4–forms (4D volumes)#
This article is part of a series of systematic calculations of combinations of the Hodge star operator and the exterior derivative.
Here, we perform these calculations for a 4-form. A generic 4-form is denoted \(G^{♭♭♭♭} = \; g \; dt ∧ dx ∧ dy ∧ dz\).
\(⋆G^{♭♭♭♭}\)#
Applying the Hodge operator results directly in:
\[⋆ G^{♭♭♭♭} = - g\]
\(dG^{♭♭♭♭}\)#
Applying the exterior derivative results directly in:
\[d G^{♭♭♭♭} = 0\]
\(d⋆G^{♭♭♭♭}\)#
Applying the exterior derivative results directly in:
\[d ⋆ G^{♭♭♭♭} = - ∂_t G dt - ∂_x G dx - ∂_y G dy - ∂_z G dz\]
\(⋆dG^{♭♭♭♭}\)#
Applying the Hodge star results directly in:
\[⋆ d G^{♭♭♭♭} = 0\]
\(⋆d⋆G^{♭♭♭♭}\)#
Applying the Hodge star results directly in:
\[\begin{split}⋆ d ⋆ G^{♭♭♭♭} = \left[ \begin{aligned}
- ∂_t G dx ∧ dy ∧ dz \\
- ∂_x G dt ∧ dy ∧ dz \\
- ∂_y G dt ∧ dz ∧ dx \\
- ∂_z G dt ∧ dx ∧ dy \\
\end{aligned} \right]\end{split}\]
\(d⋆dG^{♭♭♭♭}\)#
Applying the exterior derivative results directly in:
\[d ⋆ d G^{♭♭♭♭} = 0\]