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Special Relativity and GPS

  1. Dec 12, 2007 #1

    p4h

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    So I'm doing a report on special relativity and GPS and kinda at a loss what to do:

    1. I've actually found several sites denoting the need to calibrate for relativistic effects when calculating locations, and I'm writing a report on just the opposite, that how is the system effected / how much difference does it make. So which should I believe? My Physics / Math teacher or for example, this guy: < link to crackpot site deleted by Doc Al >
    2. If I am to prove that it has an effect, what equation should I use to see the diffirence? Should I use:
    t^1 =γ ((t - v*x)/c^2 )
    x^1 =γ (x - v*t)
    where γ is Lorentz factor.

    Thanks in advance.

    PS: Sorry for any Typos, English aint my first language.
     
    Last edited by a moderator: Dec 12, 2007
  2. jcsd
  3. Dec 12, 2007 #2

    Doc Al

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    That particular site you linked is a crank site--don't waste your time. (And it's against our policy to allow linking to such sites, so I will delete it. Nothing personal!)
     
  4. Dec 12, 2007 #3

    p4h

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    No need, edited my post, as I got the answer to the question too.. Just saw his title and figured he might be right so.
     
  5. Dec 12, 2007 #4

    mgb_phys

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    There are two effects.
    Time runs quicker for moving satelites (due to special realtivity) by 7us / day.
    Time runs slower for the satelite in lower gravity ( general relativity) by 45us/day.

    So overall the effect is the opposite to what you would expect from just the motion.

    There was a posting in the forums which compared the two effects , I can't find it but I think it was by astronuc.
    edit. found this page giving the details http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
     
    Last edited: Dec 12, 2007
  6. Dec 12, 2007 #5

    p4h

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    I've read that too, but I need the actual math involved, as I can't just write 7 ms / day without any explanation as to why it's 7 ms / day and not 200
     
  7. Dec 12, 2007 #6

    p4h

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    Well I did a search on google and more than one site popped up, so I figured there might be an actual debate on the subject, in which case I'd be screwed. Which is why I also posted it here, to see whether it was just some crack site or not.
     
  8. Dec 12, 2007 #7
    You must learn to critically read between the lines for yourself.
    Laziness. Google will always pop up more than one page (don't panic), there's no point asking questions before you've actually read through a few of them (since until then your answers are all just going to be "google finds sites that answer your question").
     
  9. Dec 12, 2007 #8

    George Jones

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    In this post, I give a simple-minded (without proper justification) analysis of GPS that is based on Project A from the book Exploring Black Holes by Taylor and Wheeler. It calculates the effect all in one go, i.e., no split into special relativistic and general relativistic effects.

    Even though the post is simple-minded, it still might be a bit too technical for the intended project. To get some idea for the size of the accumulated error, convert to a distance by multiplying by the speed of light.
     
  10. Dec 12, 2007 #9

    mgb_phys

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    Back of the envelope calcs.
    GPS orbit is around 26,000km period is around 12hours.
    So speed V = 2 pi 26.6E6 m / 12 * 3600 s = 3900 m/s

    contraction = sqrt( 1 - v^2/c^2 ) = sqrt ( 1 - 1.6e-10 ) = 0.99999999991699995
    times 24hours = 7 microsecs
     
  11. Dec 12, 2007 #10

    p4h

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    Yeah it's a bit out of my league also. But thanks :)

    So two questions, as I want to understand it also, rather than just writing it down.

    1. V = 2 pi 26.6E6 m / 12 * 3600 s: Where do you get the 2 pi from? (Might be a newbie question)

    2. And from where do you derive the formula for contraction, as it seems to look alot like the formula for Lorentz factor?
     
  12. Dec 12, 2007 #11

    mgb_phys

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    26,000 km is the orbital radius, so the distance in one orbit is the circumference ie "2 pi r"
    Length / time / mass all have the same lorentz contraction formula, it's one of the clever things about special relativity.
     
  13. Dec 12, 2007 #12

    p4h

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    Cool thanks a bundle!

    Might revive this thread later this week, but we'll see..

    And thanks again :) Really appreciate it!
     
  14. Dec 12, 2007 #13
    You typed too fast and have this backwards; relative to the earth bound GPS user;
    1)Time runs slower for moving satellites (due to SR) by 7 us/day.
    2)Time runs fast for the satelite in lower gravity at altitude (GR) by 45us/day.

    Net of 1) & 2) is 38 micro seconds fast per day for the satellites over surface with GR lower density gravity due to distance, dominating SR speed effect.

    The link you provided:
    http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
    Has this correct.
     
  15. Dec 12, 2007 #14
    The argument in physicsmyths is that GPS works by simply comparing the lag between signals from different satellites, hence drift between surface clocks and geostationary satellite clocks will not translate to positioning error. So the fact that GPS is accurate to better than 10km is not in itself evidence for GR.

    However, I remember (or imagine) a detailed article describing the positioning error that would result if the GPS switched GR corrections off. Presumably this would result from the roughly-geosynchronous satellites not being exactly geostationary, and hence very slowly losing record of their own positions (from which to triangulate from), but while the measured clock drift aboard geostationary satellites may indeed have verified GR (as the crackpot failed to note) it nonetheless is not obvious that an accurately working GPS requires distinguishing between Newtonian gravity and GR.
     
    Last edited: Dec 12, 2007
  16. Dec 12, 2007 #15

    George Jones

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    I had before never heard of
    <crank site link deleted>

    but after poking around there for a few minutes, it seems to me that this an anti-relativity crank site.
     
    Last edited by a moderator: Dec 12, 2007
  17. Dec 12, 2007 #16

    Doc Al

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    It's a notorious anti-relativity crank site. That's why I deleted the link! :grumpy:
     
  18. Dec 12, 2007 #17
    How much of this thread did you read??

    I think your unfounded suggestion that GPS would fail by 10km/day (if GR weren't true) lends the crank more credibility than any hyper-link has.
     
  19. Dec 12, 2007 #18

    George Jones

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    If I gave the link that you deleted in the original thread, I apologize. I started reading the thread after the link had been deleted.

    I scanned the whole thread. Reading more slowing, it appears that one of your posts has been deleted, so I didn't read the post that p4h quoted in post#6. Since I scanned every post, I wasn't paying attention to quotes. Without seeing either your deleted post or the original link, I, erroneously, took your post #14 seems to cite a certain crank site as an authority.

    I have been under the impression that this is a way to get a rough handle on errors. For example, Neil Ashby, noted authority on GPS (the author of the livingreview on GPS) did the same thing in his presentation to the American Association of Physics Teachers. See slide 15.

    http://www.aapt-doorway.org/TGRUTalks/Ashby/ashbyGRworkshop.pdf
     
  20. Dec 12, 2007 #19
    Sure, that calculation gives a rough handle on something, but how relevant is it? You're implying it equals the magnitude of error which GPS would accumulate if GR corrections were switched off.

    Do you really think that implication is true (if so, please explain where the *crank website* argument fails), or has the crackpot correctly identified a mistake in the way we present evidence of GR to our students?

    --
    Kind of like how those that do not understand subtleties of the photoelectric effect may end up teaching non-truths about which phenomena demonstrates that light is composed of discreet photons. In denouncing crackpots we must be careful not to become them.
     
    Last edited by a moderator: Dec 13, 2007
  21. Dec 12, 2007 #20

    mgb_phys

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    Doh - I originally wrote the effects the other way round and forgot to swap the values!
     
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