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Pythagoras's Theorem ideal

  1. Sep 23, 2013 #1
    Pythagoras's Theorum is considered to be an ideal. The accuracy of its results are dependent upon the flatness of the surface to which it is being applied. Since there is no such thing as a perfectly flat surface, Pythagoras's Theorum results can only be seen as an approximation. Is this true in its application to Lorentz transformation?
     
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  3. Sep 23, 2013 #2

    mfb

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    The theorem is exact, you just have to keep its requirements in mind. Triangles in the real world are just approximations to perfect [strike]rectangular[/strike] right triangles, but that is not part of the theorem. Real triangles are not three ideal lines connected at three ideal points anyway.

    In special relativity, space is flat. The theorem can be applied by every observer, if the (idealized) triangle has a right angle for this observer (that is frame-dependent).
     
    Last edited: Sep 23, 2013
  4. Sep 23, 2013 #3
    In General relativity space time is curved. I'm confused. Surely the flat space of SR must also be an ideal.
     
  5. Sep 23, 2013 #4

    mfb

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    Sure, special relativity is general relativity without gravity. You asked about Lorentz transformations, they are mainly a tool of special relativity.
     
  6. Sep 23, 2013 #5

    SteamKing

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    I'm still working on what a 'rectangular triangle' is.
     
  7. Sep 23, 2013 #6

    Nugatory

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    It's the other way around - Pythagoras's theorem is exact and real triangles drawn on real surfaces in the real world are approximations of that ideal.

    This will be true of just about any application of mathematics to engineering and experimental science.
     
  8. Sep 23, 2013 #7
    Pythagoras's Theorum is exact and therefore can only give an approximate
    result to a real situation in an imperfect world. I think we're saying the same thing. I'm confused again.
     
  9. Sep 23, 2013 #8

    mfb

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    I'm still working on my second language.

    There is no "therefore". Nothing can give exact results in the real world, as there are no exact values or measurements.
     
  10. Sep 23, 2013 #9
    Would you please stop saying Theorum? It's really distracting. Why not say theorem?
     
  11. Sep 23, 2013 #10
    Can we trust Lorentz transformations as being accurate? Pardon me for asking, but what is your first language?
     
  12. Sep 23, 2013 #11
    My apologies, theorem.
     
  13. Sep 23, 2013 #12

    WannabeNewton

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    What's distracting are your irrelevant and pointless comments.
     
  14. Sep 23, 2013 #13

    Dale

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    This thread is drifting into philosophy. If you wish to discuss "perfection" here then you need to start with a scientific definition of perfection.
     
  15. Sep 23, 2013 #14
    I didn't make any comments. I asked Dappy for a favor and he was happy to oblige. Thank you Dappy. Now who is making pointless comments.
     
  16. Sep 23, 2013 #15
    Thankyou

    We, I assume, accept that Pythagoras's theorem is an ideal. Is that ideal realized in the real world when calculating Lorentz transformations for time, length and relativistic mass?
     
  17. Sep 23, 2013 #16

    Dale

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    Provided gravitational effects are not significant, then yes.
     
  18. Sep 23, 2013 #17

    mfb

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    If you take gravitational effects into account, no experiment has ever found a deviation from Lorentz transformations. And there are experiments testing that with a precision of better than one part in a billion.

    German.
     
  19. Sep 23, 2013 #18
    That's almost good enough for me. Can you imagine how flat a surface would have to be for Pythagoras's theorem to be accurate to better than one part in a billion. The mirror in the Hubble telescope comes to mind. I wonder how accurate it is in that context.

    I don't know how to double quote, but German, I would never of guest. I'm dyslexic and slip up a lot more than that.

    Reagards Dappy.
     
  20. Sep 23, 2013 #19

    mfb

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    Flatter than one part in a billion? :D
    To have length deviations of one part in a billion on a perfect sphere with the size of earth, the triangle needs an area of ~0.05 km^2.

    1 part in a billion is just a random number, the quality of the tests depends on the tested quantity. The stability of the speed of light in different directions can be tested way better (wikipedia gives 1 part in 1017).
    The next generation of ring lasers is expected to measure the rotation of earth more precise than satellites and telescopes. That's a 1 part in 1 billion measurement for an already tiny effect.

    More: Modern searches for Lorentz violation
     
    Last edited: Sep 23, 2013
  21. Sep 23, 2013 #20
    On a curved 2D surface, the Pythagorean Theorem cannot be accurately applied over large distances. However, over short distances, it can provide an excellent approximation. Thus, on the surface of the earth, it will not be accurate over large variations of latitude and longitude. However, over short distances like a few miles, it typically provides an excellent approximation. In space-time, the analog of the Pythagorean theorem is the Minkowski metric. Over large spans of space-time in a region of large mass-induced curvature, it is not possible to define a set of coordinates that accurately describes distances using the Minkowski metric. However, over small local regions, it is possible.
     
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