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Help me to prove this please

  1. Nov 5, 2008 #1
    1- Aµναβ=Rµ[να]β ..... prove that the tensor is antisymmetric with respect to indices u and p
    2- Denote Sµναβ =Rµ(να)β... prove that the tensor is symmetric with respect to indices u and p
    3- if we introduce the spherical coordinate (r,θ,Φ )
    X1 =r sin θ cos Φ , X2 =r sin θ sin Φ , X3 =r cos θ
    if we denote t=X0 ,show that in these coordinate the asymptotic form of the metric in the linearzed approximation for a massive rotating approximation for a massive rotating source is
    ds2 = -(1- 2GM/r)dt2 – (4GJ sin2 θ/r)dt dΦ + (1+(2GM/r))(dr2+ r2 dω2)
    dω2 = dθ2 +sin2 θ dΦ2 is a metric on unit round sphere S2 .
     

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