I GPS clock synchronization in ECI frame

[cianfa72]

For a more formal treatment, I personally like Misner's "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043, as I've probably mentioned a few times.

The point of view there is that a line element defines the coordinates - something I'll say quickly, but probably needs deep thought to be fully appreciated. Misner gives the line element for the ECI frame as:

dτ^2 = [1 + 2(V − Φ0)/c^2]dt^2 − [1 − 2V/c^2](dx^2 + dy^2 + dz^2)/c^2

This simple expression can then be regarded as the defintion of ECI coordinates.

I read that note that employs the following metric (with Newtonian gravitational potential $V$ variable with $r$):
$$d \tau^2 = [1 + 2(V − \Phi_0)/c^2 ]dt^2 − [ 1 - 2V/c^2 ] (dx^2 + dy^2 + dz^2)/ c^2$$ it actually entails a not Euclidean spatial geometry changing with the radius $r$, I believe.

Sorry, coming back to the GPS satellites clock synchronization, in post #29 has been said that GPS clock time is in synch when sent and not when received from receivers at ground.

Just to help me in understanding: consider for instance the time indication 12:00. To be in synch when signal is sent does mean the indication 12:00 at the event of sending the signal from the GPS satellite clock (that actually encode the time value 12:00) is the same as the indication 12:00 of an imaginary 'ECI coordinate clock' spatially co-located with that event ?

Your help is really appreciated !

 

[Ibix]

As I understand it, the system works by the satellites basically sending signals saying "Satellite #56, time 23:20:00, satellite #56, time 23:20:01..." and your GPS receiver looks up where satellite #56 is at 23:20:00. If your GPS receives the 23:20:00 signal from satellite #34 1ms later than the one from #56 then it knows it's on the plane one light millisecond closer to the location of #56 than #34. If it looks up #34's position at that time too then you've got a partial fix - with a third satellite you can reduce the plane to a line, which is enough for a location if you additionally assume you're on the surface of the Earth.

That's what @PeroK meant by the signals being in sync when sent (all the satellites send at the same ECI time) but not when received (the differing flight times of the radio pulses give you the difference in distances from the satellites).

So all we need on the ground is that all of the satellites report the time when they sent the signal in some agreed time standard. We choose the ECI and adjust the satellite clocks to tick that time standard.

 

[pervect]

Neil Ashby says that the clocks in the actual GPS implementation use the ECEF (earth centered, Earth fixed) frame. https://link.springer.com/article/10.12942/lrr-2003-1

Almost all users of GPS are at fixed locations on the rotating earth, or else are moving very slowly over earth’s surface. This led to an early design decision to broadcast the satellite ephemerides in a model earth-centered, earth-fixed, reference frame (ECEF frame), in which the model Earth rotates about a fixed axis with a defined rotation rate

I haven't been able to find written confirmation of my impression that clocks in the ECEF frame co-located with clocks in the ECI frame would show the same time. If ECEF and ECI share the same time coordinate for co-located clocks as I think they do, the only difference between ECEF and ECI s in how the spatial coordinates of the clocks are reported. But it would be good to have this in writing.

From a theoretical point of view, it's not necessary to use the ECEF frame, that's just the frame that GPS historically uses. For instance, see Minser "Precis of General Relatiavity"

Each clock maintains its own proper time (but may convert this via software into
other information when it transmits). We simplify to assume it transmits
its own proper time without random or systematic errors.

Conceptually using the proper times is "simpler" with idealized clocks in that it doesn't require adoption of any coordinate system for the clocks at all. The clocks report their proper time, and the software crunches the numbers to turn the proper times into whatever coordinates one wants to use. This follows the philosophy that coordinates are a convention that one adopts for reasons of convenience.

Furthermore, one can see that the synchronization of the clocks isn't critical to the question of what coordinates one wants. The fundamental issue is what coordinates one wants to use on the rotating Earth.

Practically, though, it probably is simpler to adopt a specific reference frame as the actual implmentation does, in order to simplify dealing with non-ideal effects that exist in the real world.

 

[PeterDonis]

Neil Ashby says that the clocks in the actual GPS implementation use the ECEF

The key point of the quote you give is not about the times broadcast by the satellites (as noted below, these are in fact the same as ECI coordinate times), but the spatial coordinates. Those are given in the ECEF frame, since, as Ashby notes, that's the most natural frame for GPS receivers.

I haven't been able to find written confirmation of my impression that clocks in the ECEF frame co-located with clocks in the ECI frame would show the same time.

It's later on in the Ashby article:

This generates a “coordinate clock time” in the earth-fixed, rotating system. This time is such that at each instant the coordinate clock agrees with a fictitious atomic clock at rest in the local inertial frame, whose position coincides with the earth-based standard clock at that instant.

The only complication is that one has to include a correction for gravitational redshift, as Ashby notes in the next paragraph.

 

[pervect]

It's later on in the Ashby article

Alright, thanks. I was pretty sure it had to be that way, but I wanted to find it in writing - it prevents mistakes, and is also more convincing.

This also gives us enough information to follow the outline of Misner's suggestions in "Precis of General Relativity". That would be to find the metric in the ECEF frame via the tensor transformation laws. I thought I had a source that had already done this, but a quick search didn't find one. With the information of the metric in ECI coordinates, and the coordinate transformations, we can compute the ECEF metric, though it'd be nice if we could find it also in writing somewhere to compare.

Then we can use the ECEF metric we compute to find the null paths that light would take in terms of ECEF coordinates, and use this information to confirm that the two-way time transmission standards proposed in https://tf.nist.gov/general/pdf/836.pdf work and are understood correctly. I'd have to go through this procedure myself to confirm my understanding.

What I would expect from this whole procedure is that when we compute the null paths above, that the coordinate speed of light in the ECEF frame, there is a linear relationship between position and coordinate time, but that the associated "speed" of this linear relationship is not the same east-west and west-east. It'd be easiest to do this on the equator, but if one was ambitious, one could do it at other lattitudes, and for paths that are not "due east" and "due west". One can then appreciate why Einstein's midpoint method for clock synchronization needs adjustment if done in these non-inertial coordinates, and exactly what this adjustment entails.