The Cartan-Hodge formalism#

by Stéphane Haussler

Table of Contents

This part introduces the Cartan-Hodge formalism, which bundles the notions of tensor calculus, musical operators, Hodge duality, exterior derivative, and the free matrix representation into a comprehensive framework of notation and mathematical tools. Practical computations follow clear and concise rules, with each step presented in a compact and comprehensible manner. The objective is also to depart from the abstract index notation of tensor calculus, eliminate the need for the Levi-Civita symbol, and expand tensor equations to their component from, all while adhering to standard and widespread notations.

This is not a comprehensive introduction into these subjects, but recalls basic definitions and properties necessary to perform calculations. I assume the reader possesses a solid understanding of vector and tensor calculus, as well as a working familiarity with Cartan’s differential forms and the exterior derivative.

I also here highlight and point out the article on Hodge duality can be read as a self-contained article and may be of interest to the reader on its own.

Basis vectors#

The Euclidean basis vectors are traditionally noted \(\mathbf{e}_x\), \(\mathbf{e}_y\), and \(\mathbf{e}_z\). It can be demonstrated that the basis vectors are the partial derivatives:

\[\mathbf{e}_i = ∂_i\]

Any vector can be expressed as a linear combination of the basis vectors as:

\[a \; ∂_x + b \; ∂_y + c \; ∂_z\]

The differential applied to the basis vectors results in the Kronecker \(δ\):

\[dx^i (∂_j) = δ^i_j\]

Therefore the dual basis covectors are:

\[\mathbf{e}^i = dx^i\]

And a covariant dual covector is expressed as linear combination of the basis covectors:

\[a \; dx + b \; dy + c \; dz\]

Musicality#

With the musical flat \(♭\) and sharp \(♯\) symbols, covectors and vectors are explicetely declared. For example, a contravariant three-vector is declared with the sharp operator \(♯\) as:

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

The musical flat \(♭\) and sharp \(♯\) symbols are further utilzed as operators converting vectors to covectors and vice versa. For example, a three-vector with euclidean metric \(δ\) is flattend to a three-covector with:

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

The free matrix representation#

With the free matrix representation, vectors can be ordered into arbitray matrix, while keeping the tensor basis is explicitly included. A vector can the be expressed explicitly as:

\[\begin{split}V^♯ = \begin{bmatrix} a \; ∂_x \\ b \; ∂_y \\ c \; ∂_z \\ \end{bmatrix}\end{split}\]

The representation can be freely modified to best facilitate calculations, with the brackets acting as an operator \(\begin{bmatrix}\end{bmatrix}\) adding elements together. As an example, the following row representation of a vector is also valid in the free matrix representation:

\[V^♯ = \begin{bmatrix} a \; ∂_x & b \; ∂_y & c \; ∂_z \end{bmatrix}\]

Hodge dual of k–forms in Minkowski space#

The Hodge duality assumes a central role, transitioning tensors between spaces and their dual complements.

1–forms

\[\begin{split}⋆ dt & = dx ∧ dy ∧ dz \\ ⋆ dx & = dt ∧ dy ∧ dz \\ ⋆ dy & = dt ∧ dz ∧ dx \\ ⋆ dz & = dt ∧ dx ∧ dy \\\end{split}\]

2–forms

\[\begin{split}\begin{alignedat}{2} ⋆ dt ∧ dx &= -& dy ∧ dz \\ ⋆ dt ∧ dy &= -& dz ∧ dx \\ ⋆ dt ∧ dz &= -& dx ∧ dy \\ ⋆ dy ∧ dz &= & dt ∧ dx \\ ⋆ dz ∧ dx &= & dt ∧ dy \\ ⋆ dx ∧ dy &= & dt ∧ dz \\ \end{alignedat}\end{split}\]

3–forms

\[\begin{split}⋆ dx ∧ dy ∧ dz &= dt \\ ⋆ dt ∧ dy ∧ dz &= dx \\ ⋆ dt ∧ dz ∧ dx &= dy \\ ⋆ dt ∧ dx ∧ dy &= dz \\\end{split}\]

4–forms

\[⋆ dt ∧ dx ∧ dy ∧ dz = - 1\]