Friday, 15 June 2018

General relativity: Why don't these two differentials commute?


I encountered some metric today defined by ds2=(12GMr)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






  1. More generally, a metric tensor g  Γ(Sym2(TM))

    is a section in the symmetric tensor product Sym2(TM) = TMTM
    over the cotangent bundle TM. In other words, g is a symmetric (0,2) covariant tensor field.




  2. In a coordinate chart UM, it takes the form g|U = gμνdxμdxν,

    with the manifest rule dxμdxν = dxνdxμ,
    cf. OP's question.




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