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Torsion in GR and Einstein-Cartan theory

  1. Aug 1, 2011 #1
    I was reading something about the Cartan (vierbein) formalism in GR, in wich the connection is allowed to have torsion, and it got me interested in the Einstein-Cartan theory.
    Apparently both GR and Cartan theory with torsion should give the same experimental results in vacuum, what I would like to understand better from the differential geometry point of view is why exactly this happens.
    Why doesn't torsion make any difference in vacuum?
  2. jcsd
  3. Aug 1, 2011 #2

    Ben Niehoff

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    Mathematically, the answer is that in Einstein-Cartan theory, the equation of motion for the torsion tensor is purely algebraic (i.e., without derivatives). Hence the torsion does not propagate, and is nonzero only in the presence of matter with spin.

    In the vacuum, there is no spin, and so the torsion vanishes. So in that case the geometry is identical to the pure GR case. The torsion makes no difference because there is no torsion.
  4. Aug 2, 2011 #3
    Thanks, does this mean GR assumes matter has no spin? what would be the justification to eliminate that degree of freedom in GR?
  5. Aug 2, 2011 #4


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    Because spin is quantum mechanical in nature and GR is purely classical.
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