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I Nearly Lorentz Coordinate Systems

  1. Nov 4, 2017 #1
    Hello! I am reading Schutz A first course in GR and he introduces the Nearly Lorentz coordinate systems as ones having a metric such that ##g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}##, with h a small deviation from the normal Minkowski metric. Then he introduces the Background Lorentz transformations (this is section 8.3 in the second edition) in which all the points are transformed as ##x^{\bar\alpha}=\Lambda_\beta^{\bar\alpha}x^\beta##. Applying this transformation to g he gets in the end that h transforms as ##h_{\bar\alpha\bar\beta}=\Lambda_\mu^{\bar\alpha}\Lambda_\nu^{\bar\beta} h_{\mu\nu}## and from here he says that we can treat h as if it was a tensor in SR and this simplify the calculations a lot. Can someone explain to me why ones need all these calculations for this? I am sure I am missing something but h is the difference between g and ##\eta## so isn't it a tensor, just because it is the difference between 2 tensors? Why do you need a proof for it? Moreover Schutz says that h "it is, of course, not a tensor, but just a piece of ##g_{\alpha\beta}##". So can someone explain to me why isn't h a tensor and why my logic is flawed? Thank you!
     
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  3. Nov 4, 2017 #2

    martinbn

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    Why is ##\eta_{\alpha\beta}## a tensor?
     
  4. Nov 4, 2017 #3
    Well a tensor (in this case a ##(0,2)## tensor) is a function that turns 2 vectors into a real number. ##\eta## is a 4x4 matrix so it behaves like a ##(0,2)## tensor, when applied to a 4D vector (which is our case).
     
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