Rotations in differential form#

by Stéphane Haussler

Table of Contents

In the following pages, I express rotations in 3-dimensional Euclidean as well as in 4-dimensional Minkowski space, utilizing The Cartan-Hodge formalism. Anti-symmetries emerge from the Hodge star operator when arranging the differential forms into matrices, which exactly correspond to \(\mathfrak{so}(3)\) and \(\mathfrak{so}(1,3)\) matrices. Finally, we establish the identification of the electromagnetic field tensor.

Rotations are Linear Combinations of Bivectors

Any rotation in 3-dimensional Euclidean space is represented by a linear combination of the 3 independent planes of rotation. These are represented with 3 basis bivectors:

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

Any rotation in 4-dimensional Minkowksi space is represented by a linear combination of the 6 independent planes of rotation. These are represented with 6 basis bivectors:

\[\begin{split}R^{♯♯}= \begin{bmatrix} a \; ∂_t ∧ ∂_x \\ b \; ∂_t ∧ ∂_y \\ c \; ∂_t ∧ ∂_z \\ d \; ∂_y ∧ ∂_z \\ e \; ∂_z ∧ ∂_x \\ f \; ∂_x ∧ ∂_y \\ \end{bmatrix}\end{split}\]

Equivalently, rotations in spacetime can be represented by linear combinations of 6 basis bicovectors:

\[\begin{split}R^{♭♭} = \left[ \begin{aligned} - & a \; dt ∧ dx \\ - & b \; dt ∧ dy \\ - & c \; dt ∧ dz \\ & d \; dy ∧ dz \\ & e \; dz ∧ dx \\ & f \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

Matrice Representation of the Mixed Exterior Product

The row-major free matrix representation of any rotation in Minkowski space, expressed in a mixed form is:

\[\begin{split}R^{♭♯} = \frac{1}{2} \begin{bmatrix} & + a \; dt ∧ ∂_x & + b \; dt ∧ ∂_y & + c \; dt ∧ ∂_z \\ + a \; dx ∧ ∂_t & & + f \; dx ∧ ∂_y & - e \; dx ∧ ∂_z \\ + b \; dy ∧ ∂_t & - f \; dy ∧ ∂_x & & + d \; dy ∧ ∂_z \\ + c \; dz ∧ ∂_t & + e \; dz ∧ ∂_x & - d \; dz ∧ ∂_y & \\ \end{bmatrix}\end{split}\]

Readers well versed in the tensor formulations of electromagnetism will recognise the mixed form of the electromagnetic field tensor.

Symmetries of the Mixed Exterior Product in Minkowski Space

Expressing the mixed exterior product \(∧\) in term of tensor products \(⊗\), we demonstrate that the mixed exterior product is not fully antisymmetric in Minkowski space. However, the total number of symmetries is equal. We obtain:

Symmetry	Basis elements
Symetric	\(dt ∧ ∂_x = + dx ∧ ∂_t\)
Symetric	\(dt ∧ ∂_y = + dy ∧ ∂_t\)
Symetric	\(dt ∧ ∂_z = + dz ∧ ∂_t\)
Antisymetric	\(dy ∧ ∂_z = - dz ∧ ∂_y\)
Antisymetric	\(dz ∧ ∂_x = - dx ∧ ∂_z\)
Antisymetric	\(dx ∧ ∂_y = - dy ∧ ∂_x\)