# GPS, GR, satellites and clock speed on the satellites

#### OS Richert

I am looking over the Project A: Global Positioning System is Exploring Black Holes; introduction to General Relativity by Taylor and Wheeler. At first, they convincingly suggest that the time dilation caused by the differences in radius from earth clocks to the satelight clocks will cause the satalight clocks to run fast on the order of 50000 nanoseconds. Makes sense. But then they say this.
In addition to effects of position, we must include effects due to motion of satellite and Earth observer. In which direction will these effects influence the result? The satellite clock moves faster than the clock revolving with Earth's surface. But Special relativity tells us that (in an imprecise summary) "moving clocks runs slow"
Therefor we expect the effect of motion to reduce the amount by which the satellite clock runs fast compared to the Earth Clock. In brief, when velocity effects are taken into account, we expect the satellite clock to run faster then Earth clock by less than the estimated 50000 nanoseconds per day
Here we go again! Why does the satellite which moves faster have a clock that actually runs slower. Faster according to whose frame of reference? Can't the satellight clock simply claim that it is the earth's clock that is in motion according to the satellight's frame of reference. It seems that in this sentence that are indeed defining a prefered frame of reference. The earth clock must actually be moving slower in space the the satellight clock (since it actually expereinces time dialation when you compare the two).

OR is the trick here that we consider the earth clock to be the frame of reference because it is the one receiving the signal and doing the GPS calculations. So anything in motion with repsect to it seems to have a clock that runs slow, even though they would say the same about you. In other words, if the ground clock sent a signal to the satellight clock, it likewise would think the earth clock was running slow. And somehow this is all accounted for by time simaltaneounity

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#### russ_watters

Mentor
The satellite is launched from earth and is orbit around the earth, so there is no simple way to consider the satellite stationary and the earth moving.

And yes, they are defining a "preferred" frame of reference. Pretty much every problem must have one! But don't take that to mean that that frame is somehow universally preferred. It is just the most useful for this particular problem.

#### OS Richert

And yes, they are defining a "preferred" frame of reference. Pretty much every problem must have one! But don't take that to mean that that frame is somehow universally preferred. It is just the most useful for this particular problem.
So, am I right then in my second paragraph? We are defining are rest frame as the earth clock, and the satellight is in relative motion to this frame. Therefor we see (per SR, ignoring GR for a second) its clock run slow. Since we are doing calculations for our location in our own rest frame, then this is valid. But if the satellight were trying to determine its location per multiple signals from earth, in its rest frame it would see those clocks on earth as running slow and would calculate with those numbers.

#### russ_watters

Mentor
Not quite. The rest frame would be the center of an earth stationary wrt the fixed stars. The Earth's rotation has to be taken into account. And what I said before still applies - it is difficult for the satellite to consider itself at rest because to get where it is, it needs to fire its engines.

So for the base station on earth, the clocks on the satellites going overhead every 90 minutes would appear slowed and for the satellites going over the base station, the base station's clock would appear to be ticking fast.

#### paw

I'm not sure I see how SR applies at all. No matter what frame of reference you choose the observer, satellite or both are going to be in accelerated motion. I thought SR only applied to inertial frames. Can anyone explain this? Is it that SR provides a useful shortcut that can be used even though it's not strictly applicable?

#### MeJennifer

Two objects that are in orbit with each other are both traveling inertially.
It really does not matter if you do the calculations from the perspective of the satellite or earth, they come out the same in GR.

#### OS Richert

MeJennifer/Russ,

So for the base station on earth, the clocks on the satellites going overhead every 90 minutes would appear slowed and for the satellites going over the base station, the base station's clock would appear to be ticking fast.
Doesn't this invalidate the very core of SR. I thought in SR there was no experiment in which you could do to prove that two seperate observers in seperate inital frames, that one is traveling FASTER then the other. Motion is only relative. So an example given in "Spacetime Phyiscs" by Taylor and Wheeler, if there is three observers, one at rest, one at 0.5c, and one at 0.9c; both the rest observer and the observer at 0.9c will believe that the wristwatch time of the observer at 0.5c is running slow.

Your saying that both the earth and the satellight will agree that the satellight's clock is running slower. That means both in the perspective of the rest frame of the earth, and in the persepctive of the rest frame of the satellight, the satellight clock is running slower. Then I could do an experiment and find the rest frame where that records MAXIMUM time, and this is the rest frame of the universe that is really not in motion. This to me, invalidates SR itself (there is no prefered initial frame in any sense).

To take the simpliest of examples (and remove any questions about acceleration because of earth's rotation and etc), if a man speeds by on a train traveling at 0,7c relative to an observer standing on the ground, and they both are holding wristwatches, they both will record that the other observers watch is running slow. Neither can make any authoritative claim that the other is traveling faster then the other. The reason this is not a simple paradox is because for the observer on the ground to record the wristwatch time of the traveling observer he needs to make two observerations at two different locations (because the train is moving) and the time of these observations will not be simatanious with the times that the observer himself would record on the train. This is the example used in my Modern Phyiscs book, "Modern Physics for Scientists and Engineers" by John Taylor and Chris Zafiratos. But to me, (and I could be misunderstanding you), you are saying likewise. You are saying that both the satellight (the "moving" observer) and the ground observer will agree that the satellight clock is running slower.

#### pervect

Staff Emeritus
There isn't any "frame" which has maximum timing, because there aren't any inertial frames in curved space-time.

What really has to have the maximum timing is not a "frame", but a path. Given two points, in a curved space-time, it is fair to ask "what is the path connecting them that has the maximum elapsed proper time - (but it doesn't make sense to try and analyze the situation in terms of inertial frames).

It is necessary, but not sufficient, that this path be a geodesic. In general, there will generally be more than one geodesic that connects two points in a curved space-time. Only one of these geodesics will have the maximum proper time (for a twin paradox experiment, only one path will have the oldest twin). (See some more comments on this at the end of the article).

An analogy might help. Rather than considering Lorentz intervals and maximum proper times in a Minkowskian geometry, consider the problem of finding the shortest distance between two points in a Euclidean geometry.

The statement above becomes "It is necessary that the shortest distance between two points be a straight line (geodesic) but it is not sufficient".

An example serves to illustrate this. Suppose we have two points, A and B, on the Earth's surface, separated by a mountain

A ^^^^ B

where ^^^^^ represents the mountain.

A path from A to B going over the mountain is a straight line, but it is not the shortest route from A to B. When the mountain is high enough, it is shorter to go around the mountain, rather than over it.

The property that straight lines of being local minima of distance implies that this path around the mountain will be another "straight line" (geodesic).

This is not a paradox, it's just an annoying aspect of dealing with curved geometries.

Note that when I use the term "straight line" (to avoid frightening people) in a generally curved geometry or on a curved manifold, I really mean a geodesic.

There is an old discussion here that points out that if you consider two points being close enough together in space-time, in the case of the Earth this means a time separation less than an orbital period, that there will only be one geodesic between two points. So we can say, informally, that if two points are "close enough together" there will only be one straight line connecting them in a curved geometry, but this is not true for points arbitrarily far apart. Similar remarks apply to space. If you have a general manifold, when you get points close enough together, you can consider the manifold to be "flat", and there will be a unique straight line between two points. But in general, for distant points, there can be many straight lines (i.e. geodesics) connecting two points, and some of them can be shorter than others.

#### Ich

It is certainly true that an exact treatment would use GR and lead to the result that the satellite's clock is running faster. But this doesn't help.
Taylor & Wheeler do not use GR, but a simplification where they start with a flat spacetime (like in SR) and add GR corrections.
They also split those corrections into a "potential" part and a "speed" part.
The potential correction yields the 50000 ns that the satellite's clock will go faster.
The speed correction is purely SR.
There we have the earth's surface, which is almost at rest in a suitable inertial frame. And the satellite, which is doing a round trip in this frame.
So it's essentially the twin paradox - the one going out, turning around, and returning will experience less time.
The situation is not reciprocal, because SR prefers inertial frames over arbitrary ones.

#### OS Richert

There isn't any "frame" which has maximum timing, because there aren't any inertial frames in curved space-time.
Humar me for a second. If the universe were entirely flat spacetime (ie, the whole universe operates simply under SR), would there be a reference frame under which a clock at rest would have maximum timing any everyone else in every other frame would agree that this particular frame was maximum timing?

#### OS Richert

So it's essentially the twin paradox - the one going out, turning around, and returning will experience less time.
OKAY, I am really close to understanding then. So Taylor and Wheeler are apprioximating that the earth clock is at rest in an inertial frame (though others have pointed out it is actually accerlating to stay in circular motion, yet very slowly). The satellight is doing the same but at a greater rate. But in it, we do not assume it stays in the same initerial reference frame. You relate it to the twin paradox in which one twin changes reference frames, then changes to another reference to return. Are we apprioximating two references frames then for the satellight, one as it passes over the earth clock, and the other at is travels on the other side of the planet. And if I were to do the calculations I would see that the satellight ages less.

So my mistake was assuming the two were always in relative motion to each other in two different inertial frames. In reality, neither is ever in an inertial frame because they must accererlate to keep circular orbit, but taylor and wheeler apprioximate the earth clock as in one interial frame and the satellight as in at least two and the resulting age difference is indentical to the twin paradox?

#### MeJennifer

Due to the curvature of spacetime the difference in elapsed time for any two objects in general relativity is far more complicated than in special relativity.

In special and general relativity, a definitive measure can only be established when two objects share two spacetime events. In all other cases we have to use light signals.

In special relativity these signals only show at any point in time the difference in accumulated time but not the distribution over the two objects.

In general relativity however, the light signal paths themselves are subject to spacetime curvature, so without knowing the exact curvature, it is impossible to even determine the difference in accumulated time, let alone their distribution.

Hope this helps anybody. :shy:

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#### paw

Two objects that are in orbit with each other are both traveling inertially.
In the Earth centered frame both observer and satellite are constantly accelerating (centripetal acceleration) are they not? The same is true in the observer and satellite centered frames isn't it? I don't see a true inertial frame here.

I do see an answer to my original question further down though. SR provides a useful approximation for time dilation calculations. I expect GR would provide the more complete answer at the cost of addition complexity.

#### jtbell

Mentor
I think the key word here is "approximation." An artificial satellite has a much smaller mass than the Earth, so it is a very good approximation to consider the center of the Earth as being at rest in an inertial reference frame. You wouldn't be able to do this for something like two equal-sized masses orbiting around each other.

#### MeJennifer

In the Earth centered frame both observer and satellite are constantly accelerating (centripetal acceleration) are they not? The same is true in the observer and satellite centered frames isn't it? I don't see a true inertial frame here.
If we ignore any possible atmospheric effects then the motion of both objects is determined only by the curvature of spacetime. That implies that both objects travel inertially and are not physically accelerating.

To calculate the geodesics they are travelling on is far from trivial, both objects in isolation create a Kerr like gravitational field, but there is no simple way to "add" these two gravitational fields and to merge their spin vectors. The only thing we know is that the geodesics they are traveling on cross at some point because orbits in general relativity decay.

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#### Boustrophedon

Here we go again! Why does the satellite which moves faster have a clock that actually runs slower. Faster according to whose frame of reference? Can't the satellight clock simply claim that it is the earth's clock that is in motion according to the satellight's frame of reference. It seems that in this sentence that are indeed defining a prefered frame of reference. The earth clock must actually be moving slower in space the the satellight clock (since it actually expereinces time dialation when you compare the two).
There seems to be a fair bit of confusion here. Surely the essential point is that unlike uniform linear motion, rotation is not a relative attribute, as even Newton with his spinning bucket was aware, and consequently a rotating clock is not equivalent to a non-rotating one. It will traverse a longer closed path to return to the starting point with less elapsed time.

Thus, independantly of any GR effects, a clock carried by the surface of a rotating earth could be expected to lose time compared to one not rotating with the earth (or one at the surface of a non-rotating earth). So a satellite clock could suffer either a kinematical loss or a gain in time with respect to an earthbound clock, depending respectively on whether it is going with (eastwards) or against (westwards) the earths motion.

In 1971 Hafele and Keating flew several caesium clocks in piecewise hops around the world by scheduled commercial airliners and claimed that for the eastward flight, the kinematical loss in time overwhelmed the gravitational gain (at about 30,000 feet) and resulted in a net lag in time for the eastbound clock, whilst the westbound was ahead in time by the gravitational plus the extra kinematical gain.
However, it's only fair to say that the accuracy of Hafele & Keating's results have been disputed as their clocks showed alarming amounts of random drift and their analysis involved highly selective use of favourable data while discarding less favourable. [see www.cartesio-episteme.net/H&KPaper.htm][/URL]

The problem of rotation, especially in relativity, is difficult, profound and still controversial (see for example any number of articles on Ehrenfest's paradox or the Sagnac effect), and I am only giving the generally accepted textbook explanation of anticipated effects where experimental confirmation is not conclusive.

One obvious problem with "absolute" rotation giving rise to asymmetric non-reciprocal time dilation is that the tangential velocity will be the same for slower rotation (omega) at greater radius. Thus if we continue to consider lower and lower omega with increasingly large radius, then the periphery asymptotically approaches the constant linear velocity of SR, which should have symmetric and reciprocal time dilation. Since absolute motion cannot "merge" gradually into relative motion there would seem to be a conceptual conflict.

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#### MeJennifer

Thus, independantly of any GR effects, a clock carried by the surface of a rotating earth could be expected to lose time compared to one not rotating with the earth (or one at the surface of a non-rotating earth).
GR is a complete theory of both kinematics and dynamics, there is no such thing as a kinematics or dynamics independent of GR.

It is correct however that treatment of rotation under special relativity is not interpretation free.

For a good summary see the excellently written article: "Relativistic Rotation" - Klauber ( Foundations of Physics, Volume 37, Number 2, February 2007, pp. 198-252(55) ). Alternatively one could read: "Relativity in Rotating Frames" - Guido Rizzi (Ed.) (Kluwer - 2004).

It might be interesting to know that the interpretations of our fellow member Demystifier are mentioned in both of the above mentioned references.

Lastly, I strongly suggest that we do not start a discussion about the interpretation of rotation under special relativity in this particular topic for it is not directly relevant.

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#### Boustrophedon

You've missed the point. The SR kinematical time dilation can be (and was by H&K and subsequent authors) treated independantly of the gravitational effect, which in this instance is irrelevant to O S Richert's question concerning the supposed absence of time reciprocity between rotating clocks in apparent relative motion. [eg. http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/airtim.html] [Broken]

Klauber and Rizzi are fairly typical of dozens of authors (incl. Ruggiero, Gron, Tartaglia, Keating, Goy, de Felice, Nicolic, Weber, Selleri, Dieks, etc, etc) who have copiously and diversely applied SR to rotating systems in operationally (not just interpretively) different ways concerned mainly with length relationships. The resulting confusion has shed little light on the problem but has revealed that most authors fail to distinguish the coordinate transformations of SR from the physical transformations of Lorentz's theory.

Here we are dealing with the (slightly) cleaner problem of time relationships (apparent clock rates) in rotating systems in the context of SR, where one would normally expect equivalent reciprocal time dilation. This is precisely and directly relevant to the topic introduced in post #1.

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#### MeJennifer

Anyhow, I do not think it is appropriate to discuss this in this topic.

#### Ich

OKAY, I am really close to understanding then. So Taylor and Wheeler are apprioximating that the earth clock is at rest in an inertial frame (though others have pointed out it is actually accerlating to stay in circular motion, yet very slowly). The satellight is doing the same but at a greater rate. But in it, we do not assume it stays in the same initerial reference frame. You relate it to the twin paradox in which one twin changes reference frames, then changes to another reference to return. Are we apprioximating two references frames then for the satellight, one as it passes over the earth clock, and the other at is travels on the other side of the planet. And if I were to do the calculations I would see that the satellight ages less.

So my mistake was assuming the two were always in relative motion to each other in two different inertial frames. In reality, neither is ever in an inertial frame because they must accererlate to keep circular orbit, but taylor and wheeler apprioximate the earth clock as in one interial frame and the satellight as in at least two and the resulting age difference is indentical to the twin paradox?
That's almost right.
Neither the satellite nor the earth clock are at rest in an inertial frame, but the center of earth is.
The earth clock of course is not in orbit, but moving quite slowly wrt earth's center (max. 0.5 km/s vs 8 km/s for orbital motion). Depending on the desired accuracy, you can ignore the motion and assume the earth clock to be moving inertially.
The satellite however has a relatively high and constantly changing velocity in the inertial frame. Every second, he is at rest in a different inertial frame. That's why the problem is similar to the twin paradox.

Always remember that this treatment depends on the simplifications I mentioned before.

#### Chris Hillman

Why not cite some useful resources?

I haven't the patience to try to follow this kind of thread, but in response to questions like those of the OP, I think that someone really should have taken the trouble to cite some readable expository papers which were written by experts precisely to answer the kind of questions which are troubling the OP: http://math.ucr.edu/home/baez/RelWWW/HTML/grad.html#gps [Broken]

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#### paw

If we ignore any possible atmospheric effects then the motion of both objects is determined only by the curvature of spacetime. That implies that both objects travel inertially and are not physically accelerating.
Thanks, I just realized I was confusing the classic definition of acceleration with the relativistic. Easy enough when it's not my field of expertise.

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