2–forms (rotations)

by Stéphane Haussler

This article investigates the exterior derivative of rotations expressed in differential form. We employ the Cartan-Hodge formalism within the context of Minkowski spacetime. Following the systematic calculations presented in this article, I demonstrate in Of Maxwell Equations and Rotations that a twist in spacetime leads to the equations governing electromagnetism.

Rotations

Rotations in spacetime can occur in six independent planes. Any rotation \(R\) can be decomposed into a linear combination of basis rotations within each plane as:

\[\begin{split}R^{♯♯} = \begin{bmatrix} Q^x \; ∂_t ∧ ∂_x \\ Q^y \; ∂_t ∧ ∂_y \\ Q^z \; ∂_t ∧ ∂_z \\ R^x \; ∂_y ∧ ∂_z \\ R^y \; ∂_z ∧ ∂_x \\ R^z \; ∂_x ∧ ∂_y \\ \end{bmatrix}\end{split}\]

The representation of the rotation above is fully contravariant, given in terms of the exterior product \(∧\) of vectors corresponding to our physical understanding of space (and time). By fully flattening, we obtain the associated fully covariant differential 2–form representation:

\[\begin{split}R^{♭♭} = \left[ \begin{alignedat}{3} - & Q^x \; dt &\: ∧ &\: dx \\ - & Q^y \; dt &\: ∧ &\: dy \\ - & Q^z \; dt &\: ∧ &\: dz \\ & R^x \; dy &\: ∧ &\: dz \\ & R^y \; dz &\: ∧ &\: dx \\ & R^z \; dx &\: ∧ &\: dy \\ \end{alignedat} \right]\end{split}\]

Calculations

Flatten the rotation

\[R^{♭♭} = (R^{♯♯})^{♭♭}\]

Expand and distribute the flat operator

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; ∂_t^♭ ∧ ∂_x^♭ \\ Q^y \; ∂_t^♭ ∧ ∂_y^♭ \\ Q^z \; ∂_t^♭ ∧ ∂_z^♭ \\ R^x \; ∂_y^♭ ∧ ∂_z^♭ \\ R^y \; ∂_z^♭ ∧ ∂_x^♭ \\ R^z \; ∂_x^♭ ∧ ∂_y^♭ \\ \end{bmatrix}\end{split}\]

Expand with the Minkowski metric

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; η_{αt} \; dx^α ∧ η_{βx} \; dx^β \\ Q^y \; η_{αt} \; dx^α ∧ η_{βy} \; dx^β \\ Q^z \; η_{αt} \; dx^α ∧ η_{βz} \; dx^β \\ R^x \; η_{αy} \; dx^α ∧ η_{βz} \; dx^β \\ R^y \; η_{αz} \; dx^α ∧ η_{βx} \; dx^β \\ R^z \; η_{αx} \; dx^α ∧ η_{βy} \; dx^β \\ \end{bmatrix}\end{split}\]

The exterior product \(∧\) is bilinear. The Minkowski metric components \(η\)’s can be taken in front of the expression:

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; η_{αt} η_{βx} \; dx^α ∧ dx^β \\ Q^y \; η_{αt} η_{βy} \; dx^α ∧ dx^β \\ Q^z \; η_{αt} η_{βz} \; dx^α ∧ dx^β \\ R^x \; η_{αy} η_{βz} \; dx^α ∧ dx^β \\ R^y \; η_{αz} η_{βx} \; dx^α ∧ dx^β \\ R^z \; η_{αx} η_{βy} \; dx^α ∧ dx^β \\ \end{bmatrix}\end{split}\]

For readability, replace the \(dx^μ\) symbols by their explicit expressions:

\[\begin{split}dx^t &= dt \\ dx^x &= dx \\ dx^y &= dy \\ dx^z &= dz\end{split}\]

Identify the non-zero components of the Minkowski metric

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; η_{tt} η_{xx} \; dt ∧ dx \\ Q^y \; η_{tt} η_{yy} \; dt ∧ dy \\ Q^z \; η_{tt} η_{zz} \; dt ∧ dz \\ R^x \; η_{yy} η_{zz} \; dy ∧ dz \\ R^y \; η_{zz} η_{xx} \; dz ∧ dx \\ R^z \; η_{xx} η_{yy} \; dx ∧ dy \\ \end{bmatrix}\end{split}\]

Apply the numerical values of the Minkowski metric components

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; (+1) (-1) \; dt ∧ dx \\ Q^y \; (+1) (-1) \; dt ∧ dy \\ Q^z \; (+1) (-1) \; dt ∧ dz \\ R^x \; (-1) (-1) \; dy ∧ dz \\ R^y \; (-1) (-1) \; dz ∧ dx \\ R^z \; (-1) (-1) \; dx ∧ dy \\ \end{bmatrix}\end{split}\]

Conclude

\[\begin{split}R^{♭♭} = \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

\(⋆R^{♭♭}\)

Applying the Hodge star to the rotation 2-form, we obtain:

\[\begin{split}⋆ R^{♭♭} = \left[ \begin{aligned} & Q^x \; dy ∧ dz \\ & Q^y \; dz ∧ dx \\ & Q^z \; dx ∧ dy \\ & R^x \; dt ∧ dx \\ & R^y \; dt ∧ dy \\ & R^z \; dt ∧ dz \\ \end{aligned} \right]\end{split}\]

Calculations

Apply the Hodge star by linearity

\[\begin{split}⋆ R^{♭♭} = ⋆ \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right] = \left[ \begin{aligned} - & Q^x \; ⋆ dt ∧ dx \\ - & Q^y \; ⋆ dt ∧ dy \\ - & Q^z \; ⋆ dt ∧ dz \\ & R^x \; ⋆ dy ∧ dz \\ & R^y \; ⋆ dz ∧ dx \\ & R^z \; ⋆ dx ∧ dy \\ \end{aligned} \right]\end{split}\]

Apply the Hodge star to the basis 2-Forms

Using the tables for the Hodge dual:

\[\begin{split}⋆ R^{♭♭} = \left[ \begin{aligned} & Q^x \; dy ∧ dz \\ & Q^y \; dz ∧ dx \\ & Q^z \; dx ∧ dy \\ & R^x \; dt ∧ dx \\ & R^y \; dt ∧ dy \\ & R^z \; dt ∧ dz \\ \end{aligned} \right]\end{split}\]

\(dR^{♭♭}\)

Applying the exterior derivative to the rotation 2-form, we obtain:

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Calculations

Distribute the exterior derivative

\[\begin{split}dR^{♭♭} = \begin{bmatrix} d( - Q^x \; dt ∧ dx ) \\ d( - Q^y \; dt ∧ dy ) \\ d( - Q^z \; dt ∧ dz ) \\ d( + R^x \; dy ∧ dz ) \\ d( + R^y \; dz ∧ dx ) \\ d( + R^z \; dx ∧ dy ) \\ \end{bmatrix}\end{split}\]

Apply the exterior derivative

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{3} ∂_y (- Q^x ) \; & dy ∧ dt ∧ dx & + & ∂_z (- Q^x ) \; & dz ∧ dt ∧ dx \\ ∂_x (- Q^y ) \; & dx ∧ dt ∧ dy & + & ∂_z (- Q^y ) \; & dz ∧ dt ∧ dy \\ ∂_x (- Q^z ) \; & dx ∧ dt ∧ dz & + & ∂_y (- Q^z ) \; & dy ∧ dt ∧ dz \\ ∂_t (+ R^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ R^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^y ) \; & dy ∧ dz ∧ dx \\ ∂_t (+ R^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^z ) \; & dz ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Reorder the exterior products

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{3} ∂_y (- Q^x )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (- Q^x )(-1) \; & dt ∧ dz ∧ dx \\ ∂_x (- Q^y )(-1) \; & dt ∧ dx ∧ dy & + & ∂_z (- Q^y )(+1) \; & dt ∧ dy ∧ dz \\ ∂_x (- Q^z )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (- Q^z )(-1) \; & dt ∧ dy ∧ dz \\ ∂_t (+ R^x )(+1) \; & dt ∧ dy ∧ dz & + & ∂_x (+ R^x )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^y )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^y )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^z )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^z )(+1) \; & dx ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Simplify

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{3} ∂_y (- Q^x ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^x ) \; & dt ∧ dz ∧ dx \\ ∂_x (+ Q^y ) \; & dt ∧ dx ∧ dy & + & ∂_z (- Q^y ) \; & dt ∧ dy ∧ dz \\ ∂_x (- Q^z ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^z ) \; & dt ∧ dy ∧ dz \\ ∂_t (+ R^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ R^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^y ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^z ) \; & dx ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

\(d⋆R^{♭♭}\)

Applying in sequence the exterior derivative and the Hodge star operator to the rotation 2-form, we obtain:

\[\begin{split}d⋆R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Calculations

Take the exterior derivative

\[\begin{split}d⋆R^{♭♭} = d \begin{bmatrix} Q^x \; dy ∧ dz \\ Q^y \; dz ∧ dx \\ Q^z \; dx ∧ dy \\ R^x \; dt ∧ dx \\ R^y \; dt ∧ dy \\ R^z \; dt ∧ dz \\ \end{bmatrix}\end{split}\]

Distribute the exterior derivative

\[\begin{split}d⋆R^{♭♭} = \begin{bmatrix} d( Q^x \; dy ∧ dz) \\ d( Q^y \; dz ∧ dx) \\ d( Q^z \; dx ∧ dy) \\ d( R^x \; dt ∧ dx) \\ d( R^y \; dt ∧ dy) \\ d( R^z \; dt ∧ dz) \\ \end{bmatrix}\end{split}\]

Apply

\[\begin{split}d⋆R^{♭♭})= \left[ \begin{alignedat}{5} ∂_t (+ Q^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ Q^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^y ) \; & dy ∧ dz ∧ dx \\ ∂_t (+ Q^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^z ) \; & dz ∧ dx ∧ dy \\ ∂_y (+ R^x ) \; & dy ∧ dt ∧ dx & + & ∂_z (+ R^x ) \; & dz ∧ dt ∧ dx \\ ∂_x (+ R^y ) \; & dx ∧ dt ∧ dy & + & ∂_z (+ R^y ) \; & dz ∧ dt ∧ dy \\ ∂_x (+ R^z ) \; & dx ∧ dt ∧ dz & + & ∂_y (+ R^z ) \; & dy ∧ dt ∧ dz \\ \end{alignedat} \right]\end{split}\]

Reorder

\[\begin{split}d⋆R^{♭♭} = \left[ \begin{alignedat}{5} ∂_t (+ Q^x )(+1) \; & dt ∧ dy ∧ dz & + & ∂_x (+ Q^x )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^y )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^y )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^z )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^z )(+1) \; & dx ∧ dy ∧ dz \\ ∂_y (+ R^x )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^x )(-1) \; & dt ∧ dz ∧ dx \\ ∂_x (+ R^y )(-1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^y )(+1) \; & dt ∧ dy ∧ dz \\ ∂_x (+ R^z )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^z )(-1) \; & dt ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Apply values

\[\begin{split}d(⋆R^{♭♭}) = \left[ \begin{alignedat}{5} ∂_t (+ Q^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ Q^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^y ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^z ) \; & dx ∧ dy ∧ dz \\ ∂_y (+ R^x ) \; & dt ∧ dx ∧ dy & + & ∂_z (- R^x ) \; & dt ∧ dz ∧ dx \\ ∂_x (- R^y ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^y ) \; & dt ∧ dy ∧ dz \\ ∂_x (+ R^z ) \; & dt ∧ dz ∧ dx & + & ∂_y (- R^z ) \; & dt ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d ⋆ R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z & \: ) \; & dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y & \: ) \; & dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x & \: ) \; & dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & & \: ) \; & dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

\(⋆dR^{♭♭}\)

Applying in sequence the Hodge star and the exterior derivative operator \(d\) to the rotation 2-form, we obtain:

\[\begin{split}⋆ dR^{♭♭} = \left[ \begin{alignedat}{5} (& & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z &\:) \; dt \\ (& - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y &\:) \; dx \\ (& - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x &\:) \; dy \\ (& - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

Calculations

Apply the Hodge star

Apply the Hodge star to \(dR^{♭♭}\):

\[\begin{split}⋆dR^{♭♭} = ⋆\left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Distribute the Hodge star

\[\begin{split}⋆dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; ⋆ dx^x ∧ dx^y ∧ dx^z \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; ⋆ dx^t ∧ dx^y ∧ dx^z \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; ⋆ dx^t ∧ dx^z ∧ dx^x \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; ⋆ dx^t ∧ dx^x ∧ dx^y \\ \end{alignedat} \right]\end{split}\]

Apply the Hodge star to the basis 1-forms

Using the tables for the Hodge dual:

\[\begin{split}⋆dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; (-dt) \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; (-dx) \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; (-dy) \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; (-dz) \\ \end{alignedat} \right]\end{split}\]

Conclude

\[\begin{split}⋆dR^{♭♭} = \left[ \begin{alignedat}{5} (& & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z &\:) \; dt \\ (& - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y &\:) \; dx \\ (& - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x &\:) \; dy \\ (& - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

\(⋆d⋆R^{♭♭}\)

Applying the Hodge star to \(d⋆R^{♭♭}\), we obtain:

\[\begin{split}⋆d⋆R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dt \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dx \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dy \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

Calculations

Apply the Hodge star

Apply the Hodge star to \(d⋆R^{♭♭}\):

\[\begin{split}⋆ d⋆R^{♭♭} = ⋆ \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Distribute the Hodge star by linearity

\[\begin{split}⋆ d⋆R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; ⋆ dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; ⋆ dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; ⋆ dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; ⋆ dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Apply the Hodge star to the basis 3-forms

Using the tables for the Hodge dual:

\[\begin{split}⋆ d⋆R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dt \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dx \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dy \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

\(d⋆dR^{♭♭}\)

Applying the exterior derivative to \(⋆d R^{♭♭}\), we obtain:

\[\begin{split}d⋆d R^{♭♭} &= \left[ \begin{alignedat}{4} & ( - ∂_t^2 R^x & + ∂_x^2 R^x & & & ) \; dt ∧ dx \\ & ( - ∂_t^2 R^y & & + ∂_y^2 R^y & & ) \; dt ∧ dy \\ & ( - ∂_t^2 R^z & & & + ∂_z^2 R^z & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) \; dy ∧ dz \\ & (+ ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) \; dz ∧ dx \\ & (+ ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y Q^z & + ∂_t ∂_z Q^y & ) \; dt ∧ dx \\ & ( + ∂_t ∂_x Q^z & & - ∂_t ∂_z Q^x & ) \; dt ∧ dy \\ & ( - ∂_t ∂_x Q^y & + ∂_t ∂_y Q^x & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_z ∂_x Q^y & - ∂_x ∂_y Q^y & ) \; dy ∧ dz \\ & ( - ∂_y ∂_z Q^z & & - ∂_x ∂_y Q^x & ) \; dz ∧ dx \\ & ( - ∂_y ∂_z Q^y & - ∂_z ∂_x Q^x & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_x ∂_y R^y & + ∂_x ∂_z R^z & ) \; dt ∧ dx \\ & ( + ∂_y ∂_x R^x & & + ∂_y ∂_z R^z & ) \; dt ∧ dy \\ & ( + ∂_z ∂_x R^x & + ∂_z ∂_y R^y & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dy ∧ dz \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dz ∧ dx \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm]\end{split}\]

Calculations

Apply the exterior derivative

\[\begin{split}d⋆dR^{♭♭} = d \left[ \begin{alignedat}{5} (& & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z &\:) \; dt \\ (& - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y &\:) \; dx \\ (& - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x &\:) \; dy \\ (& - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

Apply the exterior derivative

\[\begin{split}d⋆dR^{♭♭} = d \left[ \begin{alignedat}{5} &( & - & ∂_x ∂_x R^x & - & ∂_x ∂_y R^y & - & ∂_x ∂_z R^z & ) & \; dx ∧ dt \\ &( & - & ∂_y ∂_x R^x & - & ∂_y ∂_y R^y & - & ∂_y ∂_z R^z & ) & \; dy ∧ dt \\ &( & - & ∂_z ∂_x R^x & - & ∂_z ∂_y R^y & - & ∂_z ∂_z R^z & ) & \; dz ∧ dt \\[2mm] &( & - & ∂_t ∂_t R^x & - & ∂_t ∂_y Q^z & + & ∂_t ∂_z Q^y & ) & \; dt ∧ dx \\ &( & - & ∂_y ∂_t R^x & - & ∂_y ∂_y Q^z & + & ∂_y ∂_z Q^y & ) & \; dy ∧ dx \\ &( & - & ∂_z ∂_t R^x & - & ∂_z ∂_y Q^z & + & ∂_z ∂_z Q^y & ) & \; dz ∧ dx \\[2mm] &( & - & ∂_t ∂_t R^y & + & ∂_t ∂_x Q^z & - & ∂_t ∂_z Q^x & ) & \; dt ∧ dy \\ &( & - & ∂_x ∂_t R^y & + & ∂_x ∂_x Q^z & - & ∂_x ∂_z Q^x & ) & \; dx ∧ dy \\ &( & - & ∂_z ∂_t R^y & + & ∂_z ∂_x Q^z & - & ∂_z ∂_z Q^x & ) & \; dz ∧ dy \\[2mm] &( & - & ∂_t ∂_t R^z & - & ∂_t ∂_x Q^y & + & ∂_t ∂_y Q^x & ) & \; dt ∧ dz \\ &( & - & ∂_x ∂_t R^z & - & ∂_x ∂_x Q^y & + & ∂_x ∂_y Q^x & ) & \; dx ∧ dz \\ &( & - & ∂_y ∂_t R^z & - & ∂_y ∂_x Q^y & + & ∂_y ∂_y Q^x & ) & \; dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d⋆dR^{♭♭} = d \left[ \begin{alignedat}{5} &( & - & ∂_x ∂_x R^x & - & ∂_x ∂_y R^y & - & ∂_x ∂_z R^z & ) & \; dx ∧ dt \\ &( & - & ∂_y ∂_x R^x & - & ∂_y ∂_y R^y & - & ∂_y ∂_z R^z & ) & \; dy ∧ dt \\ &( & - & ∂_z ∂_x R^x & - & ∂_z ∂_y R^y & - & ∂_z ∂_z R^z & ) & \; dz ∧ dt \\[2mm] &( & - & ∂_t ∂_t R^x & - & ∂_t ∂_y Q^z & + & ∂_t ∂_z Q^y & ) & \; dt ∧ dx \\ &( & - & ∂_t ∂_t R^y & + & ∂_t ∂_x Q^z & - & ∂_t ∂_z Q^x & ) & \; dt ∧ dy \\ &( & - & ∂_t ∂_t R^z & - & ∂_t ∂_x Q^y & + & ∂_t ∂_y Q^x & ) & \; dt ∧ dz \\[2mm] &( & - & ∂_z ∂_t R^y & + & ∂_z ∂_x Q^z & - & ∂_z ∂_z Q^x & ) & \; dz ∧ dy \\ &( & - & ∂_y ∂_t R^z & - & ∂_y ∂_x Q^y & + & ∂_y ∂_y Q^x & ) & \; dy ∧ dz \\[2mm] &( & - & ∂_z ∂_t R^x & - & ∂_z ∂_y Q^z & + & ∂_z ∂_z Q^y & ) & \; dz ∧ dx \\ &( & - & ∂_x ∂_t R^z & - & ∂_x ∂_x Q^y & + & ∂_x ∂_y Q^x & ) & \; dx ∧ dz \\[2mm] &( & - & ∂_y ∂_t R^x & - & ∂_y ∂_y Q^z & + & ∂_y ∂_z Q^y & ) & \; dy ∧ dx \\ &( & - & ∂_x ∂_t R^y & + & ∂_x ∂_x Q^z & - & ∂_x ∂_z Q^x & ) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Reorder the exterior products

\[\begin{split}d⋆dR^{♭♭} = d \left[ \begin{alignedat}{5} &( & + & ∂_x ∂_x R^x & + & ∂_x ∂_y R^y & + & ∂_x ∂_z R^z & ) & \; dt ∧ dx \\ &( & + & ∂_y ∂_x R^x & + & ∂_y ∂_y R^y & + & ∂_y ∂_z R^z & ) & \; dt ∧ dy \\ &( & + & ∂_z ∂_x R^x & + & ∂_z ∂_y R^y & + & ∂_z ∂_z R^z & ) & \; dt ∧ dz \\[2mm] &( & - & ∂_t ∂_t R^x & - & ∂_t ∂_y Q^z & + & ∂_t ∂_z Q^y & ) & \; dt ∧ dx \\ &( & - & ∂_t ∂_t R^y & + & ∂_t ∂_x Q^z & - & ∂_t ∂_z Q^x & ) & \; dt ∧ dy \\ &( & - & ∂_t ∂_t R^z & - & ∂_t ∂_x Q^y & + & ∂_t ∂_y Q^x & ) & \; dt ∧ dz \\[2mm] &( & + & ∂_z ∂_t R^y & - & ∂_z ∂_x Q^z & + & ∂_z ∂_z Q^x & ) & \; dy ∧ dz \\ &( & - & ∂_y ∂_t R^z & - & ∂_y ∂_x Q^y & + & ∂_y ∂_y Q^x & ) & \; dy ∧ dz \\[2mm] &( & - & ∂_z ∂_t R^x & - & ∂_z ∂_y Q^z & + & ∂_z ∂_z Q^y & ) & \; dz ∧ dx \\ &( & + & ∂_x ∂_t R^z & + & ∂_x ∂_x Q^y & - & ∂_x ∂_y Q^x & ) & \; dz ∧ dx \\[2mm] &( & + & ∂_y ∂_t R^x & + & ∂_y ∂_y Q^z & - & ∂_y ∂_z Q^y & ) & \; dx ∧ dy \\ &( & - & ∂_x ∂_t R^y & + & ∂_x ∂_x Q^z & - & ∂_x ∂_z Q^x & ) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Rearange

Here we ar looking for terms that belong together.

\[\begin{split}d⋆dR^{♭♭} = d \left[ \begin{alignedat}{5} &( & + & ∂_x ∂_x R^x & + & ∂_x ∂_y R^y & + & ∂_x ∂_z R^z & ) & \; dt ∧ dx \\ &( & + & ∂_y ∂_x R^x & + & ∂_y ∂_y R^y & + & ∂_y ∂_z R^z & ) & \; dt ∧ dy \\ &( & + & ∂_z ∂_x R^x & + & ∂_z ∂_y R^y & + & ∂_z ∂_z R^z & ) & \; dt ∧ dz \\[2mm] &( & - & ∂_t ∂_t R^x & & & & & ) & \; dt ∧ dx \\ &( & - & ∂_t ∂_t R^y & & & & & ) & \; dt ∧ dy \\ &( & - & ∂_t ∂_t R^z & & & & & ) & \; dt ∧ dz \\[2mm] &( & & & - & ∂_t ∂_y Q^z & + & ∂_t ∂_z Q^y & ) & \; dt ∧ dx \\ &( & & & + & ∂_t ∂_x Q^z & - & ∂_t ∂_z Q^x & ) & \; dt ∧ dy \\ &( & & & - & ∂_t ∂_x Q^y & + & ∂_t ∂_y Q^x & ) & \; dt ∧ dz \\[2mm] &( & + & ∂_z ∂_t R^y & & & & & ) & \; dy ∧ dz \\ &( & - & ∂_y ∂_t R^z & & & & & ) & \; dy ∧ dz \\ &( & - & ∂_z ∂_t R^x & & & & & ) & \; dz ∧ dx \\ &( & + & ∂_x ∂_t R^z & & & & & ) & \; dz ∧ dx \\ &( & + & ∂_y ∂_t R^x & & & & & ) & \; dx ∧ dy \\ &( & - & ∂_x ∂_t R^y & & & & & ) & \; dx ∧ dy \\[2mm] &( & & & & & + & ∂_z ∂_z Q^x & ) & \; dy ∧ dz \\ &( & & & & & + & ∂_y ∂_y Q^x & ) & \; dy ∧ dz \\ &( & & & & & + & ∂_z ∂_z Q^y & ) & \; dz ∧ dx \\ &( & & & + & ∂_x ∂_x Q^y & & & ) & \; dz ∧ dx \\ &( & & & + & ∂_y ∂_y Q^z & & & ) & \; dx ∧ dy \\ &( & & & + & ∂_x ∂_x Q^z & & & ) & \; dx ∧ dy \\[2mm] &( & & & - & ∂_z ∂_x Q^z & & & ) & \; dy ∧ dz \\ &( & & & - & ∂_y ∂_x Q^y & & & ) & \; dy ∧ dz \\ &( & & & - & ∂_z ∂_y Q^z & & & ) & \; dz ∧ dx \\ &( & & & & & - & ∂_x ∂_y Q^x & ) & \; dz ∧ dx \\ &( & & & & & - & ∂_y ∂_z Q^y & ) & \; dx ∧ dy \\ &( & & & & & - & ∂_x ∂_z Q^x & ) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Reorder

\[\begin{split}d⋆dR^{♭♭} &= \left[ \begin{alignedat}{4} ( & ∂_x ∂_x & \; R^x & \, + \, & ∂_x ∂_y & \; R^y & \, + \, & ∂_x ∂_z & \; R^z & \; ) & \; dt ∧ dx \\ ( & ∂_x ∂_y & \; R^x & \, + \, & ∂_y ∂_y & \; R^y & \, + \, & ∂_y ∂_z & \; R^z & \; ) & \; dt ∧ dy \\ ( & ∂_x ∂_z & \; R^x & \, + \, & ∂_y ∂_z & \; R^y & \, + \, & ∂_z ∂_z & \; R^z & \; ) & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & - ∂_t^2 R^x & \; dt ∧ dx \\ & - ∂_t^2 R^y & \; dt ∧ dy \\ & - ∂_t^2 R^z & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y Q^z & + ∂_t ∂_z Q^y & ) & \; dt ∧ dx \\ & ( + ∂_t ∂_x Q^z & & - ∂_t ∂_z Q^x & ) & \; dt ∧ dy \\ & ( - ∂_t ∂_x Q^y & + ∂_t ∂_y Q^x & & ) & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) & \; dy ∧ dz \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) & \; dz ∧ dx \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) & \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) & \; dy ∧ dz \\ & (+ ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) & \; dz ∧ dx \\ & (+ ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) & \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_z ∂_x Q^z & - ∂_x ∂_y Q^y & ) & \; dy ∧ dz \\ & ( - ∂_y ∂_z Q^z & & - ∂_x ∂_y Q^x & ) & \; dz ∧ dx \\ & ( - ∂_y ∂_z Q^y & - ∂_z ∂_x Q^x & & ) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d⋆d R^{♭♭} &= \left[ \begin{alignedat}{4} & ( - ∂_t^2 R^x & + ∂_x^2 R^x & & & ) \; dt ∧ dx \\ & ( - ∂_t^2 R^y & & + ∂_y^2 R^y & & ) \; dt ∧ dy \\ & ( - ∂_t^2 R^z & & & + ∂_z^2 R^z & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) \; dy ∧ dz \\ & (+ ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) \; dz ∧ dx \\ & (+ ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y Q^z & + ∂_t ∂_z Q^y & ) \; dt ∧ dx \\ & ( + ∂_t ∂_x Q^z & & - ∂_t ∂_z Q^x & ) \; dt ∧ dy \\ & ( - ∂_t ∂_x Q^y & + ∂_t ∂_y Q^x & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_z ∂_x Q^y & - ∂_x ∂_y Q^y & ) \; dy ∧ dz \\ & ( - ∂_y ∂_z Q^z & & - ∂_x ∂_y Q^x & ) \; dz ∧ dx \\ & ( - ∂_y ∂_z Q^y & - ∂_z ∂_x Q^x & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_x ∂_y R^y & + ∂_x ∂_z R^z & ) \; dt ∧ dx \\ & ( + ∂_y ∂_x R^x & & + ∂_y ∂_z R^z & ) \; dt ∧ dy \\ & ( + ∂_z ∂_x R^x & + ∂_z ∂_y R^y & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dy ∧ dz \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dz ∧ dx \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm]\end{split}\]

\(d⋆d⋆R^{♭♭}\)

Applying the exterior derivative to \(⋆d⋆R^{♭♭}\), we obtain:

\[\begin{split}d⋆d⋆ R^{♭♭} &= \left[ \begin{alignedat}{4} & ( + ∂_t^2 Q^x & - ∂_x^2 Q^x & & & ) \; dt ∧ dx \\ & ( + ∂_t^2 Q^y & & - ∂_y^2 Q^y & & ) \; dt ∧ dy \\ & ( + ∂_t^2 Q^z & & & - ∂_z^2 Q^z & ) \; dt ∧ dx \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 R^x & + ∂_z^2 R^x & ) \; dy ∧ dz \\ & ( + ∂_x^2 R^y & & + ∂_z^2 R^y & ) \; dz ∧ dx \\ & ( + ∂_x^2 R^z & + ∂_y^2 R^z & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y Q^y & - ∂_z ∂_x Q^z & ) \; dt ∧ dx \\ & ( - ∂_x ∂_y Q^x & & - ∂_y ∂_z Q^z & ) \; dt ∧ dy \\ & ( - ∂_z ∂_x Q^x & - ∂_z ∂_y Q^y & - & ) \; dt ∧ dx \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_t ∂_y Q^z & - ∂_t ∂_z Q^y & ) \; dy ∧ dz \\ & ( - ∂_t ∂_x Q^z & & + ∂_t ∂_z Q^x & ) \; dz ∧ dx \\ & ( + ∂_t ∂_x Q^y & - ∂_t ∂_y Q^x & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dt ∧ dx \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dt ∧ dy \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y R^y & - ∂_x ∂_z R^z & ) \; dy ∧ dz \\ & ( - ∂_y ∂_x R^x & & - ∂_y ∂_z R^z & ) \; dz ∧ dx \\ & ( - ∂_z ∂_x R^x & - ∂_z ∂_y R^y & & ) \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Calculations

Take the exterior derivative

\[\begin{split}d⋆d⋆ R^{♭♭} = d \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &) & \; & dt \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &) & \; & dx \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &) & \; & dy \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &) & \; & dz \\ \end{alignedat} \right]\end{split}\]

Apply the exterior derivative

\[\begin{split}d⋆d⋆ R^{♭♭} = d \left[ \begin{alignedat}{5} ( & + & ∂_x ∂_x Q^x & + & ∂_x ∂_y Q^y & + & ∂_x ∂_z Q^z & ) & \; & dx ∧ dt \\ ( & + & ∂_y ∂_x Q^x & + & ∂_y ∂_y Q^y & + & ∂_y ∂_z Q^z & ) & \; & dy ∧ dt \\ ( & + & ∂_z ∂_x Q^x & + & ∂_z ∂_y Q^y & + & ∂_z ∂_z Q^z & ) & \; & dx ∧ dt \\[2mm] ( & + & ∂_t ∂_t Q^x & - & ∂_t ∂_y R^z & + & ∂_t ∂_z R^y & ) & \; & dt ∧ dx \\ ( & + & ∂_y ∂_t Q^x & - & ∂_y ∂_y R^z & + & ∂_y ∂_z R^y & ) & \; & dy ∧ dx \\ ( & + & ∂_z ∂_t Q^x & - & ∂_z ∂_y R^z & + & ∂_z ∂_z R^y & ) & \; & dz ∧ dx \\[2mm] ( & + & ∂_t ∂_t Q^y & + & ∂_t ∂_x R^z & - & ∂_t ∂_z R^x & ) & \; & dt ∧ dy \\ ( & + & ∂_x ∂_t Q^y & + & ∂_x ∂_x R^z & - & ∂_x ∂_z R^x & ) & \; & dx ∧ dy \\ ( & + & ∂_z ∂_t Q^y & + & ∂_z ∂_x R^z & - & ∂_z ∂_z R^x & ) & \; & dz ∧ dy \\[2mm] ( & + & ∂_t ∂_t Q^z & - & ∂_t ∂_x R^y & + & ∂_t ∂_y R^x & ) & \; & dt ∧ dz \\ ( & + & ∂_x ∂_t Q^z & - & ∂_x ∂_x R^y & + & ∂_x ∂_y R^x & ) & \; & dx ∧ dz \\ ( & + & ∂_y ∂_t Q^z & - & ∂_y ∂_x R^y & + & ∂_y ∂_y R^x & ) & \; & dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d⋆d⋆ R^{♭♭} = d \left[ \begin{alignedat}{5} ( & + & ∂_x ∂_x Q^x & + & ∂_x ∂_y Q^y & + & ∂_x ∂_z Q^z & ) & \; & dx ∧ dt \\ ( & + & ∂_y ∂_x Q^x & + & ∂_y ∂_y Q^y & + & ∂_y ∂_z Q^z & ) & \; & dy ∧ dt \\ ( & + & ∂_z ∂_x Q^x & + & ∂_z ∂_y Q^y & + & ∂_z ∂_z Q^z & ) & \; & dx ∧ dt \\[2mm] ( & + & ∂_t ∂_t Q^x & - & ∂_t ∂_y R^z & + & ∂_t ∂_z R^y & ) & \; & dt ∧ dx \\ ( & + & ∂_t ∂_t Q^y & + & ∂_t ∂_x R^z & - & ∂_t ∂_z R^x & ) & \; & dt ∧ dy \\ ( & + & ∂_t ∂_t Q^z & - & ∂_t ∂_x R^y & + & ∂_t ∂_y R^x & ) & \; & dt ∧ dz \\[2mm] ( & + & ∂_z ∂_t Q^y & + & ∂_z ∂_x R^z & - & ∂_z ∂_z R^x & ) & \; & dz ∧ dy \\ ( & + & ∂_y ∂_t Q^z & - & ∂_y ∂_x R^y & + & ∂_y ∂_y R^x & ) & \; & dy ∧ dz \\[2mm] ( & + & ∂_z ∂_t Q^x & - & ∂_z ∂_y R^z & + & ∂_z ∂_z R^y & ) & \; & dz ∧ dx \\ ( & + & ∂_x ∂_t Q^z & - & ∂_x ∂_x R^y & + & ∂_x ∂_y R^x & ) & \; & dx ∧ dz \\[2mm] ( & + & ∂_y ∂_t Q^x & - & ∂_y ∂_y R^z & + & ∂_y ∂_z R^y & ) & \; & dy ∧ dx \\ ( & + & ∂_x ∂_t Q^y & + & ∂_x ∂_x R^z & - & ∂_x ∂_z R^x & ) & \; & dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Reorder the exterior products

\[\begin{split}d⋆d⋆ R^{♭♭} = d \left[ \begin{alignedat}{5} ( & - & ∂_x ∂_x Q^x & - & ∂_x ∂_y Q^y & - & ∂_x ∂_z Q^z & ) & \; & dt ∧ dx \\ ( & - & ∂_y ∂_x Q^x & - & ∂_y ∂_y Q^y & - & ∂_y ∂_z Q^z & ) & \; & dt ∧ dy \\ ( & - & ∂_z ∂_x Q^x & - & ∂_z ∂_y Q^y & - & ∂_z ∂_z Q^z & ) & \; & dt ∧ dx \\[2mm] ( & + & ∂_t ∂_t Q^x & - & ∂_t ∂_y R^z & + & ∂_t ∂_z R^y & ) & \; & dt ∧ dx \\ ( & + & ∂_t ∂_t Q^y & + & ∂_t ∂_x R^z & - & ∂_t ∂_z R^x & ) & \; & dt ∧ dy \\ ( & + & ∂_t ∂_t Q^z & - & ∂_t ∂_x R^y & + & ∂_t ∂_y R^x & ) & \; & dt ∧ dz \\[2mm] ( & - & ∂_z ∂_t Q^y & - & ∂_z ∂_x R^z & + & ∂_z ∂_z R^x & ) & \; & dy ∧ dz \\ ( & + & ∂_y ∂_t Q^z & - & ∂_y ∂_x R^y & + & ∂_y ∂_y R^x & ) & \; & dy ∧ dz \\[2mm] ( & + & ∂_z ∂_t Q^x & - & ∂_z ∂_y R^z & + & ∂_z ∂_z R^y & ) & \; & dz ∧ dx \\ ( & - & ∂_x ∂_t Q^z & + & ∂_x ∂_x R^y & - & ∂_x ∂_y R^x & ) & \; & dz ∧ dx \\[2mm] ( & - & ∂_y ∂_t Q^x & + & ∂_y ∂_y R^z & - & ∂_y ∂_z R^y & ) & \; & dx ∧ dy \\ ( & + & ∂_x ∂_t Q^y & + & ∂_x ∂_x R^z & - & ∂_x ∂_z R^x & ) & \; & dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d⋆d⋆ R^{♭♭} = d \left[ \begin{alignedat}{5} ( & - & ∂_x ∂_x Q^x & - & ∂_x ∂_y Q^y & - & ∂_x ∂_z Q^z & ) & \; & dt ∧ dx \\ ( & - & ∂_y ∂_x Q^x & - & ∂_y ∂_y Q^y & - & ∂_y ∂_z Q^z & ) & \; & dt ∧ dy \\ ( & - & ∂_z ∂_x Q^x & - & ∂_z ∂_y Q^y & - & ∂_z ∂_z Q^z & ) & \; & dt ∧ dx \\[2mm] ( & + & ∂_t ∂_t Q^x & & & & & ) & \; & dt ∧ dx \\ ( & + & ∂_t ∂_t Q^y & & & & & ) & \; & dt ∧ dy \\ ( & + & ∂_t ∂_t Q^z & & & & & ) & \; & dt ∧ dz \\[2mm] ( & & & - & ∂_t ∂_y R^z & + & ∂_t ∂_z R^y & ) & \; & dt ∧ dx \\ ( & & & + & ∂_t ∂_x R^z & - & ∂_t ∂_z R^x & ) & \; & dt ∧ dy \\ ( & & & - & ∂_t ∂_x R^y & + & ∂_t ∂_y R^x & ) & \; & dt ∧ dz \\[2mm] ( & - & ∂_z ∂_t Q^y & & & & & ) & \; & dy ∧ dz \\ ( & + & ∂_y ∂_t Q^z & & & & & ) & \; & dy ∧ dz \\ ( & + & ∂_z ∂_t Q^x & & & & & ) & \; & dz ∧ dx \\ ( & - & ∂_x ∂_t Q^z & & & & & ) & \; & dz ∧ dx \\ ( & - & ∂_y ∂_t Q^x & & & & & ) & \; & dx ∧ dy \\ ( & + & ∂_x ∂_t Q^y & & & & & ) & \; & dx ∧ dy \\[2mm] ( & & & & & + & ∂_z ∂_z R^x & ) & \; & dy ∧ dz \\ ( & & & & & + & ∂_y ∂_y R^x & ) & \; & dy ∧ dz \\ ( & & & & & + & ∂_z ∂_z R^y & ) & \; & dz ∧ dx \\ ( & & & + & ∂_x ∂_x R^y & & & ) & \; & dz ∧ dx \\ ( & & & + & ∂_y ∂_y R^z & & & ) & \; & dx ∧ dy \\ ( & & & + & ∂_x ∂_x R^z & & & ) & \; & dx ∧ dy \\[2mm] ( & & & - & ∂_z ∂_x R^z & & & ) & \; & dy ∧ dz \\ ( & & & - & ∂_y ∂_x R^y & & & ) & \; & dy ∧ dz \\ ( & & & - & ∂_z ∂_y R^z & & & ) & \; & dz ∧ dx \\ ( & & & & & - & ∂_x ∂_y R^x & ) & \; & dz ∧ dx \\ ( & & & & & - & ∂_y ∂_z R^y & ) & \; & dx ∧ dy \\ ( & & & & & - & ∂_x ∂_z R^x & ) & \; & dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d⋆d⋆ R^{♭♭} &= \left[ \begin{alignedat}{4} ( & + & ∂_t^2 Q^x & - & ∂_x ∂_x Q^x & - & ∂_x ∂_y Q^y & - & ∂_x ∂_z Q^z & ) & \; & dt ∧ dx \\ ( & + & ∂_t^2 Q^y & - & ∂_x ∂_y Q^x & - & ∂_y ∂_y Q^y & - & ∂_y ∂_z Q^z & ) & \; & dt ∧ dy \\ ( & + & ∂_t^2 Q^z & - & ∂_x ∂_z Q^x & - & ∂_z ∂_y Q^y & - & ∂_z ∂_z Q^z & ) & \; & dt ∧ dx \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} ( & \; & & \; - \; & ∂_t ∂_y R^z & \; + \; & ∂_t ∂_z R^y & ) & \; & dt ∧ dx \\ ( & + \; & ∂_t ∂_x R^z & \; \; & & \; - \; & ∂_t ∂_z R^x & ) & \; & dt ∧ dy \\ ( & - \; & ∂_t ∂_x R^y & \; + \; & ∂_t ∂_y R^x & \; \; & & ) & \; & dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} ( & \; & & \; + \; & ∂_y ∂_t Q^z & \; - \; & ∂_z ∂_t Q^y & ) & \; & dy ∧ dz \\ ( & - \; & ∂_x ∂_t Q^z & \; \; & & \; + \; & ∂_z ∂_t Q^x & ) & \; & dz ∧ dx \\ ( & + \; & ∂_x ∂_t Q^y & \; - \; & ∂_y ∂_t Q^x & \; \; & & ) & \; & dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} ( & \; & & \; + \; & ∂_y^2 R^x & \; + \; & ∂_z^2 R^x & ) & \; & dy ∧ dz \\ ( & + \; & ∂_x^2 R^y & \; \; & & \; + \; & ∂_z^2 R^y & ) & \; & dz ∧ dx \\ ( & + \; & ∂_x^2 R^z & \; + \; & ∂_y^2 R^z & \; \; & & ) & \; & dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} ( & \; & & \; - \; & ∂_y ∂_x R^y & \; - \; & ∂_z ∂_x R^z & ) & \; & dy ∧ dz \\ ( & - \; & ∂_x ∂_y R^x & \; \; & & \; - \; & ∂_z ∂_y R^z & ) & \; & dz ∧ dx \\ ( & - \; & ∂_x ∂_z R^x & \; - \; & ∂_y ∂_z R^y & \; \; & & ) & \; & dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d⋆d⋆ R^{♭♭} &= \left[ \begin{alignedat}{4} & ( + ∂_t^2 Q^x & - ∂_x^2 Q^x & & & ) \; dt ∧ dx \\ & ( + ∂_t^2 Q^y & & - ∂_y^2 Q^y & & ) \; dt ∧ dy \\ & ( + ∂_t^2 Q^z & & & - ∂_z^2 Q^z & ) \; dt ∧ dx \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 R^x & + ∂_z^2 R^x & ) \; dy ∧ dz \\ & ( + ∂_x^2 R^y & & + ∂_z^2 R^y & ) \; dz ∧ dx \\ & ( + ∂_x^2 R^z & + ∂_y^2 R^z & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y Q^y & - ∂_z ∂_x Q^z & ) \; dt ∧ dx \\ & ( - ∂_x ∂_y Q^x & & - ∂_y ∂_z Q^z & ) \; dt ∧ dy \\ & ( - ∂_z ∂_x Q^x & - ∂_z ∂_y Q^y & - & ) \; dt ∧ dx \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_t ∂_y Q^z & - ∂_t ∂_z Q^y & ) \; dy ∧ dz \\ & ( - ∂_t ∂_x Q^z & & + ∂_t ∂_z Q^x & ) \; dz ∧ dx \\ & ( + ∂_t ∂_x Q^y & - ∂_t ∂_y Q^x & & ) \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dt ∧ dx \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dt ∧ dy \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y R^y & - ∂_x ∂_z R^z & ) \; dy ∧ dz \\ & ( - ∂_y ∂_x R^x & & - ∂_y ∂_z R^z & ) \; dz ∧ dx \\ & ( - ∂_z ∂_x R^x & - ∂_z ∂_y R^y & & ) \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

\(⋆d⋆d R^{♭♭}\)

\[\begin{split}⋆d⋆d R^{♭♭} &= \left[ \begin{alignedat}{4} & ( + ∂_t^2 R^x & - ∂_x^2 R^x & & & ) \; dy ∧ dz \\ & ( + ∂_t^2 R^y & & - ∂_y^2 R^y & & ) \; dy ∧ dx \\ & ( + ∂_t^2 R^z & & & - ∂_z^2 R^z & ) \; dy ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) \; dt ∧ dx \\ & ( + ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) \; dt ∧ dy \\ & ( + ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y Q^y & - ∂_z ∂_x Q^z & ) \; dt ∧ dx \\ & ( - ∂_x ∂_y Q^x & & - ∂_y ∂_z Q^z & ) \; dt ∧ dy \\ & ( - ∂_z ∂_x Q^x & - ∂_y ∂_z Q^y & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_t ∂_y Q^z & - ∂_t ∂_z Q^y & ) \; dy ∧ dz \\ & ( - ∂_t ∂_x Q^z & & + ∂_t ∂_z Q^x & ) \; dy ∧ dx \\ & ( + ∂_t ∂_x Q^y & - ∂_t ∂_y Q^x & & ) \; dy ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dt ∧ dx \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dt ∧ dy \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y R^y & - ∂_x ∂_z R^z & ) \; dy ∧ dz \\ & ( - ∂_y ∂_x R^x & & - ∂_y ∂_z R^z & ) \; dz ∧ dx \\ & ( - ∂_z ∂_x R^x & - ∂_z ∂_y R^y & & ) \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Calculations

References

• Hodge dual of k–forms in Minkowski space

Take the Hodge star

\[\begin{split}⋆d⋆d R^{♭♭} &= ⋆ \left[ \begin{alignedat}{4} ( & - ∂_t^2 R^x & ∂_x ∂_x & \; R^x & \, + \, & ∂_y ∂_x & \; R^y & \, + \, & ∂_x ∂_z & \; R^z & \; ) & \; dt ∧ dx \\ ( & - ∂_t^2 R^y & ∂_x ∂_y & \; R^x & \, + \, & ∂_y ∂_y & \; R^y & \, + \, & ∂_y ∂_z & \; R^z & \; ) & \; dt ∧ dy \\ ( & - ∂_t^2 R^z & ∂_x ∂_z & \; R^x & \, + \, & ∂_y ∂_z & \; R^y & \, + \, & ∂_z ∂_z & \; R^z & \; ) & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ ⋆ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y Q^z & + ∂_t ∂_z Q^y & ) & \; dt ∧ dx \\ & ( + ∂_t ∂_x Q^z & & - ∂_t ∂_z Q^x & ) & \; dt ∧ dy \\ & ( - ∂_t ∂_x Q^y & + ∂_t ∂_y Q^x & & ) & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ ⋆ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) & \; dy ∧ dz \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) & \; dz ∧ dx \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) & \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ ⋆ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) & \; dy ∧ dz \\ & (+ ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) & \; dz ∧ dx \\ & (+ ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) & \; dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ ⋆ \left[ \begin{alignedat}{4} & ( & - ∂_z ∂_x Q^z & - ∂_x ∂_y Q^y & ) & \; dy ∧ dz \\ & ( - ∂_y ∂_z Q^z & & - ∂_x ∂_y Q^x & ) & \; dz ∧ dx \\ & ( - ∂_y ∂_z Q^y & - ∂_z ∂_x Q^x & & ) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Distribute the Hodge star

\[\begin{split}⋆d⋆d R^{♭♭} &= \left[ \begin{alignedat}{4} ( & - ∂_t^2 R^x & ∂_x ∂_x & \; R^x & \, + \, & ∂_y ∂_x & \; R^y & \, + \, & ∂_x ∂_z & \; R^z & \; ) & \; ⋆ dt ∧ dx \\ ( & - ∂_t^2 R^y & ∂_x ∂_y & \; R^x & \, + \, & ∂_y ∂_y & \; R^y & \, + \, & ∂_y ∂_z & \; R^z & \; ) & \; ⋆ dt ∧ dy \\ ( & - ∂_t^2 R^z & ∂_x ∂_z & \; R^x & \, + \, & ∂_y ∂_z & \; R^y & \, + \, & ∂_z ∂_z & \; R^z & \; ) & \; ⋆ dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y Q^z & + ∂_t ∂_z Q^y & ) & \; ⋆ dt ∧ dx \\ & ( + ∂_t ∂_x Q^z & & - ∂_t ∂_z Q^x & ) & \; ⋆ dt ∧ dy \\ & ( - ∂_t ∂_x Q^y & + ∂_t ∂_y Q^x & & ) & \; ⋆ dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) & \; ⋆ dy ∧ dz \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) & \; ⋆ dz ∧ dx \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) & \; ⋆ dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) & \; ⋆ dy ∧ dz \\ & (+ ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) & \; ⋆ dz ∧ dx \\ & (+ ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) & \; ⋆ dx ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_z ∂_x Q^z & - ∂_x ∂_y Q^y & ) & \; ⋆ dy ∧ dz \\ & ( - ∂_y ∂_z Q^z & & - ∂_x ∂_y Q^x & ) & \; ⋆ dz ∧ dx \\ & ( - ∂_y ∂_z Q^y & - ∂_z ∂_x Q^x & & ) & \; ⋆ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Apply the Hodge star

\[\begin{split}⋆d⋆d R^{♭♭} &= \left[ \begin{alignedat}{4} ( & + ∂_t^2 R^x & - \, & ∂_x ∂_x & \; R^x & \, - \, & ∂_y ∂_x & \; R^y & \, - \, & ∂_x ∂_z & \; R^z & \; ) & \; dy ∧ dz \\ ( & + ∂_t^2 R^y & - \, & ∂_x ∂_y & \; R^x & \, - \, & ∂_y ∂_y & \; R^y & \, - \, & ∂_y ∂_z & \; R^z & \; ) & \; dy ∧ dx \\ ( & + ∂_t^2 R^z & - \, & ∂_x ∂_z & \; R^x & \, - \, & ∂_y ∂_z & \; R^y & \, - \, & ∂_z ∂_z & \; R^z & \; ) & \; dy ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_t ∂_y Q^z & - ∂_t ∂_z Q^y & ) & \; dy ∧ dz \\ & ( - ∂_t ∂_x Q^z & & + ∂_t ∂_z Q^x & ) & \; dy ∧ dx \\ & ( + ∂_t ∂_x Q^y & - ∂_t ∂_y Q^x & & ) & \; dy ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) & \; dt ∧ dx \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) & \; dt ∧ dy \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) & \; dt ∧ dx \\ & (+ ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) & \; dt ∧ dy \\ & (+ ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) & \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_z ∂_x Q^z & - ∂_x ∂_y Q^y & ) & \; dt ∧ dx \\ & ( - ∂_y ∂_z Q^z & & - ∂_x ∂_y Q^x & ) & \; dt ∧ dy \\ & ( - ∂_y ∂_z Q^y & - ∂_z ∂_x Q^x & & ) & \; dt ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}⋆d⋆d R^{♭♭} &= \left[ \begin{alignedat}{4} & ( + ∂_t^2 R^x & - ∂_x^2 R^x & & & ) \; dy ∧ dz \\ & ( + ∂_t^2 R^y & & - ∂_y^2 R^y & & ) \; dy ∧ dx \\ & ( + ∂_t^2 R^z & & & - ∂_z^2 R^z & ) \; dy ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) \; dt ∧ dx \\ & ( + ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) \; dt ∧ dy \\ & ( + ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y Q^y & - ∂_z ∂_x Q^z & ) \; dt ∧ dx \\ & ( - ∂_x ∂_y Q^x & & - ∂_y ∂_z Q^z & ) \; dt ∧ dy \\ & ( - ∂_z ∂_x Q^x & - ∂_y ∂_z Q^y & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & + ∂_t ∂_y Q^z & - ∂_t ∂_z Q^y & ) \; dy ∧ dz \\ & ( - ∂_t ∂_x Q^z & & + ∂_t ∂_z Q^x & ) \; dy ∧ dx \\ & ( + ∂_t ∂_x Q^y & - ∂_t ∂_y Q^x & & ) \; dy ∧ dy \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dt ∧ dx \\ & ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dt ∧ dy \\ & ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dt ∧ dz \\ \end{alignedat} \right] \\[2mm] &+ \left[ \begin{alignedat}{4} & ( & - ∂_x ∂_y R^y & - ∂_x ∂_z R^z & ) \; dy ∧ dz \\ & ( - ∂_y ∂_x R^x & & - ∂_y ∂_z R^z & ) \; dz ∧ dx \\ & ( - ∂_z ∂_x R^x & - ∂_z ∂_y R^y & & ) \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

\(d⋆d⋆ - ⋆d⋆d\)

\[\begin{split}(d⋆d⋆ - ⋆d⋆d) \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right] &= \left[ \begin{alignedat}{5} & ( + ∂_t^2 Q^x & - ∂_x^2 Q^x & - ∂_y^2 Q^x & - ∂_z^2 Q^x & \: ) \; dt∧dx \\ & ( + ∂_t^2 Q^y & - ∂_x^2 Q^y & - ∂_y^2 Q^y & - ∂_z^2 Q^y & \: ) \; dt∧dy \\ & ( + ∂_t^2 Q^z & - ∂_x^2 Q^z & - ∂_y^2 Q^z & - ∂_z^2 Q^z & \: ) \; dt∧dz \\ & ( - ∂_t^2 R^x & + ∂_x^2 R^x & + ∂_y^2 R^x & + ∂_z^2 R^x & \: ) \; dy∧dz \\ & ( - ∂_t^2 R^y & + ∂_x^2 R^y & + ∂_y^2 R^y & + ∂_z^2 R^y & \: ) \; dz∧dx \\ & ( - ∂_t^2 R^z & + ∂_x^2 R^z & + ∂_y^2 R^z & + ∂_z^2 R^z & \: ) \; dx∧dy \\ \end{alignedat} \right]\end{split}\]