Einstein and the GPS

By Phil Deutchman
Reader Contributor

A hundred years ago, Einstein published his famous work on General Relativity (GR). Little did he know then that the results of his new, radical theory would be used today in the Global Positioning System (GPS).

GR needs to be taken into account in the system. It is absolutely essential for the accuracy required, especially if you want airplanes to land on the airport tarmac at the right spot. In this brief note, I can’t go into a complete analysis of the GPS because many, many factors are involved. Instead, for fun, I’d like to focus on two basic and fascinating General Relativistic effects that deal with the time rates of clocks. The first is a velocity effect and the second is a gravity effect.

What happens to clocks is important because the GPS uses atomic clocks in orbiting communications satellites; and, another clock is placed inside a receiver on Earth. A satellite sends a signal that codes the emission time, while the receiver clock times the reception of the signal. You can see that time may be of utmost importance in the operation of the GPS.

Before Einstein, it was assumed that any two perfectly-made clocks would “tick” at the same rate. But, with Einstein’s GR, this is no longer true!

For the sake of understanding the basic ideas, I’ll use somewhat loose language for what follows. Strictly speaking, in relativity, you always have to compare one clock with another one. After all, that’s where the word “relativity” comes from. For our example, the velocity of a clock is compared relative to another clock; and, the position of a clock in a gravity field is compared relative to another clock at a different position. Since it becomes cumbersome to repeat this over and over, just keep those comparisons in the back of your mind.

Now, for the velocity effect, GR predicts that a speeding clock will actually tick slower. And, for the gravity effect, a clock immersed in a gravity field will also tick slower. Furthermore, the stronger the field, the slower a clock ticks. I’m not saying that these effects are obvious, but they are two major predictions from the Theory of General Relativity.

Applying this to the GPS, a satellite clock wants to tick slower than an Earth clock. This is because the orbiting satellite moves at a higher speed than the Earth clock, which is slowly rotating with the Earth. However, the satellite clock also wants to tick faster. Since it is farther away from the Earth, it experiences a weaker field than does the Earth clock. Both clocks are ticking slower, but the Earth clock ticks more slowly because it is in a stronger field.

To summarize: A satellite clock wants to tick slower because of velocity, but it also wants to tick faster because of gravity. So … which effect dominates? Doing careful calculations using GR, it turns out that the gravity effect slightly dominates. The end result is that a satellite clock ticks slightly faster than the Earth clock. This difference is extremely small, but it can add up to a large discrepancy over long times of operation. Fortunately, today’s technology is sensitive enough to measure these small differences, and provides the accuracy needed for a successful GPS.

Do these time effects on clocks actually happen? Yes! Atomic clocks have been flown around the Earth and then compared to Earth clocks. The experimental results very much confirm the time predictions given by Einstein’s Theory of General Relativity.

So, when you pick up your GPS device, you are literally holding General Relativity in the palm of your hand.

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