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Albert Einstein has made one of the most significant contributions to physics. It started with theory of special relativity and was later extended to the theory of general relativity.
On a everyday basis we don’t realize relativistic effects but we still consider them every time we use a GPS receiver like e.g. in your car navigation system.
The basic idea of GPS is to measure, very precise, the time difference between the transmissions of a signal from a satellite to a receiver.
Considering that the signal is transmitted with a constant speed, we can determine the distance from the sender to the receiver. Using multiple distances to multiple satellites, we are able to calculate the position of the receiver.
Since we depend on a precise measurement of the time we generate the signal in the satellite using high precision atomic clocks in the satellite. This is where Einstein’s theories of special and general relativity becomes relevant.
The impact of the special relativity
Clocks that are moving with respect to an observer are running slower. That means because GPS satellites orbit quite fast with ~3.9km/s in relation to the earth, the satellite clocks are also impacted.
*Due to the special relativity, time measurement is influenced by the speed of the satellite.
To put this into perspective, see example below:
To measure our position with ± 15 Meters we will need to measure the time difference with a precision of ±0,000.000.010 seconds. This means because of the special relativity an error of 0.83×10−8 %will occur.
This might seem very low, but the error will constantly increase, generating a position error of ~1km after 12 hours.
The impact of the general relativity
The impact of the general relativity is even bigger: The higher the gravity the slower the time moves.
The further away we are from an object, the lower the gravitation of this object is (e.g. the earth). Navigation satellites orbit at approx. 20,000 km altitude above the earth, so the gravitational influence is much lower on them compared to a clock on the earth.
This means that clocks in the satellite run faster than the clocks on earth: The time runs 5.28 ×10−8 % slower in the satellite clock.
Error without considering the relativity
As we can see the effect of the special and general relativity are exact opposite. But the effect to the general relativity is approximately 6 times bigger than the effect of the special.
In sum, the result is that the atomic clocks inside the satellites will run faster 4.44 ×10−8 %.
If we calculate our position at a given time, then one second later, the error on the position would already be ~13 cm. This is caused only by relativistic effect. After 1 hour, the error would already be 500 meters.
Even though the error might seem very small, the resulting error in the position will cause the GPS system to become unusable since the error will grow over time.
How to eliminate the error
Since the error caused by relativistic effects are linear with time, we can simply reduce the frequency that the GPS clocks are working with from 1023 MHz to 1022,999.999.545 MHz.
Since the receiver thinks that he gets the signal still with 1023 MHz, the difference between the true and the assumed frequency is exactly eliminating the relativistic effects.
Therefore, without considering Einstein’s theories your navigation system won’t work.