2–forms (rotations)#
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.
While the concept might not be entirely novel, I have not yet encountered some of my observations elsewhere. If you are aware of relevant references, feel free to open an issue and I will include them. I you identify any errors, you can either open an issue, or directly submit corrections via a merge request to my GitHub repository: Theoretical Universe.
Rotations#
Rotations in spacetime can occur in six independent planes. Any rotation can be decomposed into a linear combination of basis rotations within each plane:
\[\begin{split}R^{♯♯} = \begin{bmatrix}
Q^x \; ∂_t ∧ ∂_x \\
Q^y \; ∂_t ∧ ∂_y \\
Q^z \; ∂_t ∧ ∂_z \\
R^x \; ∂_y ∧ ∂_z \\
R^y \; ∂_z ∧ ∂_x \\
R^z \; ∂_x ∧ ∂_y \\
\end{bmatrix}\end{split}\]
The generic rotation above is doubly contravariant, given in terms of the wedge product \(∧\) of vectors corresponding to our physical understanding of space (and time). By fully flattening, we obtain the associated doubly covariant differential 2-form representative of the rotation:
\[\begin{split}R^{♭♭} = \left[ \begin{aligned}
- & Q^x \; dt ∧ dx \\
- & Q^y \; dt ∧ dy \\
- & Q^z \; dt ∧ dz \\
& R^x \; dy ∧ dz \\
& R^y \; dz ∧ dx \\
& R^z \; dx ∧ dy \\
\end{aligned} \right]\end{split}\]
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:
\[\begin{split}R^{♭♭} = \begin{bmatrix}
Q^x \; η_{αt} η_{βx} \; dx^α ∧ dx^β \\
Q^y \; η_{αt} η_{βy} \; dx^α ∧ dx^β \\
Q^z \; η_{αt} η_{βz} \; dx^α ∧ dx^β \\
R^x \; η_{αy} η_{βz} \; dx^α ∧ dx^β \\
R^y \; η_{αz} η_{βx} \; dx^α ∧ dx^β \\
R^z \; η_{αx} η_{βy} \; dx^α ∧ dx^β \\
\end{bmatrix}\end{split}\]
Identify the non-zero components of the Minkowski metric
\[\begin{split}R^{♭♭} = \begin{bmatrix}
Q^x \; η_{tt} η_{xx} \; dx^t ∧ dx^x \\
Q^y \; η_{tt} η_{yy} \; dx^t ∧ dx^y \\
Q^z \; η_{tt} η_{zz} \; dx^t ∧ dx^z \\
R^x \; η_{yy} η_{zz} \; dx^y ∧ dx^z \\
R^y \; η_{zz} η_{xx} \; dx^z ∧ dx^x \\
R^z \; η_{xx} η_{yy} \; dx^x ∧ dx^y \\
\end{bmatrix}\end{split}\]
Rewrite
For readability, replace the \(dx^μ\) symbols by their explicit expressions:
\[\begin{split}dx^t &= dt \\
dx^x &= dx \\
dx^y &= dy \\
dx^z &= dz\end{split}\]
We obtain:
\[\begin{split}R^{♭♭} = \begin{bmatrix}
Q^x \; η_{tt} η_{xx} \; dt ∧ dx \\
Q^y \; η_{tt} η_{yy} \; dt ∧ dy \\
Q^z \; η_{tt} η_{zz} \; dt ∧ dz \\
R^x \; η_{yy} η_{zz} \; dy ∧ dz \\
R^y \; η_{zz} η_{xx} \; dz ∧ dx \\
R^z \; η_{xx} η_{yy} \; dx ∧ dy \\
\end{bmatrix}\end{split}\]
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:
\[\begin{split}⋆ R^{♭♭} = \left[ \begin{aligned}
& Q^x \; dy ∧ dz \\
& Q^y \; dz ∧ dx \\
& Q^z \; dx ∧ dy \\
& R^x \; dt ∧ dx \\
& R^y \; dt ∧ dy \\
& R^z \; dt ∧ dz \\
\end{aligned} \right]\end{split}\]
\(d R^{♭♭}\)#
Apply the exterior derivative to the rotation 2-form, we obtain:
\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{5}
(& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx^x ∧ dx^y ∧ dx^z \\
(& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dx^t ∧ dx^y ∧ dx^z \\
(& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dx^t ∧ dx^z ∧ dx^x \\
(& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dx^t ∧ dx^x ∧ dx^y \\
\end{alignedat} \right]\end{split}\]
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
\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{5}
(& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx^x ∧ dx^y ∧ dx^z \\
(& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dx^t ∧ dx^y ∧ dx^z \\
(& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dx^t ∧ dx^z ∧ dx^x \\
(& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dx^t ∧ dx^x ∧ dx^y \\
\end{alignedat} \right]\end{split}\]
\(d⋆ R^{♭♭}\)#
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
\[\begin{split}d ⋆ R^{♭♭} = \left[ \begin{alignedat}{5}
(& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z & \: ) \; & dx ∧ dy ∧ dz \\
(& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y & \: ) \; & dt ∧ dy ∧ dz \\
(& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x & \: ) \; & dt ∧ dz ∧ dx \\
(& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & & \: ) \; & dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
\(⋆d R^{♭♭}\)#
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:
\[\begin{split}⋆ d⋆R^{♭♭} = \left[ \begin{alignedat}{5}
(& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dt \\
(& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dx \\
(& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dy \\
(& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dz \\
\end{alignedat} \right]\end{split}\]
\(d⋆d R^{♭♭}\)#
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
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
\[\begin{split}⋆d⋆d R^{♭♭}
&= \left[ \begin{alignedat}{4}
& ( + ∂_t^2 R^x & - ∂_x^2 R^x & & & ) \; dy ∧ dz \\
& ( + ∂_t^2 R^y & & - ∂_y^2 R^y & & ) \; dy ∧ dx \\
& ( + ∂_t^2 R^z & & & - ∂_z^2 R^z & ) \; dy ∧ dy \\
\end{alignedat} \right] \\[2mm]
&+ \left[ \begin{alignedat}{4}
& ( & + ∂_y^2 Q^x & + ∂_z^2 Q^x & ) \; dt ∧ dx \\
& ( + ∂_x^2 Q^y & & + ∂_z^2 Q^y & ) \; dt ∧ dy \\
& ( + ∂_x^2 Q^z & + ∂_y^2 Q^z & & ) \; dt ∧ dz \\
\end{alignedat} \right] \\[2mm]
&+ \left[ \begin{alignedat}{4}
& ( & - ∂_x ∂_y Q^y & - ∂_z ∂_x Q^z & ) \; dt ∧ dx \\
& ( - ∂_x ∂_y Q^x & & - ∂_y ∂_z Q^z & ) \; dt ∧ dy \\
& ( - ∂_z ∂_x Q^x & - ∂_y ∂_z Q^y & & ) \; dt ∧ dz \\
\end{alignedat} \right] \\[2mm]
&+ \left[ \begin{alignedat}{4}
& ( & + ∂_t ∂_y Q^z & - ∂_t ∂_z Q^y & ) \; dy ∧ dz \\
& ( - ∂_t ∂_x Q^z & & + ∂_t ∂_z Q^x & ) \; dy ∧ dx \\
& ( + ∂_t ∂_x Q^y & - ∂_t ∂_y Q^x & & ) \; dy ∧ dy \\
\end{alignedat} \right] \\[2mm]
&+ \left[ \begin{alignedat}{4}
& ( & - ∂_t ∂_y R^z & + ∂_t ∂_z R^y & ) \; dt ∧ dx \\
& ( + ∂_t ∂_x R^z & & - ∂_t ∂_z R^x & ) \; dt ∧ dy \\
& ( - ∂_t ∂_x R^y & + ∂_t ∂_y R^x & & ) \; dt ∧ dz \\
\end{alignedat} \right] \\[2mm]
&+ \left[ \begin{alignedat}{4}
& ( & - ∂_x ∂_y R^y & - ∂_x ∂_z R^z & ) \; dy ∧ dz \\
& ( - ∂_y ∂_x R^x & & - ∂_y ∂_z R^z & ) \; dz ∧ dx \\
& ( - ∂_z ∂_x R^x & - ∂_z ∂_y R^y & & ) \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
\(d⋆d⋆ - ⋆d⋆d\)#
\[\begin{split}(d⋆d⋆ - ⋆d⋆d) \left[ \begin{aligned}
- & Q^x \; dt ∧ dx \\
- & Q^y \; dt ∧ dy \\
- & Q^z \; dt ∧ dz \\
& R^x \; dy ∧ dz \\
& R^y \; dz ∧ dx \\
& R^z \; dx ∧ dy \\
\end{aligned} \right]
&= \left[ \begin{alignedat}{5}
& ( + ∂_t^2 Q^x & - ∂_x^2 Q^x & - ∂_y^2 Q^x & - ∂_z^2 Q^x & \: ) \; dt∧dx \\
& ( + ∂_t^2 Q^y & - ∂_x^2 Q^y & - ∂_y^2 Q^y & - ∂_z^2 Q^y & \: ) \; dt∧dy \\
& ( + ∂_t^2 Q^z & - ∂_x^2 Q^z & - ∂_y^2 Q^z & - ∂_z^2 Q^z & \: ) \; dt∧dz \\
& ( - ∂_t^2 R^x & + ∂_x^2 R^x & + ∂_y^2 R^x & + ∂_z^2 R^x & \: ) \; dy∧dz \\
& ( - ∂_t^2 R^y & + ∂_x^2 R^y & + ∂_y^2 R^y & + ∂_z^2 R^y & \: ) \; dz∧dx \\
& ( - ∂_t^2 R^z & + ∂_x^2 R^z & + ∂_y^2 R^z & + ∂_z^2 R^z & \: ) \; dx∧dy \\
\end{alignedat} \right]\end{split}\]