences gravitational time dilation of 0.00006 seconds.

The GPS satellite orbits about 20 000 kilometers above the Earth's surface. At this distance ( **r** + **h** ) from the center of mass of the Earth the clock onboard experiences a time dilation of 0.000015 seconds.

So, we get a time difference of 45 microseconds (0.00006 - 0.000015 = 0.000045) per day between the time measured on Earth and the time measured on the satellite. From the first equation, we found that the clock on the fast-moving satellite lags behind the receiver's clock by 7 microseconds per day. Subtracting time dilation of 7 microseconds from 45 microseconds, we find that the receiver clock runs 38 microseconds slower per day than the clock onboard of GPS satellite (0.000045 - 0.000007 = 0.000038 sec = 38 us).

Although this does not seem to be a big difference, it is crucial that the clocks are synchronized. Because the signal propagates at the speed of light and travels long distances over a very short period of time, as a result, the signal receiver may calculate the position of the satellite incorrectly, resulting in an inaccurate estimate of the receiver's location.

Great and useful. Small typo "(0.000045-0.000007=0.000038sec=38ms)." it should be 38us not ms.

Thanks for the great explanation! Finally, I got it all sorted out. However, there's a typo in the middle formula: the r value was written in km (times 10^3) instead of meters (time 10^6)

Beautiful explanation! Thanks!