1–forms (translations)#
Translations#
A generic translation in spacetime is expressed with a 4-vector:
\[\begin{split}T^♯ = \left[ \begin{alignedat}{1}
A^t & ∂_t \\
A^x & ∂_x \\
A^y & ∂_y \\
A^z & ∂_z \\
\end{alignedat} \right]\end{split}\]
Flattened, the translation expressed in differential form is:
\[\begin{split}T^♭ = \left[ \begin{alignedat}{2}
& A^t & dt \\
- & A^x & dx \\
- & A^y & dy \\
- & A^z & dz \\
\end{alignedat} \right]\end{split}\]
Calculations
References
Flatten the translation vector
\[T^♭ = \left(T^♯\right)^♭\]
Expand
\[\begin{split}T^♭ = \left[ \begin{alignedat}{1}
A^t & ∂_t \\
A^x & ∂_x \\
A^y & ∂_y \\
A^z & ∂_z \\
\end{alignedat} \right]^♭\end{split}\]
Distribute the flat operator to the basis vectors
\[\begin{split}T^♭ = \left[ \begin{alignedat}{1}
A^t & ∂_t^♭ \\
A^x & ∂_x^♭ \\
A^y & ∂_y^♭ \\
A^z & ∂_z^♭ \\
\end{alignedat} \right]\end{split}\]
Apply the flat operators to each basis vectors
\[\begin{split}T^♭ = \left[ \begin{alignedat}{2}
A^t & η_{tμ} & dx^μ \\
A^x & η_{xμ} & dx^μ \\
A^y & η_{yμ} & dx^μ \\
A^z & η_{zμ} & dx^μ \\
\end{alignedat} \right]\end{split}\]
Identify non-zero terms
\[\begin{split}T^♭ = \left[ \begin{alignedat}{2}
A^t & η_{tt} & dx^t \\
A^x & η_{xx} & dx^x \\
A^y & η_{yy} & dx^y \\
A^z & η_{zz} & dx^z \\
\end{alignedat} \right]\end{split}\]
Apply numerical values
\[\begin{split}T^♭ = \left[ \begin{alignedat}{2}
A^t & & dx^t \\
A^x & (-) & dx^x \\
A^y & (-) & dx^y \\
A^z & (-) & dx^z \\
\end{alignedat} \right]\end{split}\]
Rearange and conclude
\[\begin{split}T^♭ = \left[ \begin{alignedat}{2}
& A^t & dt \\
- & A^x & dx \\
- & A^y & dy \\
- & A^z & dz \\
\end{alignedat} \right]\end{split}\]
\(⋆ T^♭\)#
\[\begin{split}⋆ T^♭ = \left[ \begin{alignedat}{2}
& A^t & dx ∧ dy ∧ dz \\
- & A^x & dt ∧ dy ∧ dz \\
- & A^y & dt ∧ dz ∧ dx \\
- & A^z & dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Calculations
References
Apply the Hodge star to all
\[\begin{split}⋆ T^♭ = ⋆ \left[ \begin{alignedat}{2}
& A^t & dt \\
- & A^x & dx \\
- & A^y & dy \\
- & A^z & dz \\
\end{alignedat} \right]\end{split}\]
Distribute the Hodge star to each
\[\begin{split}⋆ T^♭ = \left[ \begin{alignedat}{3}
& A^t & ⋆ & dt \\
- & A^x & ⋆ & dx \\
- & A^y & ⋆ & dy \\
- & A^z & ⋆ & dz \\
\end{alignedat} \right]\end{split}\]
Apply the Hodge star and conclude
\[\begin{split}⋆ T^♭ = \left[ \begin{alignedat}{2}
& A^t & dx ∧ dy ∧ dz \\
- & A^x & dt ∧ dy ∧ dz \\
- & A^y & dt ∧ dz ∧ dx \\
- & A^z & dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
\(d T^♭\)#
\[\begin{split}d T^♭ = \left[ \begin{alignedat}{1}
( & - & ∂_t A^x & - & ∂_x A^t) & \; dt ∧ dx \\
( & - & ∂_t A^y & - & ∂_y A^t) & \; dt ∧ dy \\
( & - & ∂_t A^z & - & ∂_z A^t) & \; dt ∧ dz \\
( & - & ∂_y A^z & + & ∂_z A^y) & \; dy ∧ dz \\
( & - & ∂_z A^x & + & ∂_x A^z) & \; dz ∧ dx \\
( & - & ∂_x A^y & + & ∂_y A^x) & \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Calculations
Take the exterior derivative
\[\begin{split}d T^♭ = d \left[ \begin{alignedat}{2}
& A^t & dt \\
- & A^x & dx \\
- & A^y & dy \\
- & A^z & dz \\
\end{alignedat} \right]\end{split}\]
Apply the exterior derivative
\[\begin{split}d T^♭ = \begin{bmatrix}
+ ∂_x A^t dx ∧ dt & + ∂_y A^t dy ∧ dt & + ∂_z A^t dz ∧ dt \\
- ∂_t A^x dt ∧ dx & - ∂_y A^x dy ∧ dx & - ∂_z A^x dz ∧ dx \\
- ∂_t A^y dt ∧ dy & - ∂_z A^y dz ∧ dy & - ∂_x A^y dx ∧ dy \\
- ∂_t A^z dt ∧ dz & - ∂_x A^z dx ∧ dz & - ∂_y A^z dy ∧ dz \\
\end{bmatrix}\end{split}\]
Rearrange and conclude
\[\begin{split}d T^♭ = \left[ \begin{alignedat}{1}
( & - & ∂_t A^x & - & ∂_x A^t) & \; dt ∧ dx \\
( & - & ∂_t A^y & - & ∂_y A^t) & \; dt ∧ dy \\
( & - & ∂_t A^z & - & ∂_z A^t) & \; dt ∧ dz \\
( & - & ∂_y A^z & + & ∂_z A^y) & \; dy ∧ dz \\
( & - & ∂_z A^x & + & ∂_x A^z) & \; dz ∧ dx \\
( & - & ∂_x A^y & + & ∂_y A^x) & \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
\(d⋆ T^♭\)#
\[d ⋆ T^♭ = \left( ∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z \right) \; dt ∧ dx ∧ dy ∧ dz\]
Calculations
References
Apply the exterior derivative
\[\begin{split}d ⋆ T^♭ = d \left[ \begin{alignedat}{2}
& A^t & dx ∧ dy ∧ dz \\
- & A^x & dt ∧ dy ∧ dz \\
- & A^y & dt ∧ dz ∧ dx \\
- & A^z & dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Expand the exterior derivative
\[\begin{split}d ⋆ T^♭ = \left[ \begin{alignedat}{2}
& ∂_t A^t & dt ∧ dx ∧ dy ∧ dz \\
- & ∂_x A^x & dx ∧ dt ∧ dy ∧ dz \\
- & ∂_y A^y & dy ∧ dt ∧ dz ∧ dx \\
- & ∂_z A^z & dz ∧ dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Reorder the exterior products
\[\begin{split}d ⋆ T^♭ = \left[ \begin{alignedat}{2}
∂_t A^t & dt ∧ dx ∧ dy ∧ dz \\
∂_x A^x & dt ∧ dx ∧ dy ∧ dz \\
∂_y A^y & dt ∧ dx ∧ dy ∧ dz \\
∂_z A^z & dt ∧ dx ∧ dy ∧ dz \\
\end{alignedat} \right]\end{split}\]
Conclude
\[d ⋆ T^♭ = \left( ∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z \right) \; dt ∧ dx ∧ dy ∧ dz\]
\(⋆d T^♭\)#
\[\begin{split}⋆ d T^♭ = \left[ \begin{alignedat}{4}
(+ & ∂_z A^y & - & ∂_y A^z & ) & \; dt ∧ dx \\
(+ & ∂_x A^z & - & ∂_z A^x & ) & \; dt ∧ dy \\
(+ & ∂_y A^x & - & ∂_x A^y & ) & \; dt ∧ dz \\
(+ & ∂_x A^t & + & ∂_t A^x & ) & \; dy ∧ dz \\
(+ & ∂_y A^t & + & ∂_t A^y & ) & \; dz ∧ dx \\
(+ & ∂_z A^t & + & ∂_t A^z & ) & \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Calculations
References
Take the Hodge star
\[\begin{split}⋆ d T^♭ = ⋆ \left[ \begin{alignedat}{1}
( & - & ∂_t A^x & - & ∂_x A^t & ) & \; dt ∧ dx \\
( & - & ∂_t A^y & - & ∂_y A^t & ) & \; dt ∧ dy \\
( & - & ∂_t A^z & - & ∂_z A^t & ) & \; dt ∧ dz \\
( & - & ∂_y A^z & + & ∂_z A^y & ) & \; dy ∧ dz \\
( & - & ∂_z A^x & + & ∂_x A^z & ) & \; dz ∧ dx \\
( & - & ∂_x A^y & + & ∂_y A^x & ) & \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Distribute the Hodge star
\[\begin{split}⋆ d T^♭ = \left[ \begin{alignedat}{1}
( & - & ∂_t A^x & - & ∂_x A^t) & \; ⋆ \left( dt ∧ dx \right) \\
( & - & ∂_t A^y & - & ∂_y A^t) & \; ⋆ \left( dt ∧ dy \right) \\
( & - & ∂_t A^z & - & ∂_z A^t) & \; ⋆ \left( dt ∧ dz \right) \\
( & - & ∂_y A^z & + & ∂_z A^y) & \; ⋆ \left( dy ∧ dz \right) \\
( & - & ∂_z A^x & + & ∂_x A^z) & \; ⋆ \left( dz ∧ dx \right) \\
( & - & ∂_x A^y & + & ∂_y A^x) & \; ⋆ \left( dx ∧ dy \right) \\
\end{alignedat} \right]\end{split}\]
Apply the Hodge star
\[\begin{split}⋆ d T^♭ = \left[ \begin{alignedat}{1}
( & - & ∂_t A^x & - & ∂_x A^t) & \; & - & \left( dy ∧ dz \right) \\
( & - & ∂_t A^y & - & ∂_y A^t) & \; & - & \left( dz ∧ dx \right) \\
( & - & ∂_t A^z & - & ∂_z A^t) & \; & - & \left( dx ∧ dy \right) \\
( & - & ∂_y A^z & + & ∂_z A^y) & \; & & \left( dt ∧ dx \right) \\
( & - & ∂_z A^x & + & ∂_x A^z) & \; & & \left( dt ∧ dy \right) \\
( & - & ∂_x A^y & + & ∂_y A^x) & \; & & \left( dt ∧ dz \right) \\
\end{alignedat} \right]\end{split}\]
Reorder and conclude
\[\begin{split}⋆ d T^♭ = \left[ \begin{alignedat}{4}
( + & ∂_z A^y & - & ∂_y A^z & ) & \; dt ∧ dx \\
( + & ∂_x A^z & - & ∂_z A^x & ) & \; dt ∧ dy \\
( + & ∂_y A^x & - & ∂_x A^y & ) & \; dt ∧ dz \\
( + & ∂_x A^t & + & ∂_t A^x & ) & \; dy ∧ dz \\
( + & ∂_y A^t & + & ∂_t A^y & ) & \; dz ∧ dx \\
( + & ∂_z A^t & + & ∂_t A^z & ) & \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
\(⋆d⋆ T^♭\)#
\[⋆d⋆ T^♭ = ∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z\]
Calculations
Apply the Hodge star
\[⋆ d ⋆ T^♭ = ⋆ \left( ∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z \right) \; dt ∧ dx ∧ dy ∧ dz\]
Conclude
\[⋆ d ⋆ T^♭ = ∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z\]
\(d⋆d T^♭\)#
\[\begin{split}d⋆d T^♭ = \left[ \begin{alignedat}{7}
( & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & + ∂_t ∂_x A^x & + ∂_t ∂_y A^y & + ∂_t ∂_z A^z & ) \; dx ∧ dx ∧ dy \\
( & + ∂_t^2 A^x & - ∂_y^2 A^x & - ∂_z^2 A^x & + ∂_t ∂_x A^t & + ∂_x ∂_y A^y & + ∂_z ∂_x A^z & ) \; dt ∧ dy ∧ dz \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & - ∂_z^2 A^y & + ∂_t ∂_y A^t & + ∂_y ∂_z A^z & + ∂_x ∂_y A^x & ) \; dt ∧ dz ∧ dx \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & + ∂_t ∂_z A^t & + ∂_z ∂_x A^x & + ∂_y ∂_z A^y & ) \; dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Calculations
Apply the exterior derivative to all
\[\begin{split}d ⋆ d T^♭ = d \left[ \begin{alignedat}{4}
(+ & ∂_z A^y & - & ∂_y A^z &) \; dt ∧ dx \\
(+ & ∂_x A^z & - & ∂_z A^x &) \; dt ∧ dy \\
(+ & ∂_y A^x & - & ∂_x A^y &) \; dt ∧ dz \\
(+ & ∂_x A^t & + & ∂_t A^x &) \; dy ∧ dz \\
(+ & ∂_y A^t & + & ∂_t A^y &) \; dz ∧ dx \\
(+ & ∂_z A^t & + & ∂_t A^z &) \; dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Collapse permutations
\[\begin{split}d ⋆ d T^♭ = Π d \left[ \begin{alignedat}{4}
(+ & ∂_z A^y & - & ∂_y A^z &) \; dt ∧ dx \\
(+ & ∂_x A^t & + & ∂_t A^x &) \; dy ∧ dz \\
\end{alignedat} \right]\end{split}\]
Apply the exterior derivative
\[\begin{split}d ⋆ d T^♭ = Π d \left[ \begin{alignedat}{4}
∂_y (+ & ∂_z A^y & - & ∂_y A^z &) \; dy ∧ dt ∧ dx \\
∂_z (+ & ∂_z A^y & - & ∂_y A^z &) \; dz ∧ dt ∧ dx \\
∂_t (+ & ∂_x A^t & + & ∂_t A^x &) \; dt ∧ dy ∧ dz \\
∂_x (+ & ∂_x A^t & + & ∂_t A^x &) \; dx ∧ dy ∧ dz \\
\end{alignedat} \right]\end{split}\]
Rearange
\[\begin{split}d ⋆ d T^♭ = Π d \left[ \begin{alignedat}{4}
(- & ∂_y^2 A^z & + & ∂_y ∂_z A^y & ) \; dt ∧ dx ∧ dy \\
(- & ∂_z^2 A^y & + & ∂_y ∂_z A^z & ) \; dt ∧ dz ∧ dx \\
(+ & ∂_t^2 A^x & + & ∂_t ∂_x A^t & ) \; dt ∧ dy ∧ dz \\
(+ & ∂_x^2 A^t & + & ∂_t ∂_x A^x & ) \; dx ∧ dy ∧ dz \\
\end{alignedat} \right]\end{split}\]
Rearange
\[\begin{split}d ⋆ d T^♭ = Π d \left[ \begin{alignedat}{4}
(+ & ∂_x^2 A^t & + & ∂_t ∂_x A^x & ) \; dx ∧ dy ∧ dz \\
(+ & ∂_t^2 A^x & + & ∂_t ∂_x A^t & ) \; dt ∧ dy ∧ dz \\
(- & ∂_z^2 A^y & + & ∂_y ∂_z A^z & ) \; dt ∧ dz ∧ dx \\
(- & ∂_y^2 A^z & + & ∂_y ∂_z A^y & ) \; dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Expand permutations
\[\begin{split}d ⋆ d T^♭ = \left[ \begin{alignedat}{4}
(+ & ∂_x^2 A^t & + & ∂_t ∂_x A^x & ) \; dx ∧ dy ∧ dz \\
(+ & ∂_y^2 A^t & + & ∂_t ∂_y A^y & ) \; dx ∧ dz ∧ dx \\
(+ & ∂_z^2 A^t & + & ∂_t ∂_z A^z & ) \; dx ∧ dx ∧ dy \\
%
(+ & ∂_t^2 A^x & + & ∂_t ∂_x A^t & ) \; dt ∧ dy ∧ dz \\
(+ & ∂_t^2 A^y & + & ∂_t ∂_y A^t & ) \; dt ∧ dz ∧ dx \\
(+ & ∂_t^2 A^z & + & ∂_t ∂_z A^t & ) \; dt ∧ dx ∧ dy \\
%
(- & ∂_z^2 A^y & + & ∂_y ∂_z A^z & ) \; dt ∧ dz ∧ dx \\
(- & ∂_x^2 A^z & + & ∂_z ∂_x A^x & ) \; dt ∧ dx ∧ dy \\
(- & ∂_y^2 A^x & + & ∂_x ∂_y A^y & ) \; dt ∧ dy ∧ dz \\
%
(- & ∂_y^2 A^z & + & ∂_y ∂_z A^y & ) \; dt ∧ dx ∧ dy \\
(- & ∂_z^2 A^x & + & ∂_z ∂_x A^z & ) \; dt ∧ dy ∧ dz \\
(- & ∂_x^2 A^y & + & ∂_x ∂_y A^x & ) \; dt ∧ dz ∧ dx \\
\end{alignedat} \right]\end{split}\]
Simplify and conclude
\[\begin{split}d ⋆ d T^♭ = \left[ \begin{alignedat}{7}
( & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & + ∂_t ∂_x A^x & + ∂_t ∂_y A^y & + ∂_t ∂_z A^z & ) \; dx ∧ dx ∧ dy \\
( & + ∂_t^2 A^x & - ∂_y^2 A^x & - ∂_z^2 A^x & + ∂_t ∂_x A^t & + ∂_x ∂_y A^y & + ∂_z ∂_x A^z & ) \; dt ∧ dy ∧ dz \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & - ∂_z^2 A^y & + ∂_t ∂_y A^t & + ∂_y ∂_z A^z & + ∂_x ∂_y A^x & ) \; dt ∧ dz ∧ dx \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & + ∂_t ∂_z A^t & + ∂_z ∂_x A^x & + ∂_y ∂_z A^y & ) \; dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
\(d⋆d⋆ T^♭\)#
\[\begin{split} d⋆d⋆ T^♭ = \left[ \begin{alignedat}{7}
( & ∂_t ∂_x A^x & \: + \: & ∂_t ∂_y A^y & \: + \: & ∂_t ∂_z A^z & ) & \; dt \\
( & ∂_t ∂_x A^t & \: + \: & ∂_x ∂_y A^y & \: + \: & ∂_z ∂_x A^z & ) & \; dx \\
( & ∂_t ∂_y A^t & \: + \: & ∂_y ∂_z A^z & \: + \: & ∂_x ∂_y A^x & ) & \; dy \\
( & ∂_t ∂_z A^t & \: + \: & ∂_z ∂_x A^x & \: + \: & ∂_y ∂_z A^y & ) & \; dz \\
\end{alignedat} \right]\end{split}\]
Calculations
Apply the exterior derivative
\[d ⋆ d ⋆ T^♭ = d (∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z)\]
Reorder
\[\begin{split} d ⋆ d ⋆ T^♭ = \left[ \begin{alignedat}{7}
d \: & (∂_t A^t & ) \\
d \: & (∂_x A^x & ) \\
d \: & (∂_y A^y & ) \\
d \: & (∂_z A^z & ) \\
\end{alignedat} \right]\end{split}\]
Apply the exterior derivative
\[\begin{split} d ⋆ d ⋆ T^♭ = \left[ \begin{alignedat}{7}
∂_x ∂_t A^t dx & \: + \: & ∂_y ∂_t A^t dy & \: + \: & ∂_z ∂_t A^t dz \\
∂_t ∂_x A^x dt & \: + \: & ∂_y ∂_x A^x dy & \: + \: & ∂_z ∂_x A^x dz \\
∂_t ∂_y A^y dt & \: + \: & ∂_x ∂_y A^y dx & \: + \: & ∂_z ∂_y A^y dz \\
∂_t ∂_z A^z dt & \: + \: & ∂_x ∂_z A^z dx & \: + \: & ∂_y ∂_z A^z dy \\
\end{alignedat} \right]\end{split}\]
Reorder, simplify and conclude
\[\begin{split} d ⋆ d ⋆ T^♭ = \left[ \begin{alignedat}{7}
( & ∂_t ∂_x A^x & \: + \: & ∂_t ∂_y A^y & \: + \: & ∂_t ∂_z A^z & ) & \; dt \\
( & ∂_x ∂_t A^t & \: + \: & ∂_x ∂_y A^y & \: + \: & ∂_x ∂_z A^z & ) & \; dx \\
( & ∂_y ∂_t A^t & \: + \: & ∂_y ∂_z A^z & \: + \: & ∂_y ∂_x A^x & ) & \; dy \\
( & ∂_z ∂_t A^t & \: + \: & ∂_z ∂_x A^x & \: + \: & ∂_z ∂_y A^y & ) & \; dz \\
\end{alignedat} \right]\end{split}\]
\(⋆d⋆d T^♭\)#
\[\begin{split}⋆d⋆d T^♭ = \left[ \begin{alignedat}{7}
( & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & + ∂_t ∂_x A^x & + ∂_t ∂_y A^y & + ∂_t ∂_z A^z & ) \; dt \\
( & + ∂_t^2 A^x & - ∂_y^2 A^x & - ∂_z^2 A^x & + ∂_t ∂_x A^t & + ∂_x ∂_y A^y & + ∂_z ∂_x A^z & ) \; dx \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & - ∂_z^2 A^y & + ∂_t ∂_y A^t & + ∂_y ∂_z A^z & + ∂_x ∂_y A^x & ) \; dy \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & + ∂_t ∂_z A^t & + ∂_z ∂_x A^x & + ∂_y ∂_z A^y & ) \; dz \\
\end{alignedat} \right]\end{split}\]
Calculations
Apply the Hodge star to all
\[\begin{split}⋆ d ⋆ d T^♭ = ⋆ \left[ \begin{alignedat}{7}
( & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & + ∂_t ∂_x A^x & + ∂_t ∂_y A^y & + ∂_t ∂_z A^z & ) \; dx ∧ dx ∧ dy \\
( & + ∂_t^2 A^x & - ∂_y^2 A^x & - ∂_z^2 A^x & + ∂_t ∂_x A^t & + ∂_x ∂_y A^y & + ∂_z ∂_x A^z & ) \; dt ∧ dy ∧ dz \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & - ∂_z^2 A^y & + ∂_t ∂_y A^t & + ∂_y ∂_z A^z & + ∂_x ∂_y A^x & ) \; dt ∧ dz ∧ dx \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & + ∂_t ∂_z A^t & + ∂_z ∂_x A^x & + ∂_y ∂_z A^y & ) \; dt ∧ dx ∧ dy \\
\end{alignedat} \right]\end{split}\]
Apply the Hodge star to each
\[\begin{split}⋆ d ⋆ d T^♭ = \left[ \begin{alignedat}{7}
( & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & + ∂_t ∂_x A^x & + ∂_t ∂_y A^y & + ∂_t ∂_z A^z & ) \; ⋆ (dx ∧ dx ∧ dy) \\
( & + ∂_t^2 A^x & - ∂_y^2 A^x & - ∂_z^2 A^x & + ∂_t ∂_x A^t & + ∂_x ∂_y A^y & + ∂_z ∂_x A^z & ) \; ⋆ (dt ∧ dy ∧ dz) \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & - ∂_z^2 A^y & + ∂_t ∂_y A^t & + ∂_y ∂_z A^z & + ∂_x ∂_y A^x & ) \; ⋆ (dt ∧ dz ∧ dx) \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & + ∂_t ∂_z A^t & + ∂_z ∂_x A^x & + ∂_y ∂_z A^y & ) \; ⋆ (dt ∧ dx ∧ dy) \\
\end{alignedat} \right]\end{split}\]
Conclude
\[\begin{split}⋆ d ⋆ d T^♭ = \left[ \begin{alignedat}{7}
( & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & + ∂_t ∂_x A^x & + ∂_t ∂_y A^y & + ∂_t ∂_z A^z & ) \; dt \\
( & + ∂_t^2 A^x & - ∂_y^2 A^x & - ∂_z^2 A^x & + ∂_t ∂_x A^t & + ∂_x ∂_y A^y & + ∂_z ∂_x A^z & ) \; dx \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & - ∂_z^2 A^y & + ∂_t ∂_y A^t & + ∂_y ∂_z A^z & + ∂_x ∂_y A^x & ) \; dy \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & + ∂_t ∂_z A^t & + ∂_z ∂_x A^x & + ∂_y ∂_z A^y & ) \; dz \\
\end{alignedat} \right]\end{split}\]
\((⋆d⋆d - d⋆d⋆) T^♭\)#
\[\begin{split}(⋆ d ⋆ d - d ⋆ d ⋆) T^♭ = \left[ \begin{alignedat}{7}
( & & + ∂_x^2 A^t & + ∂_y^2 A^t & + ∂_z^2 A^t & ) \; dt \\
( & + ∂_t^2 A^x & & - ∂_y^2 A^x & - ∂_z^2 A^x & ) \; dx \\
( & + ∂_t^2 A^y & - ∂_x^2 A^y & & - ∂_z^2 A^y & ) \; dy \\
( & + ∂_t^2 A^z & - ∂_x^2 A^z & - ∂_y^2 A^z & & ) \; dz \\
\end{alignedat} \right]\end{split}\]