The GPS paradox

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Main Question or Discussion Point

The satellites of GPS sysstem had been adjusted so they compensate for
GR and SR dilation of time due their velocity and less gravity.
This is what I found on Internet ( Not all agree ).
Is that true ?
But I´d like to know if the frequency of the ground stations that
control the satellites has been measured ( compared with its clocks )
inside the satellites.
I think this would be a real "twin paradox".
thanks in advance.
 

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  • #2
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an orbiting satellite isn't an inertial reference frame so I don't think one can apply SR
 
  • #3
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When they designed the GPS system they accounted for GR and SR. The GR effects were bigger than SR but both were calculated.
Forget GR = calculate the effects theoretically and substract them.
The question is "Which is the frequency measured by the satellites"
 
  • #4
olgranpappy
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an orbiting satellite isn't an inertial reference frame so I don't think one can apply SR
:rolleyes: that's like saying that a rotating bucket isn't an inertial reference frame so we can't apply newton's laws... planes and satelites need to take SR into account.
 
  • #5
russ_watters
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The question is "Which is the frequency measured by the satellites"
I'm not being coy here, but what frequency are you talking about? The frequency of the radio communications? Any doppler shift in that frequency is far too small to require a correction. The reason the clocks need the corrections is because the positional accuracy of the system requires nanoseconds-per-day accuracy.
 
  • #6
pervect
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Here at some references at various levels of sophistication:

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

Probably one of the more elementary treatments. I will give some brief quotes from the article to hopefully answer the original poster's questions:

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion.

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.
From the same source:

The engineers who designed the GPS system included these relativistic effects when they designed and deployed the system. For example, to counteract the General Relativistic effect once on orbit, they slowed down the ticking frequency of the atomic clocks before they were launched so that once they were in their proper orbit stations their clocks would appear to tick at the correct rate as compared to the reference atomic clocks at the GPS ground stations. Further, each GPS receiver has built into it a microcomputer that (among other things) performs the necessary relativistic calculations when determining the user's location.
I seem to recall reading that the GR corrections were switchable, and were initally switched off, but I don't have a reference for that offhand.

One wishes that this article had attributed the GR effects to the metric, rather than to curvature, but it's got the basic facts right.

Here are some more references: Neal Ashby's paper:
http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Broken]

Comments on Ashby's paper:
http://arxiv.org/abs/gr-qc/9508043

(this link is for the abstract: click on pdf to get the full paper)

Some comments by Misner about Ashby's paper. Misner and Ashby come to the same results, but Misner uses a somewhat more modern approach (less emphasis on coordinates and more emphasis on the metric as the fundamental foundation of GR).

Some (probably not all) of the past PF threads on this issue:

https://www.physicsforums.com/showthread.php?t=138481
https://www.physicsforums.com/showthread.php?t=87010

I'll close with a quick recap:

GR effects due to height make the satellite clock tick faster. This is the dominant effect. (You might look at the "Harvard tower" experiment for why higher clocks tick faster).

SR effects due to velocity make the satellite clock tick slower.

The GR effect dominates - the GPS satellite clocks tick faster than the ground clocks.
 
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  • #7
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First take a point of view of an observer on Earth (let's call him E), and suppose that GPS sattellite has a clock that is identical to the clock on Earth. From the point of view of E the clock on GPS satellite would go 7 us/day (us = microsecond) slower, because the satellite is moving with respect to E (= SR time dilation effect). In addition the GPS clock would appear as going 45 us/day faster, because the satellite is at a higher gravitational potential (= gravitational time dilation). So the net effect is that GPS clock goes 38 us/day faster than an identical clock on Earth. The operation of GPS would be impossible if GPS clock and Earth clock ticked at different rates. Therefore, GPS clocks are preset before the launch to tick 38 us/day faster. Then all involved clocks tick at the same rate, and the system can work.

Now let us take the point of view of an observer on one of the GPS satellites (let us call this observer S). Let us assume that all clocks (on Earths and on satellite) are ticking at their natural rate (no before-launch pre-setting). Observer S will conclude that a clock on Earth goes 7 us/day slower (because the clock on Earth is moving relative to the satellite). In addition, S will conclude that the clock on Earth goes 45 us/day slower than his clock, because Earth surface is at a lower gravitational potential. The net effect is that (from the point of view of S) all clocks on Earth are ticking at a rate 52 us/day slower than his own clock. So, in order to achieve synchronization, all clocks on Earths should be adjusted to run 52 us/day faster than their natural speed.

Now you see that there is no symmetry in the points of view of observers E and S. Who is correct? Both of them are correct in their own way. That's relativity. Whose prescription for clock synchronization should we accept? Certainly, we should use prescription of observer E (set GPS clocks to run 38 us/day slower than normal), because GPS is supposed to work for people on the Earth surface. Here on Earth we don't care whether or not astronauts think that clocks are synchronized.
 
  • #8
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meopemuk said:

"From the point of view of E the clock on GPS satellite would go 7 us/day (us = microsecond) slower, because the satellite is moving with respect to E (= SR time dilation effect). "

"Observer S will conclude that a clock on Earth goes 7 us/day slower (because the clock on Earth is moving relative to the satellite)."
I omitted the words where you refer to GR, because the twin paradox don't take it into account.

So both twins age at the same rate.

But I'm curious, and this is the goal of this thread, if this has been measured.

From my first post:

"But I´d like to know if the frequency of the ground stations that
control the satellites has been measured ( compared with its clocks )
inside the satellites."
 
  • #9
russ_watters
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And I asked you: frequency of what?
 
  • #10
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To russ:
your photos are very beautiful.
The frequency refers to the rate of the cessium clocks on earth, or the frequency of the modulating signal ( which must be a factor of the carrier frequency ) or any periodic signal ( = clock ) that has enough accuracy for testing SR.
 
  • #11
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meopemuk said:

"From the point of view of E the clock on GPS satellite would go 7 us/day (us = microsecond) slower, because the satellite is moving with respect to E (= SR time dilation effect). "

"Observer S will conclude that a clock on Earth goes 7 us/day slower (because the clock on Earth is moving relative to the satellite)."
I omitted the words where you refer to GR, because the twin paradox don't take it into account.

So both twins age at the same rate.
This is not exactly correct. The correct statement is that from the point of view of E, observer S ages slower. From the point of view of S, observer E ages slower. There is no such a thing as the "true" aging speed. You should always specify who is the observer.

Another question is what one would find after bringing both E and S together (e.g., by landing S on Earth) and comparing them side-by-side? The answer is that the observer who experienced acceleration (i.e., the one whose movement was non-inertial) will be younger.

But I'm curious, and this is the goal of this thread, if this has been measured.

From my first post:

"But I´d like to know if the frequency of the ground stations that
control the satellites has been measured ( compared with its clocks )
inside the satellites."
No, I haven't heard about measurements of the rate of Earth clocks performed from satellites. However, I am sure that if such measurements were done, they would be in agreement with what I wrote before. I.e., an astronaut will find that Earth clocks go 52 us/day slower than his own clock.
 
  • #12
russ_watters
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To russ:
your photos are very beautiful.
Thank you very much.
The frequency refers to the rate of the cessium clocks on earth, or the frequency of the modulating signal ( which must be a factor of the carrier frequency ) or any periodic signal ( = clock ) that has enough accuracy for testing SR.
Ok, well, yes they are of course compared to each other. The ground station periodically resynchronizes the satellites' clocks.
 
  • #13
pervect
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A few more quick comments:

The case of an actual satellite gets complicated because it involves gravity, which implies GR.

If one is just interested in understanding the twin paradox, it's much simpler to consider a rotating disk without gravity.

It's very clear in this case that a clock on the center of the disk is not accelerating, and has the longest proper time, while a clock on the outside of the disk is accelerating, and hence will be the "travelling twin", with a shorter proper time. (The "stay at home" twin, the one in the center of the disk, which does not accelerate, will have the longest proper time.).
 
  • #14
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Talking about twin paradox you can´t refer to any acceleration.

Nobody denies GR dilation of time under acceleration. Hewlett Packard
clock´s brochures state "any clock changes its rate under gravity or
acceleration". But they say nothing about relative displacement at
constant velocity.

I don´t understand why are you so fanatic about this topic.

There are other topics, older than this ( the Maxwell´s devil ) that
haven´t been explained yet.

In a rotating disk without gravity the center of the disk doesn´t feel
acceleration but the outside of the disk does.

In the GPS system, the satellites don´t feel acceleration but the ground
station does ( because acceleration = gravity, isn´t it ? ).
They are the same case.

I´m willing to learn from you, the experts.
 
  • #15
jtbell
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The modern viewpoint is that SR can handle accelerations in flat spacetime (such in pervect's rotating disk example). It's basically a matter of taking into account the varying relative velocity by applying integral calculus to the Lorentz transformation.

However, SR can't handle curved spacetime (i.e. gravitation) this way. For that you need GR.

To put it another way, it is not true that "gravitation = acceleration", in general. To describe gravitational accelerations, you need GR. To describe other kinds of acceleration, SR is sufficient.
 
  • #16
The modern viewpoint is that SR can handle accelerations in flat spacetime (such in pervect's rotating disk example). It's basically a matter of taking into account the varying relative velocity by applying integral calculus to the Lorentz transformation.

However, SR can't handle curved spacetime (i.e. gravitation) this way. For that you need GR.

To put it another way, it is not true that "gravitation = acceleration", in general. To describe gravitational accelerations, you need GR. To describe other kinds of acceleration, SR is sufficient.
Indeed you're right.

Sufficient but not perfect :wink:, thus the discrepancy between GR and QM. But we're not talking about anything close to relativistic speeds or any discrepancy, so we don't allow for it as such, classical Newtonian mechanics suffices in SR in the motion of satellites relatively.

This is something that proves that SR is correct so there is no need to get into a debate about GR. Satellites adequately prove the theory and we adequately adjust for it.
 
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  • #17
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Ok, I surrender.

I can´t argue against "modern viewpoint of SR", and I don´t know what
"QM" means.

I suppose that I must read 10 books and study 10 years before understanting
a very simple experiment.

Thanks to all for your time.
 
  • #18
Ok, I surrender.

I can´t argue against "modern viewpoint of SR", and I don´t know what
"QM" means.

I suppose that I must read 10 books and study 10 years before understanting
a very simple experiment.

Thanks to all for your time.
Or you could just spend half an hour or so looking at this site.

A neat set of lectures that explain it fairly simply. A little bit of maths knowledge might help but not much and not particularly complicated.

http://galileoandeinstein.physics.virginia.edu/lectures/spec_rel.html

QM=Quantum Mechanics.

We now come to Einstein's major insight: the Theory of Special Relativity. It is deceptively simple. Einstein first dusted off Galileo's discussion of experiments below decks on a uniformly moving ship, and restated it as :

The Laws of Physics are the same in all Inertial Frames.

Einstein then simply brought this up to date, by pointing out that the Laws of Physics must now include Maxwell's equations describing electric and magnetic fields as well as Newton's laws describing motion of masses under gravity and other forces.

Demanding that Maxwell's equations be satisfied in all inertial frames has one major consequence as far as we are concerned. As we stated above, Maxwell's equations give the speed of light to be 186,300 miles per second. Therefore, demanding that the laws of physics are the same in all inertial frames implies that the speed of any light wave, measured in any inertial frame, must be 186,300 miles per second.

This then is the entire content of the Theory of Special Relativity: the Laws of Physics are the same in any inertial frame, and, in particular, any measurement of the speed of light in any inertial frame will always give 186,300 miles per second.
You Really Can't Tell You're Moving!
A simple overview of the relevent the postulates from wiki:-

Einstein has said that all of the consequences of special relativity can be derived from examination of the Lorentz transformations.

These transformations, and hence special relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter-intuitive:

* Time dilation — the time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that his twin has aged much more).
* Relativity of simultaneity — two events happening in two different locations that occur simultaneously to one observer, may occur at different times to another observer (lack of absolute simultaneity).
* Composition of velocities — velocities (and speeds) do not simply 'add', for example if a rocket is moving at ⅔ the speed of light relative to an observer, and the rocket fires a missile at ⅔ of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. (In this example, the observer would see the missile travel with a speed of 12/13 the speed of light.)
General relativity deals more with the gravitational effects on an object, and is a bit more complicated but it is dealt with here:-

http://archive.ncsa.uiuc.edu/Cyberia/NumRel/GenRelativity.html

Gravitational Time Dilation
Einstein's Special Theory of Relativity predicted that time does not flow at a fixed rate: moving clocks appear to tick more slowly relative to their stationary counterparts. But this effect only becomes really significant at very high velocities that app roach the speed of light.

When "generalized" to include gravitation, the equations of relativity predict that gravity, or the curvature of spacetime by matter, not only stretches or shrinks distances (depending on their direction with respect to the gravitational field) but also will appear to slow down or "dilate" the flow of time.

In most circumstances in the universe, such time dilation is miniscule, but it can become very significant when spacetime is curved by a massive object such as a black hole.
 
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  • #19
pervect
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Ok, I surrender.

I can´t argue against "modern viewpoint of SR", and I don´t know what
"QM" means.

I suppose that I must read 10 books and study 10 years before understanting
a very simple experiment.

Thanks to all for your time.
If you spend your time studying the material, rather than to try and come up with "paradoxes" and argue against the theory, it doesn't take all that much time to learn SR.

(GR is a lot more complicated, but SR is fairly simple, if one has a good high school background in physics).

The biggest obstacle I've seen is that some people have pre-conceived ideas that they aren't willing to unlearn.
 
  • #20
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pervect wrote:

"If one is just interested in understanding the twin paradox, it's much simpler to consider a rotating disk without gravity"

Consider two rotating disks one clockwise and the other counterclockwise.

Each disk has a clock on the outside of the disk.

I´d like to know, because I´m not sure, if you can apply time dilation due to SR between the two clocks. ( Each half turn the clocks are approaching and
the other half they are moving away )

Time dilation due to GR must be the same, I think.

Of course, the two rotating disks are the orbits of the GPS satellites.
 
  • #21
pervect
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Let's consider three disks. One is stationary, one rotates clockwise, and one rotates clockwise.

On these three disks, we have three clocks.

Clock S is located on the stationary disk. Clock CW is located on the disk rotating clockwise, and clock CC is located on the disk rotating counter-clockwise.

Initially, the clocks S, CW, and CC are all located at the same point, and they all read the same time, t=0.

After one rotation of the disks, S, CW, and CC will all be located at the same location again.

At this time, S will have the largest reading, while CW and CC will both have identical readings which are the same, but lower than the reading on clock S.

This follows from a simple principle which always holds true in SR (and often holds true in GR, but not always - after we clear up the SR case we can talk about the GR case a bit more.). This is the principle that if different clocks take different paths through spacetime, starting out at the same point and re-uniting at some later point, the clock that did not accelerate during its journey will read the longest time.

Note that it is not in general possible to compare clocks in relativity when they are not at the same location in space without specifying the means of comparison.

Thus in our example, in order to compare S, CW and CC when they are not at the same place, we'd have to specify how the comparison was being made. I can get into that if it's necessary, but I'd rather avoid it to keep things simple.

When S, CW, and CC are all at the same location (which happens periodically in this example) we don't have to specify the means of comparison, and everyone will agree that S accumulates the most time, while CW and CC accumulate slightly less time because of their velocity.
 
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  • #22
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So, if one twin takes off from North Pole direction North and the other takes off from South Pole direction South.
They accelerate at the same rate, until they spend the first tank of fuel (say one day )
They continue their voyage whit the engines off ( say one year, mesured by their own clock ).
They switch on their engines to return to earth, two days, two more tanks of fuel.
They continue their voyage towards earth whit the engines off .
They land on earth after another year, one day, another tank of fuel.

Their clocks have the same reading ??
 
  • #23
jtbell
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If the two ships have the same "acceleration profiles" then yes, their clocks will show the same amount of elapsed time when they return to Earth.
 
  • #24
HallsofIvy
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So, if one twin takes off from North Pole direction North and the other takes off from South Pole direction South.
How can you go North from the North pole or South from the South Pole? Did you mean "North Pole direction South" and "South Pole Direction North"? That would be ambiguous since from the North Pole every direction is South and from the South pole every direction is North!

They accelerate at the same rate, until they spend the first tank of fuel (say one day )
They continue their voyage whit the engines off ( say one year, mesured by their own clock ).
They switch on their engines to return to earth, two days, two more tanks of fuel.
They continue their voyage towards earth whit the engines off .
They land on earth after another year, one day, another tank of fuel.

Their clocks have the same reading ??
 
  • #25
jtbell
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How can you go North from the North pole or South from the South Pole?
I interpreted it to mean, "straight away from the earth along a radial line, starting from the North Pole', and "straight away from the earth along a radial line, starting from the South Pole'"
 

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