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Spacetime interval and the metrric

  1. Jan 13, 2015 #1
    This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other?

    Thanks!
     
  2. jcsd
  3. Jan 13, 2015 #2

    Matterwave

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    They aren't exactly equivalent, but they are closely related. The space time interval is ##ds^2=g_{ab}dx^a dx^b##. Because the metric ##g_{ab}## is a tensor, contracting it with (infinitesimal) vectors ##dx^{a}## will yield an invariant scalar ##ds^2##.
     
  4. Jan 13, 2015 #3
    Thanks! A little confused why you referred to ##ds^2## as a scalar. Isn't it a four-vector?
     
  5. Jan 13, 2015 #4

    bapowell

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    Nope. It's the scalar product of (differential) four-vectors.
     
  6. Jan 13, 2015 #5

    Matterwave

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    Why would it be a 4 vector? What are the 4 components that you are thinking of? It is a scalar because it's just 1 single number, which does not change under an arbitrary Lorentz transformation.
     
  7. Jan 13, 2015 #6
    Got it now. Thanks very much.
     
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