# What relativistic effects to GPS receivers compensate for?

## Main Question or Discussion Point

So I understand the basics of the general and special relativistic effects on the speed of the satellite clocks onboard the GPS satellites, and why they are rigged to calculate time slower than an atomic clock at sea level.

But then, in several articles, I read that GPS receivers are also designed to make relativistic calculations as well. What all do the receivers have to compensate for? Just the relativistic doppler effect? None of the articles I came across actually specify anything, let alone everything, the receivers have to calculate relativistically. Any enlightenment would be much appreciated.

## Answers and Replies

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GPS satellites are both moving at a fast enough speed and have a large enough difference in altitude that both gravitational time dilation and velocity-caused time dilation need to be compensated for.

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
In total, relativistic time dilation causes the clocks onboard the satellites to measure about 38 microseconds more in a day than sea-level clocks.

EDIT: Oh, excuse me, I misread the question. Give me a moment.
I apologize for misreading the question, I'll keep what I already wrote for the sake of it and will try and find out the answer to your question, but in the meantime someone else could resolve it.

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This is what I've found so far:
…if the orbit is eccentric, an additional correction arises from a combination of varying gravitational and motional frequency shifts as the satellite's distance from earth varies. This correction is periodic and is proportional to the orbit eccentricity. For an eccentricity of .01, the amplitude of this term is 23ns. Due to a shortage of computer resources on satellites in the early days of GPS, it was decided that this latter correction was to be the responsibility of software in GPS receivers. It is a correction which must be applied to the broadcast time of signal transmission, to obtain the coordinate time epoch of the transmission event in the ECI frame.
http://www.phys.lsu.edu/mog/mog9/node9.html

pervect
Staff Emeritus
So I understand the basics of the general and special relativistic effects on the speed of the satellite clocks onboard the GPS satellites, and why they are rigged to calculate time slower than an atomic clock at sea level.

But then, in several articles, I read that GPS receivers are also designed to make relativistic calculations as well. What all do the receivers have to compensate for? Just the relativistic doppler effect? None of the articles I came across actually specify anything, let alone everything, the receivers have to calculate relativistically. Any enlightenment would be much appreciated.
Wiki has this to say:

Sagnac distortion

GPS observation processing must also compensate for the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system, a system in which simultaneity is not uniquely defined. A Lorentz transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east-west error on the order of hundreds of nanoseconds, or tens of meters in position.[16]
This sounds like it might be what you're looking for...

bcrowell
Staff Emeritus
Gold Member
The atomic clocks aboard the satellites are tuned to a frequency of 10.22999999543 MHz, which is perceived on the ground as 10.23 MHz (on average). The difference is due to kinematic and gravitational time dilation, which are constant effects. There are also time-varying effects: a Sagnac effect and a kinematic Doppler effect.

There is a review article by Ashby: http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Broken]

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The atomic clocks aboard the satellites are tuned to a frequency of 10.22999999543 MHz, which is perceived on the ground as 10.23 MHz (on average). The difference is due to kinematic and gravitational time dilation, which are constant effects. There are also time-varying effects: a Sagnac effect and a kinematic Doppler effect.

There is a review article by Ashby: http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Broken]
One question... the satellites do not display any kinematic time dilation among themselves, in spite of having proper velocities w.r.t. one another. What is the SR explanation for this?

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jbriggs444
Homework Helper
2019 Award
One question... the satellites do not display any kinematic time dilation among themselves, in spite of having proper velocities w.r.t. one another. What is the SR explanation for this?
They're synchronized in an earth-centered inertial frame and are moving at a nearly constant speed at a nearly constant altitude in that frame. Naturally, this means that they stay synchronized in that frame.

When reckoned against an accelerating frame in which one of the satellites is at rest, the remaining satellites do exhibit kinematic time dilation.

bcrowell
Staff Emeritus
Gold Member
the satellites do not display any kinematic time dilation among themselves,
Why do you claim that this is true? I see no reason to believe that it is. In fact, it's probably not a well-defined statement. You can't necessarily take the various relativistic effects and unambiguously separate them into various effects to be added up. Depending on one's frame of reference or choice of coordinates, the contributions of the different effects will in general be different.

When reckoned against an accelerating frame in which one of the satellites is at rest, the remaining satellites do exhibit kinematic time dilation.
...as in, the clocks of the remaining satellites go out of synch w.r.t. the satellite considered at rest? That does not happen.

Consider two satellites when they are very close together, but travelling in different planes. If we take the point of view of one of them (considering it to be at rest), it will see the other satellite travel away for a while, and then come back. When it is back very close again, a comparison between their clocks will not show any relative time dilation (or it would have been noticed on Earth too!).

Why do you claim that this is true? I see no reason to believe that it is. In fact, it's probably not a well-defined statement. You can't necessarily take the various relativistic effects and unambiguously separate them into various effects to be added up. Depending on one's frame of reference or choice of coordinates, the contributions of the different effects will in general be different.
Since the satellites are all at the same altitude, there is no difference of gravitational potential among them, so relative gravitational time dilation is ruled out. Therefore, we can talk about velocity time dilation effect independently here.

Based on discussions in various other threads, I suspect this lack of mutual velocity time dilation among GPS satellites probably has a similar explanation to the scenario of two twins travelling away from Earth and returning later, in which case both would age equally slower than someone at rest on Earth.

However, this scenario seems slightly different as the twins never return, but their clock time dilation (w.r.t. Earth surface) is detected because of signals received from them.

The reason I asked the question is that, looking at it from SR perspective, we should be able to analyze the mutual time dilation of two satellites w.r.t. each other without considering a third body (Earth), but that doesn't somehow seem possible (or at least cannot lead to a measurable time dilation between the satellites).

bcrowell
Staff Emeritus
Gold Member
Since the satellites are all at the same altitude, there is no difference of gravitational potential among them, so relative gravitational time dilation is ruled out. Therefore, we can talk about velocity time dilation effect independently here.
I don't think the second sentence follows from the first sentence. I don't think the second sentence is even making a well-defined statement.

I don't think the second sentence follows from the first sentence. I don't think the second sentence is even making a well-defined statement.
I thought the statements were quite clear, but if you think otherwise, I will not quibble. In fact, I will just drop the whole thing right here, because perhaps this discussion is going off topic from the OP's question.