GPS Satellite System. Source: https://celebrating200years.noaa.gov/transformations/gps/Figure_1.html

How Einstein’s Theories Keep Our Maps Accurate

You’re standing on a busy street, late to meet a friend. You pull out your phone, pull up Maps, type in your destination, and seconds later a blue line is guiding you from your location to where you need to be. This seems like a clever piece of software, but hidden behind the display is the dependence of this everyday convenience on something deeply unintuitive; time doesn’t tick at the same rate everywhere. Without Einstein’s theories of special and general relativity, that blue dot showing your location would drift off course by kilometres every day.

GPS doesn’t just know where you are—it figures it out by measuring time. Your phone receives signals from at least four satellites orbiting Earth. Each satellite sends a time-stamped signal, and your phone calculates how long each signal took to arrive. From those time differences, it triangulates your location.

For this to work, the clocks on the satellites and in your phone must be incredibly precise—accurate to a few billionths of a second. And that’s where Einstein comes in.

Special Relativity

Visualisation of stationary and moving light beam reflections. Source: http://www.thestargarden.co.uk/Special-relativity.html

According to special relativity, time doesn’t tick at the same rate for everyone. The faster you move, the slower time passes for you from the perspective of someone watching you move – an effect called time dilation.

This seems unintuitive, so let’s consider a classic thought experiment to illustrate this, as Einstein was fond of doing.

Imagine a clock that tells time by bouncing a beam of light between two parallel mirrors, one at the top and one at the bottom. To a person standing still next to the clock, the light can be seen moving straight up and down. But now imagine the clock is moving sideways, like on a satellite. The same observer on Earth, the light now traces out a diagonal path as it moves between the mirrors due to the system’s overall sideways motion.

This diagonal path is longer than the previous straight up and down path considered. However, as the speed of light is constant, what must change to make that longer path occur? As distance is equal to speed multiplied by time, and speed must remain constant at c, the time taken must change.

GPS satellites move at approximately 14,000 km/h. Due to this high speed, time dilation causes their onboard atomic clocks to run slower than identical clocks on Earth. This results in a loss of around 7 microseconds per day.^[1]

General Relativity

While special relativity tells us how motion affects time, general relativity goes further and considers the effect of gravity on time.

Einstein discovered that massive objects like the Earth warp the fabric of spacetime, causing a curvature – like putting a heavy object on a trampoline. This curvature means objects moving through spacetime travel along curved paths around massive objects, where in flat spacetime they would travel along a straight path. Everything in spacetime must follow these curved paths, including light. Again thinking of our light beam travelling through spacetime, it now follows a longer curved path close to a massive object when compared to a path in flat spacetime. Remembering that the speed of light is constant, this extra distance must again be made up by a change in time.

The closer you are to a massive object, the more curved the spacetime around it, and the slower time passes for you relative to an observer in flat spacetime, as your ‘path’ is longer. As GPS satellites orbit the Earth at about 20,200 km, the spacetime they move through is less curved, meaning time passes faster for them than here on Earth. This gravitational time dilation results in a gain for the satellites of around 45 microseconds per day.^[1]

Putting it all Together

These two effects – special relativity making the satellite clocks tick slower, and general relativity making them tick faster – don’t cancel out, they result in a net gain of about 38.6 microseconds per day.^[2]

This may not sound like much, but if not corrected for this gain would result in errors of around 11.6 km per day^[2], making the map on your phone useless. To prevent this, the clocks on satellites are pre-adjusted before launch so that once in orbit, the effects of relativity bring them in sync with clocks on Earth.

References

[1] Ashby, N. (2003). Relativity in the Global Positioning System. Living Reviews in Relativity, 6(1). https://doi.org/10.12942/lrr-2003-1

[2] https://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System