GPS system and general relativity

[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.

 

[cianfa72]

The important thing is the physical definition of the simultaneity surfaces.

Sorry, I believe I lost that physical definition of simultaneity surfaces you were talking about.

 

[PeterDonis]

I believe I lost that physical definition of simultaneity surfaces you were talking about.

I gave it in post #52.

 

[cianfa72]

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.

Yes, this is the definition of ECI simultaneity convention let me say in spacetime mathematical model. Now, from a physical point of view, what does it mean?

 

[PeterDonis]

this is the definition of ECI simultaneity convention let me say in spacetime mathematical model.

Not only that. Orthogonality of simultaneity surfaces to worldlines has a physical meaning--actually more than one. It means the congruence of worldlines is non-rotating. It also means that the simultaneity surfaces are the ones that would be established by Einstein clock synchronization between the worldlines.

 

[Mordred]

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

Sorry didn't see this post earlier. Yes I saw the expression and was able to find a NASA article that includes details on k20 as well as J20. Still studying that article.

Quick question isn't ECEF also oft called the geocentric coordinate reference system? If so then I already have the related mathematics for that coordinate system. Either way I was able to find the mathematical details for ECI and ECEF inclusive with the quadrupole moment.

Edit NASA article was specific to STEP program disregard the J20 and K20 above. Article still is useful otherwise.

 

[cianfa72]

Orthogonality of simultaneity surfaces to worldlines has a physical meaning--actually more than one. It means the congruence of worldlines is non-rotating. It also means that the simultaneity surfaces are the ones that would be established by Einstein clock synchronization between the worldlines.

You mean take the timelike geodesic worldline of Earth's center (it is geodesic since the center is in free-fall) and a set of massive objects whose timelike worldlines do not intersect (a congruence), do not rotate (i.e. hypersurface orthogonal) and are at rest w.r.t. the Earth's center worldline (as measured by constant travel time of bouncing light beams exchanged between Earth's center worldline and each of the congruence's worldlines).

What does physically mean the above conditions to be non-rotating/irrotational ? From Sachs and Wu section 5.3 geodesic and irrotational conditions are equivalent (iff) to locally proper synchronizable.

Then two observers in the congruence, communicating via radar (bouncing light beams), can check the above consistency condition that is basically the Einstein's clock synchronization convention.

 

[Ibix]

Given this, and the signal from one satellite, it can determine its position to a circle.

I don't think it's quite that simple - the receiver would need to know the signal flight time for that, for which it would need an atomic clock ticking ECI time. Without such a clock it needs two satellites and the time difference between signal arrivals for any kind of fix, which places it on something like a paraboloid, which maps to a line on the geoid. A third satellite gives you another pair of paraboloids from the extra pairwise differences, the intersection of which should be a point (except in certain special cases).

Either that or I'm misunderstanding how the system works.

 

[PeterDonis]

the receiver would need to know the signal flight time for that, for which it would need an atomic clock ticking ECI time.

If less than four signals are received, the receiver must use its own clock (which of course is not an atomic clock and is nowhere near that accurate, except for some special cases) to determine signal flight times. However, whenever four signals are in view, the receiver can update its clock, since four signals allows determination of all four coordinates of the reception event in the ECI frame. The latter is the normal mode of operation for receivers.

 

[cianfa72]

What does physically mean the above conditions to be non-rotating/irrotational ? From Sachs and Wu section 5.3 geodesic and irrotational conditions are equivalent (iff) to locally proper synchronizable.

Reading again Sachs & Wu section 2.3 I believe the difference between locally synchronizable vs synchronizable is about the domain of definition of $C^{\infty}$ functions $h$ and $t$ such that the differential 1-form $\omega$ (that defines the 3D spacelike distribution orthogonal to the congruence's timelike worldlines at each point) is written as $\omega = - hdt$.

In the latter both functions are globally defined on spacetime manifold, while in the former only locally (in an open neighborhood of each point).

 

[PeterDonis]

You mean take the timelike geodesic worldline of Earth's center (it is geodesic since the center is in free-fall) and a set of massive objects whose timelike worldlines do not intersect (a congruence), do not rotate (i.e. hypersurface orthogonal) and are at rest w.r.t. the Earth's center worldline (as measured by constant travel time of bouncing light beams exchanged between Earth's center worldline and each of the congruence's worldlines).

Yes.

What does physically mean the above conditions to be non-rotating/irrotational ?

Physically it means there are no "fictitious forces" in the frame, in the Newtonian sense, i.e., no centrifugal or Coriolis force. In the absence of gravity we would say that meant the worldlines of free-falling objects are straight lines in the frame, but since gravity is present (due to the Earth), we can't say that. So the ECI frame is really only "inertial" in the Newtonian sense (where gravity is not considered a "fictitious force" and it's OK for the worldlines of objects free-falling under gravity to not be straight lines in the frame). It is not "inertial" in the GR sense (and in the GR sense it is impossible to define a single inertial frame that covers the entire Earth and its neighborhood, as the ECI frame does).

From Sachs and Wu section 5.3 geodesic and irrotational conditions are equivalent (iff) to locally proper synchronizable.

The geodesic condition is not met since only one worldline at rest in the frame (that of the Earth's center) is a geodesic. The congruence that defines the ECI frame is, in Sachs and Wu terminology, locally synchronizable (a common set of simultaneity surfaces can be defined that are everywhere orthogonal to the congruence) but not locally proper time synchronizable (gravitational time dilation is present).

 

[cianfa72]

Physically it means there are no "fictitious forces" in the frame, in the Newtonian sense, i.e., no centrifugal or Coriolis force. In the absence of gravity we would say that meant the worldlines of free-falling objects are straight lines in the frame, but since gravity is present (due to the Earth), we can't say that. So the ECI frame is really only "inertial" in the Newtonian sense (where gravity is not considered a "fictitious force" and it's OK for the worldlines of objects free-falling under gravity to not be straight lines in the frame). It is not "inertial" in the GR sense (and in the GR sense it is impossible to define a single inertial frame that covers the entire Earth and its neighborhood, as the ECI frame does).

You mean non-rotating/zero vorticity timelike congruences have the feature that in the frame/chart in which they are "at rest", worldlines of free-falling objects, in case of flat spacetime, turn out to be straight lines (i.e. their coordinates are linearly dependent from the proper time parameter along each of them).

The geodesic condition is not met since only one worldline at rest in the frame (that of the Earth's center) is a geodesic. The congruence that defines the ECI frame is, in Sachs and Wu terminology, locally synchronizable (a common set of simultaneity surfaces can be defined that are everywhere orthogonal to the congruence) but not locally proper time synchronizable (gravitational time dilation is present).

Ok, so in Sachs & Wu terminology $\omega \wedge d\omega = 0$, i.e. locally $\omega = -hdt$ for some functions $h$ and $t$ defined both in an open neighborhood of any point.

 

[PeterDonis]

You mean non-rotating timelike congruences have the feature that in the frame/chart in which they are "at rest", worldlines of free-falling objects, in case of flat spacetime, turn out to be straight lines (i.e. their coordinates are linearly dependent from the proper time along each of them).

Non-rotating geodesic timelike congruences have this property in flat spacetime.

Ok, so in Sachs & Wu terminology $\omega \wedge d\omega = 0$, i.e. locally $\omega = -hdt$ for some functions $h$ and $t$ defined both in an open neighborhood of any point.

Yes.

 

[cianfa72]

Non-rotating geodesic timelike congruences have this property in flat spacetime.

And what if one considers just non-rotating timelike congruences without the requirement to be geodesics (in case of either flat or curved spacetime) ?

 

[PeterDonis]

what if one considers just non-rotating timelike congruences without the requirement to be geodesics (in case of either flat or curved spacetime) ?

Then you can no longer say that free-falling worldlines will be straight in the chart derived from the congruence. An obvious example is the Rindler congruence in Minkowski spacetime, which is irrotational but not geodesic. Free-falling worldlines are not straight in the Rindler chart derived from this congruence.

 

[cianfa72]

Then you can no longer say that free-falling worldlines will be straight in the chart derived from the congruence.

Therefore, what is the characteristic feature of the chart derivated from a generic non-rotating timelike congruence (in case of either flat or curved spacetime) ?

 

[PeterDonis]

Therefore, what is the characteristic feature of the chart derivated from a generic non-rotating timelike congruence (in case of either flat or curved spacetime) ?

What do you think it is? And why do you feel you need to ask?

 

[cianfa72]

What do you think it is?

I know that non-rotating/zero-vorticity congruence is equivalent to say hypersurface orthogonal. I'd better understand what it means from a physical perspective and what features have charts derivated from them (i.e. coordinate charts where they are "at rest").

 

[PeterDonis]

I know that non-rotating/zero-vorticity congruence is equivalent to say hypersurface orthogonal.

Yes. This is a consequence of the Frobenius theorem.

I'd better understand what it means from a physical perspective

Do you understand what having simultaneity surfaces that are orthogonal to the worldlines in the congruence means physically? More basically, do you understand what an orthonormal basis of vectors means physically?

what features have charts derivated from them (i.e. coordinate charts where they are "at rest").

Do you understand what hypersurface orthogonality implies about a chart that is adapted to that feature?

 

[cianfa72]

Do you understand what having simultaneity surfaces that are orthogonal to the worldlines in the congruence means physically?

No, I don't.

More basically, do you understand what an orthonormal basis of vectors means physically?

Can you kindly clarify it ?

Do you understand what hypersurface orthogonality implies about a chart that is adapted to that feature?

Yes, it implies there exists a chart where the mixed metric components $g_{t\alpha}, \alpha =1,2,3$ vanish.

 

[PeterDonis]

No, I don't.
Can you kindly clarify it ?

Think of the simplest case: a standard inertial frame in flat spacetime. Consider how such a frame is constructed, using "measuring rods and clocks" as Einstein described it. The measuring rods and clocks define an orthonormal basis at each point of spacetime.

For a non-geodesic case, consider the Rindler congruence in Minkowski spacetime and the Rindler chart adapted to it. The Rindler chart can also be realized with "measuring rods and clocks" in Einstein fashion, but accelerated ones instead of inertial (freely falling) ones. Again, those measuring rods and clocks define an orthonormal basis at each point of the patch of spacetime covered by the Rindler chart (but now a different orthonormal basis from the inertial one).

(For extra credit you can also consider the Schwarzschild chart in the exterior region of Schwarzschild spacetime.)

Having a hypersurface orthogonal congruence and using the orthogonal hypersurfaces as simultaneity surfaces allows a chart to be constructed that works like the above examples. And that is also the way we intuitively expect a chart to work.

it implies there exists a chart where the mixed metric components $g_{t\alpha}, \alpha =1,2,3$ vanish.

Yes. Which, again, is the way we intuitively expect a chart to work.