# If relativity is symmetric, do the satellites in the GPS system see the earthbound clocks as running slow due to relativistic time dilation?

If relativity is symmetric, do the satellites in the GPS system see the earthbound clocks as running slow due to relativistic time dilation?

How much slower does the satellite see the earthbound clock running as it passes overhead?

• Sorry, but the way you worded this makes it a moot question. The GPS satellites never 'see' the earthbound clocks at all. It is our clocks that take readings from multiple satellites and use the TOA from each to solve the location questions. It simplifies to a 2nd-degree approximation so the GPS unit needs to be smart enough to decide which answer is on the earth's surface. And yes, there is a lot of physics involving moving clocks but fortunately the speeds are not a high enough percentage of C to require serious math for our use - but also explains why we don't get 1 cm precision. Commented May 14, 2017 at 4:42
• So you are saying that the satellites aren't moving fast enough for relativistic effects to come into play? @SDsolar You write, "fortunately the speeds are not a high enough percentage of C to require serious math for our use." Really? Commented May 14, 2017 at 4:48
• Maybe I worded it awkwardly, but I meant to convey that relativistic effects introduce errors into the simplistic receivers we use. Last time I hooked up my u-Blox antenna monitoring 12 sats I could watch as it took readings could view them in 3D, and there is a lot of error in all 3 dimensions. It works well enough to keep in your lane on a major highway, but we'll never get 10 cm resolution unless we start using receivers that take relativistic effects into account like the military does. That's how they pick a particular window or chimney. But in our cars we have very simple receivers. Commented May 14, 2017 at 8:24
• The effect is not symmetric: if it was purely special relativity then it would be, but as in fact the dominant effect arises from GR, it is not.
– user107153
Commented May 14, 2017 at 9:40

The GPS takes into account both special and general relativity effects to give the correct location:

To achieve this level of precision, the clock ticks from the GPS satellites must be known to an accuracy of 20-30 nanoseconds. However, because the satellites are constantly moving relative to observers on the Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy.

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion .

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these two relativistic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time.

Italics mine.

The effect is symmetric, identical clocks on the ground will be slower .

• You write, "The GPS takes into account both special and general relativity effects to give the correct location." Yes I know! My question is, does the satellite see the earthbound clocks running slower due to translational velocity time dilation? Commented May 14, 2017 at 4:28
• as I said at the end, by algebra the effect is symmetric in comparing the two clocks. If A frame is fast in comparison with B , B will be slower in comparison with A Commented May 14, 2017 at 5:33
• So then each time the satellite clock passes overhead above the earthbound clock, the satellite clock will see the earthbound clock running slower. With each complete orbit, the satellite clock, which considers itself to be stationary, will look down and see the moving earthbound clock falling further and further behind. Each time the satellite clock passes overhead, it will see the earthbound clock further and further behind. Commented May 14, 2017 at 17:11
• Yes, that is what the algebra tells us and it is correct because GPS would not work without these relativistic corrections. Commented May 14, 2017 at 17:40
• I am NOT ignoring general relativity because, if you had read the quote in my answer, general relativity induces the effect of slowing for the earth and fast for the satellite. Special relativity works the other way. All my answers to your comments are with the combined effect, and, I am sorry, but I will not continue in this discussion. Commented May 16, 2017 at 4:00