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The GPS paradox

  1. Jul 3, 2007 #1
    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.
     
  2. jcsd
  3. Jul 3, 2007 #2
    an orbiting satellite isn't an inertial reference frame so I don't think one can apply SR
     
  4. Jul 3, 2007 #3
    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"
     
  5. Jul 3, 2007 #4

    olgranpappy

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    :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.
     
  6. Jul 3, 2007 #5

    russ_watters

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    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.
     
  7. Jul 3, 2007 #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:

    From the same source:

    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/

    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.
     
    Last edited: Jul 3, 2007
  8. Jul 3, 2007 #7
    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.
     
  9. Jul 6, 2007 #8
    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."
     
  10. Jul 6, 2007 #9

    russ_watters

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    And I asked you: frequency of what?
     
  11. Jul 6, 2007 #10
    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.
     
  12. Jul 6, 2007 #11
    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.

    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.
     
  13. Jul 6, 2007 #12

    russ_watters

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    Thank you very much.
    Ok, well, yes they are of course compared to each other. The ground station periodically resynchronizes the satellites' clocks.
     
  14. Jul 8, 2007 #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.).
     
  15. Jul 8, 2007 #14
    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.
     
  16. Jul 8, 2007 #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.
     
  17. Jul 8, 2007 #16
    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.
     
    Last edited: Jul 8, 2007
  18. Jul 9, 2007 #17
    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.
     
  19. Jul 10, 2007 #18
    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.

    A simple overview of the relevent the postulates from wiki:-

    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

     
    Last edited: Jul 10, 2007
  20. Jul 10, 2007 #19

    pervect

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    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.
     
  21. Jul 11, 2007 #20
    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.
     
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