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Question involving curvature tensor

  1. Apr 29, 2006 #1

    I'm working out some of the mathematical relations between Yang-Mill theory and GR. I'm having difficulty working out a somewhat trivial thing, I was wondering if anyone here could help me.

    To keep things concrete, I'll stick to the case D=4, but I'd like to be able to generalise to higher dimensional cases afterwards.

    Consider the Riemann curvature tensor. It has 20 independent components. The vacuĆ¼m Einstein equations amount to setting the Ricci tensor to zero, which imposes 10 constraints on the Riemann tensor. The additional 10 components are then described the the Weyl-tensor.

    Now imaging that we do not impose the Einstein equations. How many constraints would the equation

    [tex]D^{\mu} R_{\mu\nu\sigma\tau}=0[/tex] (1)

    impose, together with the second bianchi identity

    [tex]D_{\left[\rho\right.}R_{\left.\mu\nu\right]\sigma\tau}=&0[/tex] (B2)

    The reason I'm asking is because I've proven that (1)+(B2) implies Einstein equations. I'm wondering if the reverse if true, and if not, what additional constraints must be applied.

    Thanks in advance.
  2. jcsd
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