Musicality

by Stéphane Haussler

Musical declarations

Traditionally, vectors are noted with a bold font as \(\mathbf{V}\), or with an arrow symbol \(\overrightarrow{V}\). These notations precludes the advent of tensor calculus and the discovery of the dual covectors, thus leaving no notation for these dual objects. The abstract index notation of Ricci calculus brings some improvement, with three-vectors expressed as \(v^i\), and the dual covectors expressed with lower indices as \(v_i\). With the musical flat \(♭\) and sharp \(♯\) symbols, covectors and vectors are explicitly declared. For example, a contravariant three-vector is noted with the sharp symbol \(♯\):

\[V^♯\]

Whereas the dual covariant three-covector is declared with the flat symbol \(♭\):

\[V^♭\]

Musical operations

Considering a vector in 3 dimensional Euclidean space expressed as a linear combination of the basis vectors \(∂_i\):

\[V^♯ = a \; ∂_x + b \; ∂_y + c \; ∂_z\]

The musical flat \(♭\) symbol is further utilized as an operator converting vectors to covectors. For example, the three-vector \(V^♯\) is flattend to a three-covector using the Euclidean metric \(δ\) with:

\[\begin{split}V^♭ &= (V^♯)^♭ \\ &= a \; ∂_x^♭ + b \; ∂_y^♭ + c \; ∂_z^♭ \\ &= a \; δ_{xi} \; dx^i + b \; δ_{yi} \; dx^i + c \; δ_{zi} \; dx^i \\ &= a \; δ_{xx} \; dx^x + b \; δ_{yy} \; dx^y + c \; δ_{zz} \; dx^z \\ &= a \; dx + b \; dy + c \; dz \\\end{split}\]

Likewise, the musical sharp \(♯\) is utilized as an operatro converting covectors to vecotrs. For example, the three-covector \(V^♭\) is sharpened to a three-vector using:

\[\begin{split}V^♯ &= (V^♭)^♯ \\ &= a \; dx^♯ + b \; dy^♯ + c \; dz^♯ \\ &= a \; δ_{xi} \; ∂_i + b \; δ_{yi} \; ∂_i + c \; δ_{zi} \; ∂_i \\ &= a \; δ_{xx} \; ∂_x + b \; δ_{yy} \; ∂_y + c \; δ_{zz} \; ∂_z \\ &= a \; ∂_x + b \; ∂_y + c \; ∂_z \\\end{split}\]

The notation can be further utilzed to raise or lower the indices of a tensor T of any rank. For example, a doubly contravariant tensor in Minkowski space \(T^{♯♯}\), also noted \(T^{μν}\) in abstract index notation is lowered using the metric component \(η_{μν}\) with:

\[\begin{split}T^{♭♯} &= (T^{♯♯})^{♭♯} \\ &= T^{μν} \; (∂_μ ⊗ ∂_ν)^{♭♯} \\ &= T^{μν} \; ∂_μ^♭ ⊗ ∂_ν^♯ \\ &= T^{μν} \; η_{μγ} \; dx^{γ} ⊗ ∂_ν\end{split}\]

Compared to Ricci calculus, musicality serves as a mean to clearly articulate the tensor basis alongside its components. Those familiar with abstract index notation may initially perceive the musical notation with explicit tensor bases as merely introducing additional symbols. However, the practical utility of this notation becomes evident when conducting real calculations with all elements of the Cartan-Hodge formalism.