Rotations in Minkowski space

Rotations in Minkowski space#

by Stéphane Haussler

On this page, I systematically explore the geometric differential representation of rotations in minkowski spacetime. Rotations are first expressed as linear combination of rotation planes with bivectors. The dual differential 2-forms are then derived, as well as the mixed tensor representation in form of matrices.

We work out the Lie algebra of the Lorentz group as well as the full matrix representation of rotations. The mixed wedge products are fully expressed in terms of tensor products and their symmetries highlighted.

The \(♯♯\) rotation tensor#

General rotations can be expressed as linear combinations of rotations in each planes. For Minkowski space with 4 dimensions, this result in 6 planes of rotations. Each plance is expressed as the wedge product \(∧\) of basis vectors as \(∂_x ∧ ∂_y\). A rotation in spacetime is represented by the linear combination of all basis rotations in all six available planes.

The sharp symbol \(\sharp\) indicates that the components are doubly contravariant tensor components. Using the antisymmetric property of the exterior product \(∂_μ ∧ ∂_ν = - ∂_ν ∧ ∂_μ\), all terms can be rewritten as \(∂_μ ∧ ∂_ν = \frac{1}{2}( ∂_μ ∧ ∂_ν - ∂_ν ∧ ∂_μ)\):

\[\begin{split}R^{♯♯} = \begin{bmatrix} a \; \frac{1}{2} (∂_t ∧ ∂_x - ∂_x ∧ ∂_t) \\ b \; \frac{1}{2} (∂_t ∧ ∂_y - ∂_y ∧ ∂_t) \\ c \; \frac{1}{2} (∂_t ∧ ∂_z - ∂_z ∧ ∂_t) \\ d \; \frac{1}{2} (∂_y ∧ ∂_z - ∂_z ∧ ∂_y) \\ e \; \frac{1}{2} (∂_z ∧ ∂_x - ∂_x ∧ ∂_z) \\ f \; \frac{1}{2} (∂_x ∧ ∂_y - ∂_y ∧ ∂_x) \\ \end{bmatrix}\end{split}\]

With minor rearangement, we get:

\[\begin{split}R^{♯♯} = \frac{1}{2} \begin{bmatrix} a \; ∂_t ∧ ∂_x - a \; ∂_x ∧ ∂_t) \\ b \; ∂_t ∧ ∂_y - b \; ∂_y ∧ ∂_t) \\ c \; ∂_t ∧ ∂_z - c \; ∂_z ∧ ∂_t) \\ d \; ∂_y ∧ ∂_z - d \; ∂_z ∧ ∂_y) \\ e \; ∂_z ∧ ∂_x - e \; ∂_x ∧ ∂_z) \\ f \; ∂_x ∧ ∂_y - f \; ∂_y ∧ ∂_x) \\ \end{bmatrix}\end{split}\]

The free matrix representation permits to state the obvious, in that we can rewrite the rotation above utilizing a row/column matrix representation:

The \(♭♭\) rotation tensor#

Using the Minkowski metric, we flatten a basis vector with the flat operator \(♭\):

\[(∂_μ)^♭ = η_{μν} dx^ν\]

Likewise any index of the doubly contravariant exterior product can be flattened:

\[\begin{split}\begin{matrix} (∂_μ ∧ ∂_ν)^{♭♯} &= η_{γμ} dx^γ ∧ ∂_ν \\ (∂_μ ∧ ∂_ν)^{♯♭} &= η_{γν} ∂_μ ∧ dx^γ \\ (∂_μ ∧ ∂_ν)^{♭♭} &= η_{δμ} η_{γν} dx^δ ∧ dx^γ \\ \end{matrix}\end{split}\]

To obtain the doubly covariant representation of rotations in spacetime, we apply the flat operators \(♭♭\) to each components \(R^{♭♭} = (R^{♯♯})^{♭♭}\):

Calculations

Apply the flat operators

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

Distribute the musical operators

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

Expand

\[\begin{split}R^{♭♭} = \begin{bmatrix} a \; η_{tμ} d^μ ∧ η_{xμ} d^μ \\ b \; η_{tμ} d^μ ∧ η_{yμ} d^μ \\ c \; η_{tμ} d^μ ∧ η_{zμ} d^μ \\ d \; η_{yμ} d^μ ∧ η_{zμ} d^μ \\ e \; η_{zμ} d^μ ∧ η_{xμ} d^μ \\ f \; η_{xμ} d^μ ∧ η_{yμ} d^μ \\ \end{bmatrix}\end{split}\]

Identify non-zero terms

\[\begin{split}R^{♭♭} = \begin{bmatrix} a \; η_{tt} dt ∧ η_{xx} dx \\ b \; η_{tt} dt ∧ η_{yy} dy \\ c \; η_{tt} dt ∧ η_{zz} dz \\ d \; η_{yy} dy ∧ η_{zz} dz \\ e \; η_{zz} dz ∧ η_{xx} dx \\ f \; η_{xx} dx ∧ η_{yy} dy \\ \end{bmatrix}\end{split}\]

Apply numerical values

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

The \(♭♯\) rotation tensor#

In this section, I flattne the first component using the free matrix representation. The mixed tensor is obtained by applying the flatternig operator \(\flat\):

\[\begin{split}R^{♭♯} = \begin{bmatrix} a \; ∂_t ∧ ∂_x \\ b \; ∂_t ∧ ∂_y \\ c \; ∂_t ∧ ∂_z \\ d \; ∂_y ∧ ∂_z \\ e \; ∂_z ∧ ∂_x \\ f \; ∂_x ∧ ∂_y \\ \end{bmatrix}^{♭♯} = \begin{bmatrix} + a \; dx^x ∧ ∂_t \\ + b \; dx^y ∧ ∂_t \\ + c \; dx^z ∧ ∂_t \\ - d \; dx^z ∧ ∂_y \\ - e \; dx^x ∧ ∂_z \\ - f \; dx^y ∧ ∂_x \\ \end{bmatrix}\end{split}\]

Calculations

Apply the musical operators

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

Distribute the musical operators to each matrix elements

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

Distribute the musical operators

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

Apply the musical operators

\[\begin{split}B^{♭♯} = \begin{bmatrix} a \; η_{tγ} dx^γ ∧ ∂_x^♯ \\ b \; η_{tγ} dx^γ ∧ ∂_y^♯ \\ c \; η_{tγ} dx^γ ∧ ∂_z^♯ \\ d \; η_{yγ} dx^γ ∧ ∂_z^♯ \\ e \; η_{zγ} dx^γ ∧ ∂_x^♯ \\ f \; η_{xγ} dx^γ ∧ ∂_y^♯ \\ \end{bmatrix}\end{split}\]

Identify the non-zero terms of the Minkowski metric

\[\begin{split}B^{♭♯} = \begin{bmatrix} a \; η_{tt} dx^t ∧ ∂_x \\ b \; η_{tt} dx^t ∧ ∂_y \\ c \; η_{tt} dx^t ∧ ∂_z \\ d \; η_{yy} dx^y ∧ ∂_z \\ e \; η_{zz} dx^z ∧ ∂_x \\ f \; η_{xx} dx^x ∧ ∂_y \\ \end{bmatrix}\end{split}\]

Use the numerical values of the Minkowski metric

\[\begin{split}B^{♭♯} = \begin{bmatrix} + a \; dx^t ∧ ∂_x \\ + b \; dx^t ∧ ∂_y \\ + c \; dx^t ∧ ∂_z \\ - d \; dx^y ∧ ∂_z \\ - e \; dx^z ∧ ∂_x \\ - f \; dx^x ∧ ∂_y \\ \end{bmatrix}\end{split}\]

Taking into account the symetric property of \(dx^t ∧ ∂_x\), \(dx^t ∧ ∂_y\), and \(dx^t ∧ ∂_z\), as well the antisymetric property of \(dx^x ∧ ∂_y\), \(dx^y ∧ ∂_z\), and \(dx^z ∧ ∂_x\) demonstrated above, this results in:

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

The \(♯♭\) rotation tensor#

In this section, I raise the indice using the free matrix notaion. The mixed tensor is obtained by applying the flatternig operator \(\flat\):

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

Calculations

Apply the musical operators

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

Distribute the musical operators to each matrix elements

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

Distribute the musical operators

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

Apply and expand

\[\begin{split}B^{♯♭} = \begin{bmatrix} a \; ∂_t ∧ η_{xγ} dx^γ \\ b \; ∂_t ∧ η_{yγ} dx^γ \\ c \; ∂_t ∧ η_{zγ} dx^γ \\ d \; ∂_y ∧ η_{zγ} dx^γ \\ e \; ∂_z ∧ η_{xγ} dx^γ \\ f \; ∂_x ∧ η_{yγ} dx^γ \\ \end{bmatrix}\end{split}\]

The metric tensor can be taken out due to mulilinearity

\[\begin{split}B^{♯♭} = \begin{bmatrix} a \; η_{xγ} ∂_t ∧ dx^γ \\ b \; η_{yγ} ∂_t ∧ dx^γ \\ c \; η_{zγ} ∂_t ∧ dx^γ \\ d \; η_{zγ} ∂_y ∧ dx^γ \\ e \; η_{xγ} ∂_z ∧ dx^γ \\ f \; η_{yγ} ∂_x ∧ dx^γ \\ \end{bmatrix}\end{split}\]

Most terms of the Minkowski metric are zero

\[\begin{split}R^{♯♭} = \begin{bmatrix} a \; η_{xx} ∂_t ∧ dx^x \\ b \; η_{yy} ∂_t ∧ dx^y \\ c \; η_{zz} ∂_t ∧ dx^z \\ d \; η_{zz} ∂_y ∧ dx^z \\ e \; η_{xx} ∂_z ∧ dx^x \\ f \; η_{yy} ∂_x ∧ dx^y \\ \end{bmatrix}\end{split}\]

Use the numerical values of the Minkowski metric

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

Taking into account the symetric property of \(∂_t ∧ dx^x\), \(∂_t ∧ dx^y\), and \(∂_t ∧ dx^z\), as well the antisymetric property of \(∂_x ∧ dx^y\), \(∂_ey ∧ dx^z\), and \(∂_z ∧ dx^x\) demonstrated above, this results in:

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

Symmetries of rotations in \(♭♯\) form#

The purpose here is to determine the symmetries of the mixed exterior product. Calculations are tedious, but permit to verify that everything works as it should as the quantities are encountered when Deriving the Faraday Tensor from the 1865 Maxwell’s Equations. The discussion is often avoided, but it is nice to settle it. This is important when performing matrix multiplications since per convention, matrices are \(♯♭\) tensors organized in tables following the row-column convention. This is not critical when using Free matrix representation, but permits to fall back to this familiar framework.

Applying the \(♭♯\) operators to flatten the first index of each basis bivectors, we obtain:

\[\begin{split}\begin{alignedat}{4} (∂_t ∧ ∂_x)^{♭♯} =& + dt ∧ ∂_x &\qquad& (∂_x ∧ ∂_t)^{♭♯} =& - dx ∧ ∂_t \\ (∂_t ∧ ∂_y)^{♭♯} =& + dt ∧ ∂_y &\qquad& (∂_y ∧ ∂_t)^{♭♯} =& - dy ∧ ∂_t \\ (∂_t ∧ ∂_z)^{♭♯} =& + dt ∧ ∂_z &\qquad& (∂_z ∧ ∂_t)^{♭♯} =& - dz ∧ ∂_t \\ (∂_y ∧ ∂_z)^{♭♯} =& - dy ∧ ∂_z &\qquad& (∂_y ∧ ∂_x)^{♭♯} =& - dy ∧ ∂_x \\ (∂_z ∧ ∂_x)^{♭♯} =& - dz ∧ ∂_x &\qquad& (∂_z ∧ ∂_y)^{♭♯} =& - dz ∧ ∂_y \\ (∂_x ∧ ∂_y)^{♭♯} =& - dx ∧ ∂_y &\qquad& (∂_x ∧ ∂_z)^{♭♯} =& - dx ∧ ∂_z \\ \end{alignedat}\end{split}\]

Calculations

Distribute musical operators

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♭♯} &= (∂_t^♭ ∧ ∂_x^♯) &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& (∂_x^♭ ∧ ∂_t^♯) \\ (∂_t ∧ ∂_y)^{♭♯} &= (∂_t^♭ ∧ ∂_y^♯) &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& (∂_y^♭ ∧ ∂_t^♯) \\ (∂_t ∧ ∂_z)^{♭♯} &= (∂_t^♭ ∧ ∂_z^♯) &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& (∂_z^♭ ∧ ∂_t^♯) \\ (∂_x ∧ ∂_y)^{♭♯} &= (∂_x^♭ ∧ ∂_y^♯) &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& (∂_y^♭ ∧ ∂_x^♯) \\ (∂_y ∧ ∂_z)^{♭♯} &= (∂_y^♭ ∧ ∂_z^♯) &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& (∂_z^♭ ∧ ∂_y^♯) \\ (∂_z ∧ ∂_x)^{♭♯} &= (∂_z^♭ ∧ ∂_x^♯) &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& (∂_x^♭ ∧ ∂_z^♯) \\ \end{alignedat}\end{split}\]

Apply musical operators

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♭♯} &= η_{tγ} dx^γ ∧ ∂_x &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& η_{xγ} dx^γ ∧ ∂_t \\ (∂_t ∧ ∂_y)^{♭♯} &= η_{tγ} dx^γ ∧ ∂_y &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& η_{yγ} dx^γ ∧ ∂_t \\ (∂_t ∧ ∂_z)^{♭♯} &= η_{tγ} dx^γ ∧ ∂_z &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& η_{zγ} dx^γ ∧ ∂_t \\ (∂_x ∧ ∂_y)^{♭♯} &= η_{xγ} dx^γ ∧ ∂_y &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& η_{yγ} dx^γ ∧ ∂_x \\ (∂_y ∧ ∂_z)^{♭♯} &= η_{yγ} dx^γ ∧ ∂_z &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& η_{zγ} dx^γ ∧ ∂_y \\ (∂_z ∧ ∂_x)^{♭♯} &= η_{zγ} dx^γ ∧ ∂_x &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& η_{xγ} dx^γ ∧ ∂_z \\ \end{alignedat}\end{split}\]

Identify non-zero elements

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♭♯} &= η_{tt} dx^t ∧ ∂_x &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& η_{xx} dx^x ∧ ∂_t \\ (∂_t ∧ ∂_y)^{♭♯} &= η_{tt} dx^t ∧ ∂_y &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& η_{yy} dx^y ∧ ∂_t \\ (∂_t ∧ ∂_z)^{♭♯} &= η_{tt} dx^t ∧ ∂_z &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& η_{zz} dx^z ∧ ∂_t \\ (∂_x ∧ ∂_y)^{♭♯} &= η_{xx} dx^x ∧ ∂_y &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& η_{yy} dx^y ∧ ∂_x \\ (∂_y ∧ ∂_z)^{♭♯} &= η_{yy} dx^y ∧ ∂_z &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& η_{zz} dx^z ∧ ∂_y \\ (∂_z ∧ ∂_x)^{♭♯} &= η_{zz} dx^z ∧ ∂_x &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& η_{xx} dx^x ∧ ∂_z \\ \end{alignedat}\end{split}\]

Apply numerical values

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♭♯} &= + dt ∧ ∂_x &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& - dx ∧ ∂_t \\ (∂_t ∧ ∂_y)^{♭♯} &= + dt ∧ ∂_y &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& - dy ∧ ∂_t \\ (∂_t ∧ ∂_z)^{♭♯} &= + dt ∧ ∂_z &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& - dz ∧ ∂_t \\ (∂_x ∧ ∂_y)^{♭♯} &= - dx ∧ ∂_y &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& - dy ∧ ∂_x \\ (∂_y ∧ ∂_z)^{♭♯} &= - dy ∧ ∂_z &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& - dz ∧ ∂_y \\ (∂_z ∧ ∂_x)^{♭♯} &= - dz ∧ ∂_x &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& - dx ∧ ∂_z \\ \end{alignedat}\end{split}\]

We can then identify the expressions for the mixed wedge product explicitely in terms of tensor products:

\[\begin{split}\begin{alignedat}{8} (∂_t ∧ ∂_x)^{♭♯} =& + dt ⊗ ∂_x &+& dx ⊗ ∂_t &\qquad& (∂_x ∧ ∂_t)^{♭♯} =& - dx ⊗ ∂_t &-& dt ⊗ ∂_x \\ (∂_t ∧ ∂_y)^{♭♯} =& + dt ⊗ ∂_y &+& dy ⊗ ∂_t &\qquad& (∂_y ∧ ∂_t)^{♭♯} =& - dy ⊗ ∂_t &-& dt ⊗ ∂_y \\ (∂_t ∧ ∂_z)^{♭♯} =& + dt ⊗ ∂_z &+& dz ⊗ ∂_t &\qquad& (∂_z ∧ ∂_t)^{♭♯} =& - dz ⊗ ∂_t &-& dt ⊗ ∂_z \\ (∂_y ∧ ∂_z)^{♭♯} =& - dy ⊗ ∂_z &+& dz ⊗ ∂_y &\qquad& (∂_z ∧ ∂_y)^{♭♯} =& - dz ⊗ ∂_y &+& dy ⊗ ∂_z \\ (∂_z ∧ ∂_x)^{♭♯} =& - dz ⊗ ∂_x &+& dx ⊗ ∂_z &\qquad& (∂_x ∧ ∂_z)^{♭♯} =& - dx ⊗ ∂_z &+& dz ⊗ ∂_x \\ (∂_x ∧ ∂_y)^{♭♯} =& - dx ⊗ ∂_y &+& dy ⊗ ∂_x &\qquad& (∂_y ∧ ∂_x)^{♭♯} =& - dy ⊗ ∂_x &+& dx ⊗ ∂_y \\ \end{alignedat}\end{split}\]

Calculations

Expand exterior products to their tensor expressions

\[\begin{split}\begin{alignedat}{7} (∂_t ∧ ∂_x)^{♭♯} =& (∂_t ⊗ ∂_x &-& ∂_x ⊗ ∂_t)^{♭♯} &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& (∂_x ⊗ ∂_t &-& ∂_t ⊗ ∂_x)^{♭♯} \\ (∂_t ∧ ∂_y)^{♭♯} =& (∂_t ⊗ ∂_y &-& ∂_y ⊗ ∂_t)^{♭♯} &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& (∂_y ⊗ ∂_t &-& ∂_t ⊗ ∂_y)^{♭♯} \\ (∂_t ∧ ∂_z)^{♭♯} =& (∂_t ⊗ ∂_z &-& ∂_z ⊗ ∂_t)^{♭♯} &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& (∂_z ⊗ ∂_t &-& ∂_t ⊗ ∂_z)^{♭♯} \\ (∂_y ∧ ∂_z)^{♭♯} =& (∂_y ⊗ ∂_z &-& ∂_z ⊗ ∂_y)^{♭♯} &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& (∂_z ⊗ ∂_y &-& ∂_y ⊗ ∂_z)^{♭♯} \\ (∂_z ∧ ∂_x)^{♭♯} =& (∂_z ⊗ ∂_x &-& ∂_x ⊗ ∂_z)^{♭♯} &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& (∂_x ⊗ ∂_z &-& ∂_z ⊗ ∂_x)^{♭♯} \\ (∂_x ∧ ∂_y)^{♭♯} =& (∂_x ⊗ ∂_y &-& ∂_y ⊗ ∂_x)^{♭♯} &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& (∂_y ⊗ ∂_x &-& ∂_x ⊗ ∂_y)^{♭♯} \\ \end{alignedat}\end{split}\]

Distribute musical operators

\[\begin{split}\begin{alignedat}{7} (∂_t ∧ ∂_x)^{♭♯} =& ∂_t^♭ ⊗ ∂_x^♯ - ∂_x^♭ ⊗ ∂_t^♯ &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& ∂_x^♭ ⊗ ∂_t^♯ - ∂_t^♭ ⊗ ∂_x^♯ \\ (∂_t ∧ ∂_y)^{♭♯} =& ∂_t^♭ ⊗ ∂_y^♯ - ∂_y^♭ ⊗ ∂_t^♯ &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& ∂_y^♭ ⊗ ∂_t^♯ - ∂_t^♭ ⊗ ∂_y^♯ \\ (∂_t ∧ ∂_z)^{♭♯} =& ∂_t^♭ ⊗ ∂_z^♯ - ∂_z^♭ ⊗ ∂_t^♯ &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& ∂_z^♭ ⊗ ∂_t^♯ - ∂_t^♭ ⊗ ∂_z^♯ \\ (∂_y ∧ ∂_z)^{♭♯} =& ∂_y^♭ ⊗ ∂_z^♯ - ∂_z^♭ ⊗ ∂_y^♯ &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& ∂_z^♭ ⊗ ∂_y^♯ - ∂_y^♭ ⊗ ∂_z^♯ \\ (∂_z ∧ ∂_x)^{♭♯} =& ∂_z^♭ ⊗ ∂_x^♯ - ∂_x^♭ ⊗ ∂_z^♯ &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& ∂_x^♭ ⊗ ∂_z^♯ - ∂_z^♭ ⊗ ∂_x^♯ \\ (∂_x ∧ ∂_y)^{♭♯} =& ∂_x^♭ ⊗ ∂_y^♯ - ∂_y^♭ ⊗ ∂_x^♯ &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& ∂_y^♭ ⊗ ∂_x^♯ - ∂_x^♭ ⊗ ∂_y^♯ \\ \end{alignedat}\end{split}\]

Apply musical operators

\[\begin{split}\begin{alignedat}{7} (∂_t ∧ ∂_x)^{♭♯} &= η_{tγ} dx^γ ⊗ ∂_x - η_{xγ} dx^γ ⊗ ∂_t &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& η_{xγ} dx^γ ⊗ ∂_t - η_{tγ} dx^γ ⊗ ∂_x \\ (∂_t ∧ ∂_y)^{♭♯} &= η_{tγ} dx^γ ⊗ ∂_y - η_{yγ} dx^γ ⊗ ∂_t &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& η_{yγ} dx^γ ⊗ ∂_t - η_{tγ} dx^γ ⊗ ∂_y \\ (∂_t ∧ ∂_z)^{♭♯} &= η_{tγ} dx^γ ⊗ ∂_z - η_{zγ} dx^γ ⊗ ∂_t &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& η_{zγ} dx^γ ⊗ ∂_t - η_{tγ} dx^γ ⊗ ∂_z \\ (∂_y ∧ ∂_z)^{♭♯} &= η_{yγ} dx^γ ⊗ ∂_z - η_{zγ} dx^γ ⊗ ∂_y &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& η_{zγ} dx^γ ⊗ ∂_y - η_{yγ} dx^γ ⊗ ∂_z \\ (∂_z ∧ ∂_x)^{♭♯} &= η_{zγ} dx^γ ⊗ ∂_x - η_{xγ} dx^γ ⊗ ∂_z &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& η_{xγ} dx^γ ⊗ ∂_z - η_{zγ} dx^γ ⊗ ∂_x \\ (∂_x ∧ ∂_y)^{♭♯} &= η_{xγ} dx^γ ⊗ ∂_y - η_{yγ} dx^γ ⊗ ∂_x &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& η_{yγ} dx^γ ⊗ ∂_x - η_{xγ} dx^γ ⊗ ∂_y \\ \end{alignedat}\end{split}\]

Identify non-zero metric elements

\[\begin{split}\begin{alignedat}{7} (∂_t ∧ ∂_x)^{♭♯} &= η_{tt} dx^t ⊗ ∂_x - η_{xx} dx^x ⊗ ∂_t &\qquad& (∂_x ∧ ∂_t)^{♭♯} &=& η_{xx} dx^x ⊗ ∂_t - η_{tt} dx^t ⊗ ∂_x \\ (∂_t ∧ ∂_y)^{♭♯} &= η_{tt} dx^t ⊗ ∂_y - η_{yy} dx^y ⊗ ∂_t &\qquad& (∂_y ∧ ∂_t)^{♭♯} &=& η_{yy} dx^y ⊗ ∂_t - η_{tt} dx^t ⊗ ∂_y \\ (∂_t ∧ ∂_z)^{♭♯} &= η_{tt} dx^t ⊗ ∂_z - η_{zz} dx^z ⊗ ∂_t &\qquad& (∂_z ∧ ∂_t)^{♭♯} &=& η_{zz} dx^z ⊗ ∂_t - η_{tt} dx^t ⊗ ∂_z \\ (∂_y ∧ ∂_z)^{♭♯} &= η_{yy} dx^y ⊗ ∂_z - η_{zz} dx^z ⊗ ∂_y &\qquad& (∂_z ∧ ∂_y)^{♭♯} &=& η_{zz} dx^z ⊗ ∂_y - η_{yy} dx^y ⊗ ∂_z \\ (∂_z ∧ ∂_x)^{♭♯} &= η_{zz} dx^z ⊗ ∂_x - η_{xx} dx^x ⊗ ∂_z &\qquad& (∂_x ∧ ∂_z)^{♭♯} &=& η_{xx} dx^x ⊗ ∂_z - η_{zz} dx^z ⊗ ∂_x \\ (∂_x ∧ ∂_y)^{♭♯} &= η_{xx} dx^x ⊗ ∂_y - η_{yy} dx^y ⊗ ∂_x &\qquad& (∂_y ∧ ∂_x)^{♭♯} &=& η_{yy} dx^y ⊗ ∂_x - η_{xx} dx^x ⊗ ∂_y \\ \end{alignedat}\end{split}\]

Apply numerical values

\[\begin{split}\begin{alignedat}{6} (∂_t ∧ ∂_x)^{♭♯} & = + dt ⊗ ∂_x & + dx ⊗ ∂_t & \qquad & (∂_x ∧ ∂_t)^{♭♯} & = & - dx ⊗ ∂_t &-& dt ⊗ ∂_x \\ (∂_t ∧ ∂_y)^{♭♯} & = + dt ⊗ ∂_y & + dy ⊗ ∂_t & \qquad & (∂_y ∧ ∂_t)^{♭♯} & = & - dy ⊗ ∂_t &-& dt ⊗ ∂_y \\ (∂_t ∧ ∂_z)^{♭♯} & = + dt ⊗ ∂_z & + dz ⊗ ∂_t & \qquad & (∂_z ∧ ∂_t)^{♭♯} & = & - dz ⊗ ∂_t &-& dt ⊗ ∂_z \\ (∂_y ∧ ∂_z)^{♭♯} & = - dy ⊗ ∂_z & + dz ⊗ ∂_y & \qquad & (∂_z ∧ ∂_y)^{♭♯} & = & - dz ⊗ ∂_y &+& dy ⊗ ∂_z \\ (∂_z ∧ ∂_x)^{♭♯} & = - dz ⊗ ∂_x & + dx ⊗ ∂_z & \qquad & (∂_x ∧ ∂_z)^{♭♯} & = & - dx ⊗ ∂_z &+& dz ⊗ ∂_x \\ (∂_x ∧ ∂_y)^{♭♯} & = - dx ⊗ ∂_y & + dy ⊗ ∂_x & \qquad & (∂_y ∧ ∂_x)^{♭♯} & = & - dy ⊗ ∂_x &+& dx ⊗ ∂_y \\ \end{alignedat}\end{split}\]

We can then identify the expressions for the mixed wedge product explicitely in terms of tensor products:

\[\begin{split}\begin{alignedat}{9} dt ∧ ∂_x = & + dt ⊗ ∂_x & + & dx ⊗ ∂_t & \qquad & dx ∧ ∂_t & = & + dt ⊗ ∂_x & + & dx ⊗ ∂_t \\ dt ∧ ∂_y = & + dt ⊗ ∂_y & + & dy ⊗ ∂_t & \qquad & dy ∧ ∂_t & = & + dt ⊗ ∂_y & + & dy ⊗ ∂_t \\ dt ∧ ∂_z = & + dt ⊗ ∂_z & + & dz ⊗ ∂_t & \qquad & dz ∧ ∂_t & = & + dt ⊗ ∂_z & + & dz ⊗ ∂_t \\ dy ∧ ∂_z = & + dy ⊗ ∂_z & - & dz ⊗ ∂_y & \qquad & dz ∧ ∂_y & = & - dy ⊗ ∂_z & + & dz ⊗ ∂_y \\ dz ∧ ∂_x = & + dz ⊗ ∂_x & - & dx ⊗ ∂_z & \qquad & dx ∧ ∂_z & = & - dz ⊗ ∂_x & + & dx ⊗ ∂_z \\ dx ∧ ∂_y = & + dx ⊗ ∂_y & - & dy ⊗ ∂_x & \qquad & dy ∧ ∂_x & = & - dx ⊗ ∂_y & + & dy ⊗ ∂_x \\ \end{alignedat}\end{split}\]

Taken together, we get:

Symmetries of rotations in \(♯♭\) form#

The calculations in this section repeat the calculations of the previous sections. The results servers as a test with respect to the former calculations as the results should be fully consistent. We show this is indeed the case. We apply the \(♯♭\) operators to flatten the second index of each basis bivectors and obtain:

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} & = - ∂_t ∧ dx & \qquad & (∂_x ∧ ∂_t)^{♯♭} & = & + ∂_x ∧ dt \\ (∂_t ∧ ∂_y)^{♯♭} & = - ∂_t ∧ dy & \qquad & (∂_y ∧ ∂_t)^{♯♭} & = & + ∂_y ∧ dt \\ (∂_t ∧ ∂_z)^{♯♭} & = - ∂_t ∧ dz & \qquad & (∂_z ∧ ∂_t)^{♯♭} & = & + ∂_z ∧ dt \\ (∂_y ∧ ∂_z)^{♯♭} & = - ∂_y ∧ dz & \qquad & (∂_z ∧ ∂_y)^{♯♭} & = & - ∂_z ∧ dy \\ (∂_z ∧ ∂_x)^{♯♭} & = - ∂_z ∧ dx & \qquad & (∂_x ∧ ∂_z)^{♯♭} & = & - ∂_x ∧ dz \\ (∂_x ∧ ∂_y)^{♯♭} & = - ∂_x ∧ dy & \qquad & (∂_y ∧ ∂_x)^{♯♭} & = & - ∂_y ∧ dx \\ \end{alignedat}\end{split}\]

Calculations

Distribute the musical operators

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= ∂_t^♯ ∧ ∂_x^♭ &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& ∂_x^♯ ∧ ∂_t^♭ \\ (∂_t ∧ ∂_y)^{♯♭} &= ∂_t^♯ ∧ ∂_y^♭ &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& ∂_y^♯ ∧ ∂_t^♭ \\ (∂_t ∧ ∂_z)^{♯♭} &= ∂_t^♯ ∧ ∂_z^♭ &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& ∂_z^♯ ∧ ∂_t^♭ \\ (∂_y ∧ ∂_z)^{♯♭} &= ∂_y^♯ ∧ ∂_z^♭ &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& ∂_z^♯ ∧ ∂_y^♭ \\ (∂_z ∧ ∂_x)^{♯♭} &= ∂_z^♯ ∧ ∂_x^♭ &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& ∂_x^♯ ∧ ∂_z^♭ \\ (∂_x ∧ ∂_y)^{♯♭} &= ∂_x^♯ ∧ ∂_y^♭ &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& ∂_y^♯ ∧ ∂_x^♭ \\ \end{alignedat}\end{split}\]

Apply the musical operators

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= ∂_t ∧ η_{xγ} dx^γ &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& ∂_x ∧ η_{γt} dx^γ \\ (∂_t ∧ ∂_y)^{♯♭} &= ∂_t ∧ η_{yγ} dx^γ &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& ∂_y ∧ η_{γt} dx^γ \\ (∂_t ∧ ∂_z)^{♯♭} &= ∂_t ∧ η_{zγ} dx^γ &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& ∂_z ∧ η_{γt} dx^γ \\ (∂_y ∧ ∂_z)^{♯♭} &= ∂_y ∧ η_{zγ} dx^γ &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& ∂_z ∧ η_{γy} dx^γ \\ (∂_z ∧ ∂_x)^{♯♭} &= ∂_z ∧ η_{xγ} dx^γ &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& ∂_x ∧ η_{γz} dx^γ \\ (∂_x ∧ ∂_y)^{♯♭} &= ∂_x ∧ η_{yγ} dx^γ &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& ∂_y ∧ η_{γx} dx^γ \\ \end{alignedat}\end{split}\]

Identify the non-zero metric components:

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= η_{xx} ∂_t ∧ dx^x &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& ∂_x ∧ η_{tt} dx^t \\ (∂_t ∧ ∂_y)^{♯♭} &= η_{yy} ∂_t ∧ dx^y &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& ∂_y ∧ η_{tt} dx^t \\ (∂_t ∧ ∂_z)^{♯♭} &= η_{zz} ∂_t ∧ dx^z &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& ∂_z ∧ η_{tt} dx^t \\ (∂_y ∧ ∂_z)^{♯♭} &= η_{zz} ∂_y ∧ dx^z &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& ∂_z ∧ η_{yy} dx^y \\ (∂_z ∧ ∂_x)^{♯♭} &= η_{xx} ∂_z ∧ dx^x &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& ∂_x ∧ η_{zz} dx^z \\ (∂_x ∧ ∂_y)^{♯♭} &= η_{yy} ∂_x ∧ dx^y &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& ∂_y ∧ η_{xx} dx^x \\ \end{alignedat}\end{split}\]

Simplify

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= η_{xx} ∂_t ∧ dx &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& η_{tt} ∂_x ∧ dt \\ (∂_t ∧ ∂_y)^{♯♭} &= η_{yy} ∂_t ∧ dy &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& η_{tt} ∂_y ∧ dt \\ (∂_t ∧ ∂_z)^{♯♭} &= η_{zz} ∂_t ∧ dz &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& η_{tt} ∂_z ∧ dt \\ (∂_y ∧ ∂_z)^{♯♭} &= η_{zz} ∂_y ∧ dz &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& η_{yy} ∂_z ∧ dy \\ (∂_z ∧ ∂_x)^{♯♭} &= η_{xx} ∂_z ∧ dx &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& η_{zz} ∂_x ∧ dz \\ (∂_x ∧ ∂_y)^{♯♭} &= η_{yy} ∂_x ∧ dy &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& η_{xx} ∂_y ∧ dx \\ \end{alignedat}\end{split}\]

Apply numerical values:

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} & = - ∂_t ∧ dx & \qquad & (∂_x ∧ ∂_t)^{♯♭} & = & + ∂_x ∧ dt \\ (∂_t ∧ ∂_y)^{♯♭} & = - ∂_t ∧ dy & \qquad & (∂_y ∧ ∂_t)^{♯♭} & = & + ∂_y ∧ dt \\ (∂_t ∧ ∂_z)^{♯♭} & = - ∂_t ∧ dz & \qquad & (∂_z ∧ ∂_t)^{♯♭} & = & + ∂_z ∧ dt \\ (∂_y ∧ ∂_z)^{♯♭} & = - ∂_y ∧ dz & \qquad & (∂_z ∧ ∂_y)^{♯♭} & = & - ∂_z ∧ dy \\ (∂_z ∧ ∂_x)^{♯♭} & = - ∂_z ∧ dx & \qquad & (∂_x ∧ ∂_z)^{♯♭} & = & - ∂_x ∧ dz \\ (∂_x ∧ ∂_y)^{♯♭} & = - ∂_x ∧ dy & \qquad & (∂_y ∧ ∂_x)^{♯♭} & = & - ∂_y ∧ dx \\ \end{alignedat}\end{split}\]

We can then identify the expressions for the mixed wedge product explicitely in terms of tensor products:

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} & = - ∂_t ⊗ dx - ∂_x ⊗ dt & \qquad & (∂_x ∧ ∂_t)^{♯♭} & = & + ∂_x ⊗ dt + ∂_t ⊗ dx \\ (∂_t ∧ ∂_y)^{♯♭} & = - ∂_t ⊗ dy - ∂_y ⊗ dt & \qquad & (∂_y ∧ ∂_t)^{♯♭} & = & + ∂_y ⊗ dt + ∂_t ⊗ dy \\ (∂_t ∧ ∂_z)^{♯♭} & = - ∂_t ⊗ dz - ∂_z ⊗ dt & \qquad & (∂_z ∧ ∂_t)^{♯♭} & = & + ∂_z ⊗ dt + ∂_t ⊗ dz \\ (∂_y ∧ ∂_z)^{♯♭} & = - ∂_y ⊗ dz + ∂_z ⊗ dy & \qquad & (∂_z ∧ ∂_y)^{♯♭} & = & - ∂_z ⊗ dy + ∂_y ⊗ dz \\ (∂_z ∧ ∂_x)^{♯♭} & = - ∂_z ⊗ dx + ∂_x ⊗ dz & \qquad & (∂_x ∧ ∂_z)^{♯♭} & = & - ∂_x ⊗ dz + ∂_z ⊗ dx \\ (∂_x ∧ ∂_y)^{♯♭} & = - ∂_x ⊗ dy + ∂_y ⊗ dx & \qquad & (∂_y ∧ ∂_x)^{♯♭} & = & - ∂_y ⊗ dx + ∂_x ⊗ dy \\ \end{alignedat}\end{split}\]

Calculations

Expand in terms of tensor product

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= (∂_t ⊗ ∂_x - ∂_x ⊗ ∂_t)^{♯♭} &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& (∂_x ⊗ ∂_t - ∂_t ⊗ ∂_x)^{♯♭} \\ (∂_t ∧ ∂_y)^{♯♭} &= (∂_t ⊗ ∂_y - ∂_y ⊗ ∂_t)^{♯♭} &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& (∂_y ⊗ ∂_t - ∂_t ⊗ ∂_y)^{♯♭} \\ (∂_t ∧ ∂_z)^{♯♭} &= (∂_t ⊗ ∂_z - ∂_z ⊗ ∂_t)^{♯♭} &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& (∂_z ⊗ ∂_t - ∂_t ⊗ ∂_z)^{♯♭} \\ (∂_y ∧ ∂_z)^{♯♭} &= (∂_y ⊗ ∂_z - ∂_z ⊗ ∂_y)^{♯♭} &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& (∂_z ⊗ ∂_y - ∂_y ⊗ ∂_z)^{♯♭} \\ (∂_z ∧ ∂_x)^{♯♭} &= (∂_z ⊗ ∂_x - ∂_x ⊗ ∂_z)^{♯♭} &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& (∂_x ⊗ ∂_z - ∂_z ⊗ ∂_x)^{♯♭} \\ (∂_x ∧ ∂_y)^{♯♭} &= (∂_x ⊗ ∂_y - ∂_y ⊗ ∂_x)^{♯♭} &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& (∂_y ⊗ ∂_x - ∂_x ⊗ ∂_y)^{♯♭} \\ \end{alignedat}\end{split}\]

Distribute the musical operators

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= ∂_t^♯ ⊗ ∂_x^♭ - ∂_x^♯ ⊗ ∂_t^♭ &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& ∂_x^♯ ⊗ ∂_t^♭ - ∂_t^♯ ⊗ ∂_x^♭ \\ (∂_t ∧ ∂_y)^{♯♭} &= ∂_t^♯ ⊗ ∂_y^♭ - ∂_y^♯ ⊗ ∂_t^♭ &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& ∂_y^♯ ⊗ ∂_t^♭ - ∂_t^♯ ⊗ ∂_y^♭ \\ (∂_t ∧ ∂_z)^{♯♭} &= ∂_t^♯ ⊗ ∂_z^♭ - ∂_z^♯ ⊗ ∂_t^♭ &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& ∂_z^♯ ⊗ ∂_t^♭ - ∂_t^♯ ⊗ ∂_z^♭ \\ (∂_y ∧ ∂_z)^{♯♭} &= ∂_y^♯ ⊗ ∂_z^♭ - ∂_z^♯ ⊗ ∂_y^♭ &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& ∂_z^♯ ⊗ ∂_y^♭ - ∂_y^♯ ⊗ ∂_z^♭ \\ (∂_z ∧ ∂_x)^{♯♭} &= ∂_z^♯ ⊗ ∂_x^♭ - ∂_x^♯ ⊗ ∂_z^♭ &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& ∂_x^♯ ⊗ ∂_z^♭ - ∂_z^♯ ⊗ ∂_x^♭ \\ (∂_x ∧ ∂_y)^{♯♭} &= ∂_x^♯ ⊗ ∂_y^♭ - ∂_y^♯ ⊗ ∂_x^♭ &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& ∂_y^♯ ⊗ ∂_x^♭ - ∂_x^♯ ⊗ ∂_y^♭ \\ \end{alignedat}\end{split}\]

Apply musical operators

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= ∂_t ⊗ η_{xγ} dx^γ - η_{tγ} ∂_x ⊗ dx^γ &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& ∂_x ⊗ η_{γt} dx^γ - ∂_t ⊗ η_{γx} dx^γ \\ (∂_t ∧ ∂_y)^{♯♭} &= ∂_t ⊗ η_{yγ} dx^γ - η_{tγ} ∂_y ⊗ dx^γ &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& ∂_y ⊗ η_{γt} dx^γ - ∂_t ⊗ η_{γy} dx^γ \\ (∂_t ∧ ∂_z)^{♯♭} &= ∂_t ⊗ η_{zγ} dx^γ - η_{tγ} ∂_z ⊗ dx^γ &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& ∂_z ⊗ η_{γt} dx^γ - ∂_t ⊗ η_{γz} dx^γ \\ (∂_y ∧ ∂_z)^{♯♭} &= ∂_y ⊗ η_{zγ} dx^γ - η_{yγ} ∂_z ⊗ dx^γ &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& ∂_z ⊗ η_{γy} dx^γ - ∂_y ⊗ η_{γz} dx^γ \\ (∂_z ∧ ∂_x)^{♯♭} &= ∂_z ⊗ η_{xγ} dx^γ - η_{zγ} ∂_x ⊗ dx^γ &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& ∂_x ⊗ η_{γz} dx^γ - ∂_z ⊗ η_{γx} dx^γ \\ (∂_x ∧ ∂_y)^{♯♭} &= ∂_x ⊗ η_{yγ} dx^γ - η_{xγ} ∂_y ⊗ dx^γ &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& ∂_y ⊗ η_{γx} dx^γ - ∂_x ⊗ η_{γy} dx^γ \\ \end{alignedat}\end{split}\]

Identify the non-zero components

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= ∂_t ⊗ η_{xx} dx - ∂_x ⊗ η_{tt} dt &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& ∂_x ⊗ η_{tt} dt - ∂_t ⊗ η_{xx} dx \\ (∂_t ∧ ∂_y)^{♯♭} &= ∂_t ⊗ η_{yy} dy - ∂_y ⊗ η_{tt} dt &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& ∂_y ⊗ η_{tt} dt - ∂_t ⊗ η_{yy} dy \\ (∂_t ∧ ∂_z)^{♯♭} &= ∂_t ⊗ η_{zz} dz - ∂_z ⊗ η_{tt} dt &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& ∂_z ⊗ η_{tt} dt - ∂_t ⊗ η_{zz} dz \\ (∂_y ∧ ∂_z)^{♯♭} &= ∂_y ⊗ η_{zz} dz - ∂_z ⊗ η_{yy} dy &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& ∂_z ⊗ η_{yy} dy - ∂_y ⊗ η_{zz} dz \\ (∂_z ∧ ∂_x)^{♯♭} &= ∂_z ⊗ η_{xx} dx - ∂_x ⊗ η_{zz} dz &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& ∂_x ⊗ η_{zz} dz - ∂_z ⊗ η_{xx} dx \\ (∂_x ∧ ∂_y)^{♯♭} &= ∂_x ⊗ η_{yy} dy - ∂_y ⊗ η_{xx} dx &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& ∂_y ⊗ η_{xx} dx - ∂_x ⊗ η_{yy} dy \\ \end{alignedat}\end{split}\]

Apply numerical values

\[\begin{split}\begin{alignedat}{5} (∂_t ∧ ∂_x)^{♯♭} &= - ∂_t ⊗ dx - ∂_x ⊗ dt &\qquad& (∂_x ∧ ∂_t)^{♯♭} &=& + ∂_x ⊗ dt + ∂_t ⊗ dx \\ (∂_t ∧ ∂_y)^{♯♭} &= - ∂_t ⊗ dy - ∂_y ⊗ dt &\qquad& (∂_y ∧ ∂_t)^{♯♭} &=& + ∂_y ⊗ dt + ∂_t ⊗ dy \\ (∂_t ∧ ∂_z)^{♯♭} &= - ∂_t ⊗ dz - ∂_z ⊗ dt &\qquad& (∂_z ∧ ∂_t)^{♯♭} &=& + ∂_z ⊗ dt + ∂_t ⊗ dz \\ (∂_y ∧ ∂_z)^{♯♭} &= - ∂_y ⊗ dz + ∂_z ⊗ dy &\qquad& (∂_z ∧ ∂_y)^{♯♭} &=& - ∂_z ⊗ dy + ∂_y ⊗ dz \\ (∂_z ∧ ∂_x)^{♯♭} &= - ∂_z ⊗ dx + ∂_x ⊗ dz &\qquad& (∂_x ∧ ∂_z)^{♯♭} &=& - ∂_x ⊗ dz + ∂_z ⊗ dx \\ (∂_x ∧ ∂_y)^{♯♭} &= - ∂_x ⊗ dy + ∂_y ⊗ dx &\qquad& (∂_y ∧ ∂_x)^{♯♭} &=& - ∂_y ⊗ dx + ∂_x ⊗ dy \\ \end{alignedat}\end{split}\]

We can then identify the expressions for the mixed wedge product explicitely in terms of tensor products:

\[\begin{split}\begin{alignedat}{4} ∂_t ∧ dx & = + ∂_t ⊗ dx + ∂_x ⊗ dt & \qquad & ∂_x ∧ dt & = + ∂_x ⊗ dt + ∂_t ⊗ dx \\ ∂_t ∧ dy & = + ∂_t ⊗ dy + ∂_y ⊗ dt & \qquad & ∂_y ∧ dt & = + ∂_y ⊗ dt + ∂_t ⊗ dy \\ ∂_t ∧ dz & = + ∂_t ⊗ dz + ∂_z ⊗ dt & \qquad & ∂_z ∧ dt & = + ∂_z ⊗ dt + ∂_t ⊗ dz \\ ∂_y ∧ dz & = + ∂_y ⊗ dz - ∂_z ⊗ dy & \qquad & ∂_z ∧ dy & = + ∂_z ⊗ dy - ∂_y ⊗ dz \\ ∂_z ∧ dx & = + ∂_z ⊗ dx - ∂_x ⊗ dz & \qquad & ∂_x ∧ dz & = + ∂_x ⊗ dz - ∂_z ⊗ dx \\ ∂_x ∧ dy & = + ∂_x ⊗ dy - ∂_y ⊗ dx & \qquad & ∂_y ∧ dx & = + ∂_y ⊗ dx - ∂_x ⊗ dy \\ \end{alignedat}\end{split}\]

Taken together, we get the result consistent with the symmetries obtained for the \(♭♯\), thus strongly suggesting the results regarding the symmetries of the mixeed exterior product are correct.

\(\mathfrak{so}(1,3)\) Lie algebra of the Lorentz group#

Matrices are organized in column of vectors and therefore type \(♯♭\) tensors, written \(M_μ{}^ν\) in abstract index notation. The objects can take vectors \(v^ν\) as input and output transformed vectors \(M_γ{}^ν \: v^γ\).

\[\begin{split}M = \begin{pmatrix} \vdots & \vdots & \vdots & \vdots \\ v_0^ν & v_1^ν & v_2^ν & v_3^{ν} \\ \vdots & \vdots & \vdots & \vdots \\ \end{pmatrix}\end{split}\]

The type \(♯♭\) row/column matrix representation of rotations is:

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

Taking out the basis bivectors from Free matrix representation, we trivially obtain the representation of the Lorentz group, as well as the interpretation as a rotation in spacetime:

\[\begin{split}R^{♭♯} = \frac{1}{2} \begin{bmatrix} & + a & + b & + c \\ + a & & + f & - e \\ + b & - f & & + d \\ + c & + e & - d & \\ \end{bmatrix}\end{split}\]