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Stress-energy tensor explicitly in terms of the metric tensor

  1. Feb 26, 2014 #1
    I am trying to write the Einstein field equations
    $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$
    in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$ using the equations $$R=g^{\mu\nu}R_{\mu\nu}$$ (relating the scalar curvature to the trace of the Ricci curvature tensor) and $$R_{\mu\nu}=R^\lambda_{\mu\lambda\nu}$$ (relating the Ricci curvature tensor to the trace of the Riemann curvature tensor). Would anyone be willing to give recommendations on how to proceed, or already know the equation?
     
  2. jcsd
  3. Feb 26, 2014 #2

    bcrowell

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    Look up the standard expressions for the Riemann tensor in terms of the Christoffel symbols, and the Christoffel symbols in terms of the metric. It's going to be extremely messy. Consider using a computer algebra system for this; probably either Cadabra or Maxima could do it, and both are free and open source.
     
  4. Feb 27, 2014 #3
    Stress-Energy tensor deals with the enegy content of space. It's the Einstein tensor ##G_{\mu \nu} \equiv R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R## that you want to write in terms of the metric tensor. Anyway, as Ben said it's going to be extremely messy. See:
    http://en.wikipedia.org/wiki/Einstein_tensor#Explicit_form
     
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