Is the Tensor Aµναβ Antisymmetric with Respect to Indices u and p?

[tri3phi]

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 .

 

[Jhero]

1. To prove that the tensor Aµναβ is antisymmetric with respect to indices u and p, we need to show that Aµναβ = -Aαβµν. We can rewrite the given equation as Aµναβ + Aαβµν = 0. This can be further simplified as Aµναβ - Aµναβ = 0, which proves that the tensor is indeed antisymmetric with respect to indices u and p.

2. To prove that the tensor Sµναβ is symmetric with respect to indices u and p, we need to show that Sµναβ = Sαβµν. Using the given equation, we can rewrite it as Sµναβ - Sαβµν = 0. This can be further simplified as Sµναβ + Sµναβ = 0, which proves that the tensor is symmetric with respect to indices u and p.

3. In spherical coordinates, the metric is given by ds2 = gµνdxµdxν, where xµ = (t, r, θ, Φ). We can write this metric in the linearized approximation as ds2 = -(1+2Φ)dt2 + (1-2Φ)dr2 + r2(dθ2 + sin2θdΦ2), where Φ is the gravitational potential. Plugging in the values for t and r in terms of the spherical coordinates given in the post, we get ds2 = -[1-2GM/r + 4GJsin2θ/(r2sin2θ)]dt2 + [1+2GM/r](dr2 + r2dθ2 + r2sin2θdΦ2). Rearranging this equation, we get the desired metric form with the asymptotic form of the metric for a massive rotating source. The term dω2 represents the metric on the unit round sphere S2, which is given by dω2 = dθ2 + sin2θdΦ2.

I hope this helps to clarify the proofs and the metric form in spherical coordinates. Let me know if you have any further questions.