I encountered some metric today defined by ds2=−(1−2GMr)dv2+(dvdr+drdv)+r2dΩ2
In all education I've done until now dvdr=drdv.
Why is this no longer the case? I suspect this has something to do with tensors, but I am not sure why.
Answer
More generally, a metric tensor g ∈ Γ(Sym2(T∗M))
is a section in the symmetric tensor product Sym2(T∗M) = T∗M⊙T∗Mover the cotangent bundle T∗M. In other words, g is a symmetric (0,2) covariant tensor field.In a coordinate chart U⊆M, it takes the form g|U = gμνdxμ⊙dxν,
with the manifest rule dxμ⊙dxν = dxν⊙dxμ,cf. OP's question.
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