Raising and lowering of indices

Let $𝑀$ be a Semi-Riemannian manifold. The metric $𝑔 𝑎 𝑏$ on $𝑀$ specifies an isomorphism between the space $𝔛 (𝑀)$ of vector fields and the space $Ω 1 (𝑀)$ 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 $𝑣 𝑎 \in 𝔛 (𝑀)$ , we can define

𝑣 𝑎 : = 𝑔 𝑎 𝑏 𝑣 𝑏 \in Ω 1 (𝑀),

and similarly for a 1-form $𝜔 \in Ω 1 (𝑀)$ , we can define

𝜔 𝑎 : = 𝑔 𝑎 𝑏 𝜔 𝑏 .

This is consistent since by definition $𝑔 𝑎 𝑏 𝑔 𝑏 𝑐 = 𝛿 𝑎 𝑐$ . 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

(𝑣 𝑎) ♭ 𝑏 = 𝑣 𝑏 (𝜔 𝑎) ♯ 𝑏

#state/tidy | #lang/en | #SemBr