I GPS system and general relativity

[cianfa72]

TL;DR Summary

How GPS system works in the context of general relativity

Hi, we had a thread some time ago about GPS satellite system.

One starts considering the ECI coordinate system in which the Earth's center is at rest with axes pointing towards fixed stars. One may assume it is an inertial frame in which the Earth's surface undergoes circular motion.

Clocks on Earth's surface and on GPS geostationary satellites are at different gravitational potential in the Earth's gravitational field. Hence there is a gravitational time dilation between them.

First question: does GPS system employ Schwarzschild spacetime model of Earth's gravitational field to evaluate the above time dilation?

Second question: what is the role of ECI ?

Thanks.

 

[Vanadium 50]

Third question - what have you done to answer this question? Where have you looked, what did you find, and where should we start?

 

[cianfa72]

From the Exploring Black Holes book - 2nd edition, the answer to my first question is positive. Clocks on the Earth's surface "see" time dilation for the rate of clocks sitting on GPS satellites (namely there is a gravitational blue shift that gives an observer on the Earth the "impression" that clocks on GPS satellites "run fast" compared with his own wristwatch). Maybe there should be another component due to the motion of GPS satellites.

 

[russ_watters]

The Wikipedia article on GPS has a short section on this question and a link to a more detailed one...

 

[FactChecker]

Without corrections for relativity, GPS locations would be wrong by over 6 miles per day. GPS satellite clocks would run faster by 38 ms[EDIT]$\mu s$/day.

They would run 45ms[EDIT]$\mu s$ faster due to GR space curvature and 7ms[EDIT]$\mu s$ slower due to SR velocity.

This might be a place to start. It gives numbers, but does not explain the derivation in detail.
(see http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html )

 

[phinds]

Maybe there should be another component due to the motion of GPS satellites.

"Maybe" there should be? Since there IS, I'd say you are right.

 

[cianfa72]

Without corrections for relativity, GPS locations would be wrong by over 6 miles per day. GPS satellite clocks would run faster by 38 ms/day.

They would run 45ms faster due to GR space curvature and 7ms slower due to SR velocity.

You mean the overall GPS satellite clock correction is actually given by the difference 45ms/day - 7ms/day = 38 ms/day.

So the above is, let me say, the overall gravitational blue shift of a GPS clock compared to an Earth's based clock.

Edit: which is the role of ECI coordinate system ?

 

[FactChecker]

You mean the overall GPS satellite clock correction is actually given by the difference 45ms/day - 7ms/day = 38 ms/day.

Yes. My understanding is that the GPS satellite clocks are actually set so that the counted number of molecular oscillations is short [EDIT] increased be the amount that represents 38 ms[EDIT]$\mu s$/day.

So the above is, let me say, the overall gravitational blue shift of a GPS clock compared to an Earth's based clock.

That is beyond my knowledge. Does "gravitational blue shift" include the -7 ms[EDIT]$\mu s$/day? It doesn't seem like it does, since it is the gravitational effect.

Edit: which is the role of ECI coordinate system ?

I am only a casual amateur, so that is beyond my knowledge.

 

[Nugatory]

Second question: what is the role of ECI ?

We have to choose some coordinate system if we’re going to calculate anything.

ECI coordinate positions and velocities correspond to our natural understanding of positions and speeds near the surface of the earth; that’s exactly what we want when we’re navigating so that’s what we choose.

 

[JT Smith]

Not that it matters for the sake of this discussion but I think it's 38µs/day not 38ms/day.

 

[PeterDonis]

You mean the overall GPS satellite clock correction is actually given by the difference 45ms/day - 7ms/day = 38 ms/day.

So the above is, let me say, the overall gravitational blue shift of a GPS clock compared to an Earth's based clock.

No, it's the overall blueshift, period. The +45 is the "gravitational" part and the -7 is the "motion" part. (Note that, as @JT Smith pointed out, these should be microseconds per day.) The split between "gravitational" and "motion" is frame-dependent; those numbers are for the ECI frame.

The best reference online that I know of for GPS and relativity is this Living Reviews article:

https://link.springer.com/article/10.12942/lrr-2003-1

 

[DaveC426913]

Not that it matters for the sake of this discussion but I think it's 38µs/day not 38ms/day.

Indeed. 38m/s is long enough to be human-detectable. (I mean, not as a discrepancy over a whole day, but we can actually perceive a .038 second interval).


Say, doesn't this mean the entire universe is flashing by our eyes overhead at 38µs/day faster than we perceive it down here in our gravity hole? Do astrophysicists ever have to factor that in? I dunno, say, to the rotating pulses of quasars and stuff?

 

[cianfa72]

ECI coordinate positions and velocities correspond to our natural understanding of positions and speeds near the surface of the earth

Not sure to understand: do you mean perform a transformation from Schwarzschild coordinates to ECI coordinates ?

 

[FactChecker]

Not that it matters for the sake of this discussion but I think it's 38µs/day not 38ms/day.

Right. Thanks. I stand corrected. I'll correct my posts.

 

[PeterDonis]

doesn't this mean the entire universe is flashing by our eyes overhead at 38µs/day faster than we perceive it down here in our gravity hole?

No. First, the 38 microseconds per day faster is for the "uncorrected" rates of the GPS clocks in their particular orbits. It says nothing about any other objects.

Second, if we left the GPS clocks uncorrected, we would see them ticking faster than ground clocks. The correction applied is to slow them down so they tick (effectively) at the same rate as ground clocks. So it never makes sense to say "they are ticking faster than we perceive them to tick".

Do astrophysicists ever have to factor that in? I dunno, say, to the rotating pulses of quasars and stuff?

I don't think any astrophysical phenomena are observd accurately enough to make one part in a billion differences in clock rate observable. But if such differences were observable, again, it would not be a case of such phenomena happening "faster than we perceive them to happen". We would perceive them to happen faster than would be the case if there were no time dilation effects.

 

[cianfa72]

Second, if we left the GPS clocks uncorrected, we would see them ticking faster than ground clocks. The correction applied is to slow them down so they tick (effectively) at the same rate as ground clocks. So it never makes sense to say "they are ticking faster than we perceive them to tick".

Ok, so the correction is applied directly on the "adjusted rate" of GPS satellite clocks. In this way the overall blue shift is automatically "compensated" and the GPS clock is "seen" on the Earth as synthonous w.r.t. Earth's based clocks.

 

[PeterDonis]

the overall blue shift is automatically "compensated" and the GPS clock is "seen" on the Earth as synthonous w.r.t. Earth's based clocks.

Yes.

 

[Nugatory]

Not sure to understand: do you mean perform a transformation from Schwarzschild coordinates to ECI coordinates ?

No, just specify positions using ECI position coordinates. It's what navigators have been doing since forever, or at least since it was known that the earth is round.

 

[PeterDonis]

No, just specify positions using ECI position coordinates. It's what navigators have been doing since forever, or at least since it was known that the earth is round.

Not really that long, since ECI coordinates do not rotate with the Earth. Navigators since forever or at least since it was known the earth is round have been using what we now call ECEF (Earth Centered Earth Fixed) coordinates.

 

[Nugatory]

Not really that long, since ECI coordinates do not rotate with the Earth. Navigators since forever or at least since it was known the earth is round have been using what we now call ECEF (Earth Centered Earth Fixed) coordinates.

Good point

 

[DaveC426913]

No. First, the 38 microseconds per day faster is for the "uncorrected" rates of the GPS clocks in their particular orbits. It says nothing about any other objects.

The 38 microseconds is due to GR, right? Which means it would affect any observations of anything outside our gravity well, no?

Second, if we left the GPS clocks uncorrected, we would see them ticking faster than ground clocks. The correction applied is to slow them down so they tick (effectively) at the same rate as ground clocks. So it never makes sense to say "they are ticking faster than we perceive them to tick".

Right, so all phenomena happening off-planet are slightly sped up from our perspective (albeit by a small amount).

 

[PeterDonis]

The 38 microseconds is due to GR, right?

It is a GR calculation for the GPS satellites in their particular orbits.

For other satellites in other orbits, the correction will be different. Nor will it always be a speedup compared to Earth. For low earth orbits like that of the ISS, it is a slowdown compared to Earth.

Once you get outside the near Earth region and have to consider other gravity wells, things get much, much more complicated. See further comments below.

Which means it would affect any observations of anything outside our gravity well, no?

Not the 38 microseconds specifically, no. See above and further comments below.

Right, so all phenomena happening off-planet are slightly sped up from our perspective (albeit by a small amount).

No. It's much more complicated than that. You appear to be thinking that the Earth's gravity well is the only significant effect. For objects close enough to the Earth, like satellites, that's a good enough approximation. But that only covers a very small number of cases.

For example: consider the space probes that were sent to explore Mercury and Venus. Those are closer to the Sun than the Earth is and the net effect on clocks on the probes would be a slowdown compared to Earth, because the Sun's gravity well is a much more significant effect.

Or consider stars closer to the center of the Milky Way than our solar system. The galaxy's gravity well is an even more significant effect than the Sun's so the net effect on clocks in those star systems would be even more of a slowdown compared to Earth.

And once we start talking about distant quasars powered by supermassive black holes, and have to factor in the expansion of the universe, it gets even more complicated than that.

 

[FactChecker]

The 38 microseconds is due to GR, right? Which means it would affect any observations of anything outside our gravity well, no?

Anything stationary at the altitude would show a 45 microsecond speedup due to GR.

Right, so all phenomena happening off-planet are slightly sped up from our perspective (albeit by a small amount).

That would need to be adjusted if it was moving very fast. The GR and SR effects oppose each other. For GPS satellites, the GR effect dominates.

 

[PeterDonis]

The GR and SR effects

This is not really correct. "SR" implies flat spacetime, and the spacetime in question is not flat. As I said in post #22, the correct calculation is a GR calculation. In the ECI frame, the overall effect can be split up into the 45 microseconds per day due to altitude, and -7 microseconds per day due to speed, but as I said in post #11, that split is frame dependent.

 

[cianfa72]

This is not really correct. "SR" implies flat spacetime, and the spacetime in question is not flat. As I said in post #22, the correct calculation is a GR calculation. In the ECI frame, the overall effect can be split up into the 45 microseconds per day due to altitude, and -7 microseconds per day due to speed, but as I said in post #11, that split is frame dependent.

The overall 38 microseconds/day due to the overall blueshift is frame-invariant (one can calculate it in any coordinate chart and get the same result).

Could you be more specific about its split in the ECI frame ? I'm confused how ECI frame/coordinates are actually related to Schwarzschild coordinates.

 

[Nugatory]

Could you be more specific about its split in the ECI frame ? I'm confused how ECI frame/coordinates are actually related to Schwarzschild coordinates.

They aren’t. Once we have the times properly adjusted we calculate the distances from light travel time without further relativistic correction. This works because the distance correction is small compared to the time correction.

 

[cianfa72]

They aren’t. Once we have the times properly adjusted we calculate the distances from light travel time without further relativistic correction. This works because the distance correction is small compared to the time correction.

So the point is: by means of "rate adjustment" on GPS satellite clocks the (frame-invariant) blueshift is totally "compensated" (i.e. the "rate" of GPS satellite clocks is "seen" as synthonous w.r.t. Earth's based clocks).

What distances are you talking about ? Is it the distance of GPS satellites from the GPS receivers on Earth (like my smartphone's GPS receiver) ?

 

[Nugatory]

What distances are you talking about ? Is it the distance of GPS satellites from the GPS receivers on Earth (like my smartphone's GPS receiver) ?

Yes. Once we know the distance between us and several satellites we can calculate our position relative to those satellites and then, because we already know the positions of the satellites relative to the earth, our position relative to the earth. And that's basically what your GPS receiver is doing.

 

[FactChecker]

This is not really correct. "SR" implies flat spacetime, and the spacetime in question is not flat. As I said in post #22, the correct calculation is a GR calculation. In the ECI frame, the overall effect can be split up into the 45 microseconds per day due to altitude, and -7 microseconds per day due to speed, but as I said in post #11, that split is frame dependent.

Thanks! As a casual amateur, I was not aware that the GR model/calculations included the time dilation of relatively moving objects that are in SR.

 

[PeterDonis]

I am marvelling the at the very simple fact that, when we look out of our gravity well, we looking at a universe that is (on average) observed to be ticking by slightly faster than us.

No. This statement is wrong.

they will all start with that effect caused by our own gravity well.

Yes, but...

Other gravity wells will be applied (+ive or -ive) on top of that.

Yes, and in both cases I described, the effect of climbing out of other gravity wells (Sun and galaxy) is a slowdown which is larger than the speedup caused by falling into Earth's gravity well. So the net effect is a slowdown.

Also, as I said, gravity wells are not the only factor. When you're looking at a quasar billions of light years away, the redshift due to the expansion of the universe swamps any gravity well effect in either direction.

I was simplistically assuming that the effect is most pronounced very close by

If by "the effect" you mean the net gain relative to Earthbound clocks, considering only the Earth's gravity well, then you have it backwards: the effect gets larger as you get further away. (But how much larger it gets is bounded, and the upper bound is small compared to other effects in the rest of the universe.) And as you get closer, it flips sign, so that, as I already told you, clocks in low Earth orbit, such as on the ISS, run slower than Earthbound clocks.

Our baseline would be flat space, away from any mass

There is no such place anywhere in our actual universe. The spacetime of our actual universe is not asymptotically flat. An asymptotically flat model is an approximation we use when we only need to consider one gravity well and can ignore the rest of the universe. But the sweeping claim you made about looking at the rest of the universe cannot be analyzed using any such approximation.

Of course, that too is factored in.

Not in your claim, it isn't. See above.

The component effects (both +ive ad -ive) are cumulative. That does not mean any given factor disappears.

I never said it did. I said that the effect of Earth's gravity well is too small to overcome the other effects I described--not all of which are gravity wells at all.

you are over-analyzing this for my benefit.

No, you are under-analyzing and basing your claim on invalid assumptions. See above.

 

[FactChecker]

And as you get closer, it flips sign, so that, as I already told you, clocks in low Earth orbit, such as on the ISS, run slower than Earthbound clocks.

Interesting. So there is a switch between the ISS at about 200 miles altitude (Low Earth Orbit) and GPS satellites at about 13k miles (Mid Earth Orbit). Of course, the velocities are also very different.

 

[PeterDonis]

So there is a switch between the ISS at about 200 miles altitude (Low Earth Orbit) and GPS satellites at about 13k miles (Mid Earth Orbit).

The switching altitude (i.e., the altitude at which the clock tick rate is the same as for a clock at rest on the geoid of the rotating Earth--the altitude and speed effects cancel) is about 1.5 Earth radii (from the center of the Earth, or about 0.5 Earth radii above the surface).

 

[cianfa72]

It is a GR calculation for the GPS satellites in their particular orbits.

Ok, so 38 microseconds/day is actually frame-invariant for GPS satellites in their specific orbits. One can calculate it, for example, in Schwarzschild coordinates in which one must take in account both the GPS satellite coordinate velocity ${d\phi} / {dt}$ and GPS receivers's velocity on Earth's surface (like my smartphone).

 

[PeterDonis]

38 microseconds/day is actually frame-invariant for GPS satellites in their specific orbits.

Yes.

One can calculate it, for example, in Schwarzschild coordinates

Not really. In the Newtonian approximation, which is what the ECI frame really uses, the radial coordinate is actual radial distance, not areal radius. Effects due to spatial curvature of surfaces of constant coordinate time are ignored (they are too small to matter anyway).

Also, to get an accurate enough value, you need to take into account the Earth's quadrupole moment, which is not included in the metric in Schwarzschild coordinates (or in any coordinate chart on Schwarzschild spacetime).

one must take in account both the GPS satellite coordinate velocity ${d\phi} / {dt}$ and GPS receivers's velocity on Earth's surface (like my smartphone).

And the altitude difference between them.

 

[phinds]

Anything stationary at the altitude would show a 45 microsecond speedup due to GR.

Yes, for the gravity part, but it would not BE stationary except briefly since that is not a geosynchronous orbit.

 

[jbriggs444]

Yes, for the gravity part, but it would not BE stationary except briefly since that is not a geosynchronous orbit.

In a geostationary orbit a hypothetical satellite would still not be stationary in the ECI frame.

 

[PeterDonis]

it would not BE stationary except briefly

Unless it used rocket thrust to remain stationary.

since that is not a geosynchronous orbit.

No free-fall orbit is stationary in the ECI frame.

 

[Mordred]

Also, to get an accurate enough value, you need to take into account the Earth's quadrupole moment,

You wouldn't perchance have any detailed studies on Earths quadrupole moment perchance ?
If not its Not a problem I can search for some good studies but if you have one handy it would be a time saver

 

[Ibix]

Yes, for the gravity part, but it would not BE stationary except briefly since that is not a geosynchronous orbit.

You're mixing up the Earth Centered Inertial (ECI - non-rotating) and the Earth Centered Earth Fixed (ECEF - rotates once per day) systems, I think. Geosynchronous satellites are stationary in the latter, but not the former.

 

[PeterDonis]

You wouldn't perchance have any detailed studies on Earths quadrupole moment perchance ?

Not detailed studies, no.

 

[Ibix]

The best reference online that I know of for GPS and relativity is this Living Reviews article:

https://link.springer.com/article/10.12942/lrr-2003-1

I've been reading this. Stripped of implementation detail, the GPS satellites send messages saying "I am satellite #6. At the time I sent this message my internal clock reads 21:47:00.00...0 GMT, and my coordinates in the ECI are (x,y,z)". Receiving any pair of these messages from different satellites some time $\delta t$ apart places me on a surface that is $c\delta t$ closer to the first satellite than the second. Three messages places me on a line that intersects the Earth's surface somewhere and four localises my altitude too.

Right?

 

[PeterDonis]

Right?

Not quite. 3 signals is sufficient for a 3-dimensional spatial location (lat/long and altitude). 4 signals is sufficient to adjust the receiver's clock to be in sync with the GPS clocks as well.

Note also that GPS uses an ECEF frame (i.e., rotating with the Earth), not the ECI frame.

 

[Ibix]

Not quite. 3 signals is sufficient for a 3-dimensional spatial location (lat/long and altitude).

Ah yes. Note that there are certain symmetrical cases where three signals is not enough, but such situations will be fleeting (due to satellite motion) if they can exist at all in practice.

 

[cianfa72]

In the Newtonian approximation, which is what the ECI frame really uses, the radial coordinate is actual radial distance, not areal radius. Effects due to spatial curvature of surfaces of constant coordinate time are ignored (they are too small to matter anyway).

You mean that the spatial curvature of surfaces of constant Schwarzschild coordinate time $t$ is neglected/ignored. Hence, assuming flat spacetime, ECI radial coordinate is actually the proper distance of GPS satellite at time $t$ w.r.t. the center of the Earth.

Sorry, so GPS system actually uses ECEF frame and not ECI frame ?

 

[PeterDonis]

You wouldn't perchance have any detailed studies on Earths quadrupole moment perchance ?

Equation 13 in the Living Reviews paper I referenced earlier is an approximate expression for the gravitational potential around Earth including its quadrupole moment.

 

[PeterDonis]

You mean that the spatial curvature of surfaces of constant Schwarzschild coordinate time $t$ is neglected/ignored.

Actually, looking at Section 3 of the Living Reviews paper I referenced earlier, it's not completely ignored. There is an $r$-dependent correction factor in the spatial part of the metric. But the coordinates used are isotropic coordinates, so the correction is applied to the entire spatial part of the metric, not just to $g_{rr}$. In other words, the $r$ coordinate is not exactly equal to radial proper distance, but it's not the areal radius either. It's somewhat in between.

Hence, assuming flat spacetime

No, spacetime is not assumed to be flat.

ECI radial coordinate is actually the proper distance of GPS satellite at time $t$ w.r.t. the center of the Earth.

Yes.

so GPS system actually uses ECEF frame and not ECI frame ?

Yes. More precisely, it uses a rotating frame fixed to the Earth but with the simultaneity convention of the ECI frame. The Living Reviews paper I referenced earlier discusses this.

 

[Mordred]

Not detailed studies, no.

Thanks I'm still going to dig around for one. It's sometimes surprising when research of this nature can be useful as I found out nearly a decade ago on another unrelated study.

 

[cianfa72]

No, spacetime is not assumed to be flat.

Of course, my fault (space is assumed flat).

In other words, the $r$ coordinate is not exactly equal to radial proper distance, but it's not the areal radius either. It's somewhat in between.

You mean the $r$ coordinate of the rotating frame fixed to the Earth as defined below.

More precisely, it uses a rotating frame fixed to the Earth but with the simultaneity convention of the ECI frame. The Living Reviews paper I referenced earlier discusses this.

Sorry, but the simultaneity convention of the ECI frame given by its time $t$, is actually the same as the Schwarzschild coordinate time ?

 

[PeterDonis]

You mean the $r$ coordinate of the rotating frame fixed to the Earth as defined below.

Yes.

Sorry, but the simultaneity convention of the ECI frame given by its time $t$, is actually the same as the Schwarzschild coordinate time ?

The simultaneity convention of the ECI frame is that the simultaneity surfaces are orthogonal to the worldline of the Earth's center of mass, and to an imaginary congruence of non-rotating worldlines at rest relative to the Earth's center of mass.

It is true that Schwarzschild coordinates use a similar definition of simultaneity, but so do other charts. Your focus on Schwarzschild coordinates here is misplaced. The important thing is the physical definition of the simultaneity surfaces.

 

[Vanadium 50]

I have never seen so much in the way of unnecessary complication.

Fundamentally, each satellite says "This is my location and this is what my clock reads."
A receiver on or near the surface of the earth knows it is on (or near) a sphere centered on the earth. Given this, and the signal from one satellite, it can determine its position to a circle. Two satellites and its two points. Three and its one point plus altitude.
There are SR and GR corrections to these clocks. There are also corrections because the earth is not spherical. There are no GR corrections to the distance because they are tiny (smaller than the distance between antenna and receiver).

One can talk about various frames, and whether it is better to do the corrections in the transmitter or receiver, but the setup is not nearly as complicated as you make it: get some times, correct for relativity, and use them to fix your position.