# GPS and time dilation

1. Nov 15, 2009

### cfrogue

GPS satellites are preprogrammed before launch due to relativistic effects mainly time dilation and gravity.

Yet, once in orbit, the satellite should view the earth frame as "moving" and thus apply time dilation to the earth frame meaning the earth clocks should beat slower. This does not happen.

Questions, why does time dilation only apply one way with GPS satellites.

The effects are emphasized for several different orbit radii of particular interest. For a low earth orbiter such as the Space Shuttle, the velocity is so great that slowing due to time dilation is the dominant effect, while for a GPS satellite clock, the gravitational blueshift is greater. The effects cancel at . The Global Navigation Satellite System GALILEO, which is currently being designed under the auspices of the European Space Agency, will have orbital radii of approximately 30,000 km.

http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Broken]
See chapter 5.

Last edited by a moderator: May 4, 2017
2. Nov 15, 2009

### JesseM

The satellites aren't programmed to calculate things with respect to a separate coordinate system where they (the satellites) are at rest, and in GR unlike in SR there wouldn't be a unique coordinate system where this was true anyway (so your statement that they should view Earth clocks as running slower isn't really correct, you could come up with a non-inertial coordinate system where a given satellite was at rest but where clocks on Earth were ticking faster). All the calculations on board the satellite are done with respect to the Earth-centered coordinate system where the satellites are in motion. Remember, there is no physical requirement that a given observer calculate things relative to a frame where they are at rest, any observer can use any coordinate system they please for making calculations.

3. Nov 15, 2009

### cfrogue

I am confused.
If you look at the diagram in the link I posted, it shows the speed relative to the earth frame influences the clock beat rates in the satellites.

Thus, the satellite clocks are adjusted according to their orbit according to that chart.

There are two main adjustments according to the article, one for time dilation and one for gravity.

Do you agree this is correct or am I reading the article wrong.

4. Nov 15, 2009

### JesseM

You are correct, but all these statements refer to the Earth-centered coordinate system--when the satellites have a higher speed in this system, the rate of an unadjusted clock aboard the satellite will slow down relative to coordinate time in this system, so the rate has to be adjusted so they keep pace with coordinate time. No coordinate systems other than the Earth-centered one need to be used here.

5. Nov 15, 2009

### cfrogue

OK, so it is correct that time dilation is a factor for the satellite clock.

Is this correct?

6. Nov 15, 2009

### JesseM

Yes.

7. Nov 15, 2009

### cfrogue

So, why does reciprocal time dilation not apply from the POV of the satellite since the earth is moving relative to it and the satellite is at rest. Thus, the earth clocks would seem to have to beat slower and thus the satellite would progressively get more out of sync over time.

8. Nov 15, 2009

### JesseM

It seems like you are equating "the POV of the satellite" with "the POV of a coordinate system where the satellite is at rest". If so, this is incorrect--any observer is free to adopt any coordinate system they like to define a "point of view" for doing calculations, and in this case the satellites are programmed to calculate things from the perspective of the Earth-centered coordinate system. It might be true that if the satellite instead did calculations from the perspective of a satellite-centered coordinate system then they would conclude Earth clocks were running slower (though as I said, since we are talking about non-inertial coordinate systems this need not necessarily be true, time dilation is certainly not guaranteed to be 'reciprocal' in this way when dealing with two non-inertial systems), but in actual fact this is not the coordinate system the satellites are programmed to do their calculations in.

9. Nov 15, 2009

### cfrogue

The article uses two terms, time dilation and gravity.

It makes the following statement.
the velocity is so great that slowing due to time dilation is the dominant effect

What do you think this means?

10. Nov 15, 2009

### JesseM

I think they are talking about velocity-based time dilation vs. gravitational time dilation, the latter being a GR effect which isn't based on velocity.

11. Nov 15, 2009

### cfrogue

OK, and is the velocity relative velocity between the earth and satellite?

12. Nov 15, 2009

### mgb_phys

Yes it's the relative velocity of the satelite and the ground station.
Just to add some extra complexity GPS clocks are actually deliberately slowed down - the General relativistic time going faster in lower gravity is bigger than the special relativistic time slowing down at high speed.

13. Nov 15, 2009

### cfrogue

That is what the article says. But, at some altitude they cancel. At low altitudes the relative v is the stronger influence according to the article.

So, if this is a relative v, why doesn't the earth frame beat slower when the satellite is in orbit? It seems that the satellite would consider itself at rest and the earth frame moving. Thus, the satellite would believe the earth frame clocks should beat slower.

Why doesn't this happen?

14. Nov 15, 2009

### JesseM

There is only one type of velocity used--the velocity of the satellite in the Earth-centered coordinate system. It's a "relative" velocity in the sense that you're measuring the satellite's velocity relative to this particular coordinate system, but "relative" does not imply that anyone is calculating the velocity of the Earth clocks in a satellite-centered frame. You could do this on your own if you wanted to, but it is not part of the GPS calculations, the only frame that any GPS computers are using is the Earth-centered one.

Do you understand that in relativity you are never required to use multiple frames, you can always get the answer to any physical question you're interested in using a single frame, even if the situation you are analyzing involves objects which are not at rest in that frame?

15. Nov 15, 2009

### pervect

Staff Emeritus
I think you'd be best off understanding the twin paradox in flat space-time first. The key problem here I think its that you are imagining that the satellite has some well-defined coordinate system that covers all of space. That is generally not true for an accelerating observer - one can construct various coordinate systems for them, but they are in general all local.

To avoid all these complexities and gain some insight, I'd again suggest that you consider the flat space-time non-rotating twin paradox, which is being beaten to death in a number of threads around here...

16. Nov 15, 2009

### cfrogue

Yea, this is fine.

Maybe I should ask it this way.

Is the earth moving relative to the satellite?

17. Nov 15, 2009

### cfrogue

I am imagining the satellite is at virtual rest in its own system of coordinates. I say virtual because of the orbit.

Also I am imagining the earth is moving relative to the satellite. I am not assuming some universal coordinate system. I don't have any idea what that would mean.

18. Nov 15, 2009

### JesseM

When people talk about X moving relative to Y in SR, they really mean X is moving in the inertial rest frame of Y--there is no way to define "motion" except relative to a particular coordinate system. Since in GR there is no "standard" way to define the non-inertial rest frame of a given object, your question isn't really specific enough to have a definite answer, you need to define what type of coordinate system you want to use. There could be some non-inertial coordinate systems where the satellite was at rest and the Earth clocks were in motion, and others where both the satellite and the Earth clocks were at rest (at least for some period of time).

19. Nov 15, 2009

### cfrogue

I am ignoring all those complexities of GR and just focusing in on the relative v part as used in GPS.

Also, I would like to use the coords of the satellite since that is legal.

So, is the earth moving relative to the satellite?

20. Nov 15, 2009

### JesseM

But that's a v relative to a particular Earth-centered coordinate system. You can only talk about the v of anything relative to some coordinate system, velocity has no coordinate-independent meaning.
But as I already told you, there is no single coordinate system that qualifies as "the" frame of the satellite. Since we are dealing with curved spacetime we can't use an inertial frame, and there are an infinite number of different possible non-inertial coordinate systems where the satellite is at rest, all of which are equally valid in GR (see the discussion in http://www.aei.mpg.de/einsteinOnline/en/spotlights/background_independence/index.html [Broken] about the principle of 'diffeomorphism invariance' which says the laws of GR work in any kind of arbitrary coordinate system you can come up with).
In some satellite-centered coordinate systems it would be, in others it wouldn't.

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