Differential geometry MOC

Raising and lowering of indices

Let 𝑀 be a Semi-Riemannian manifold. The metric 𝑔𝑎𝑏 on 𝑀 specifies an isomorphism between the space 𝔛(𝑀) of vector fields and the space Ω1(𝑀) of 1-forms by so-called raising and lowering of indices.

Note

We will work in abstract index notation, but the same process works once a local frame is chosen.

Given a vector field 𝑣𝑎 ∈𝔛(𝑀), we can define

𝑣𝑎:=𝑔𝑎𝑏𝑣𝑏∈Ω1(𝑀),

and similarly for a 1-form 𝜔 ∈Ω1(𝑀), we can define

𝜔𝑎:=𝑔𝑎𝑏𝜔𝑏.

This is consistent since by definition 𝑔𝑎𝑏 𝑔𝑏𝑐 =𝛿𝑎𝑐. Iterating this process, we can raise and lower arbitrary indices of any tensor field.

Musical notation

These isomorphism given here is sometimes called the musical isomorphism where we use the notation

(𝑣𝑎)♭𝑏=𝑣𝑏(𝜔𝑎)♯𝑏

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