DifferentialGeometry/Tensor/BivectorSolderForm - Maple Help

Tensor[BivectorSolderForm] - construct the bivector solder form defined by a solder form

Calling Sequences

BivectorSolderForm(sigma, spinorType, indexlist)

Parameters

sigma      - a solder form

spinorType - a string, either "spinor" or "barspinor"

indexlist  - (optional) the keyword argument indexlist = ind, where ind is a list of 4 index types "con" or "cov"

Description

 • A bivector is a skew-symmetric, rank 2 contravariant tensor. On a 4-dimensional manifold with solder form $\mathrm{σ}$ there is a 1-1 correspondence between bivectors and symmetric rank 2 spinors.  This correspondence is explicitly furnished by the bivector solder forms and $\stackrel{‾}{S}$ which are defined in terms of the solder form s by

and

.

 • The tensor indices of the bivector solder forms are raised and lowered with the metric $g$ defined by $\mathrm{σ}$
 • The keyword argument indexlist = ind allows the user to specify the index structure for the bivector solder form. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form  is returned.
 • The bivector soldering forms satisfy a large number of identities, some of which are illustrated in Examples 2 - 4.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BivectorSolderForm(...) only after executing the commands with(DifferentialGeometry; with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-BivectorSolderForm.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

First create a vector bundle over with base coordinates  and fiber coordinates .

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Define a metric g on M. Note that our spinor conventions have the metric with signature

 > $g≔\mathrm{evalDG}\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

Define an orthonormal frame on M with respect to the metric g.

 > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.3)

Calculate the solder form sigma from the frame F.

 > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.4)

Calculate the bivector solder form S from sigma.

 > $S≔\mathrm{BivectorSolderForm}\left(\mathrm{\sigma },"spinor"\right)$
 ${S}{:=}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}$ (2.5)

Example 2.

The contraction of two bivector solder forms on their tensor indices can be expressed in terms of the Kronecker delta spinor.

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

 > $\mathrm{S1}≔\mathrm{BivectorSolderForm}\left(\mathrm{\sigma },"spinor"\right)$
 ${\mathrm{S1}}{:=}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}$ (2.6)
 > $\mathrm{S2}≔\mathrm{BivectorSolderForm}\left(\mathrm{\sigma },"spinor",\mathrm{indextype}=\left["con","con","cov","cov"\right]\right)$
 ${\mathrm{S2}}{:=}{\mathrm{D_t}}{}{\mathrm{D_x}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{\mathrm{D_t}}{}{\mathrm{D_x}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{D_t}}{}{\mathrm{D_y}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{I}{}{\mathrm{D_t}}{}{\mathrm{D_y}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{\mathrm{D_t}}{}{\mathrm{D_z}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{D_t}}{}{\mathrm{D_z}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{\mathrm{D_x}}{}{\mathrm{D_t}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{D_x}}{}{\mathrm{D_t}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_t}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_t}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{D_z}}{}{\mathrm{D_t}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{D_z}}{}{\mathrm{D_t}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.7)
 > $\mathrm{LHS}≔\mathrm{ContractIndices}\left(\mathrm{S1},\mathrm{S2},\left[\left[1,1\right],\left[2,2\right]\right]\right)$
 ${\mathrm{LHS}}{:=}{8}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{8}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.8)

To calculate the right-hand side we construct the symmetrized tensor product of 2 Kronecker delta spinors and multiply by 8 (because SymmetrizeIndices will include a factor of 1/2).

 > $\mathrm{\delta }≔\mathrm{KroneckerDeltaSpinor}\left("spinor"\right)$
 ${\mathrm{δ}}{:=}{\mathrm{D_z1}}{}{\mathrm{dz1}}{+}{\mathrm{D_z2}}{}{\mathrm{dz2}}$ (2.9)
 > $E≔\mathrm{RearrangeIndices}\left(\mathrm{\delta }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\delta },\left[\left[2,3\right]\right]\right)$
 ${E}{:=}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.10)
 > $\mathrm{RHS}≔8\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{SymmetrizeIndices}\left(E,\left[1,2\right],"Symmetric"\right)$
 ${\mathrm{RHS}}{:=}{8}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{8}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.11)

Check that the LHS and RHS are the same.

 > $\mathrm{LHS}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{RHS}$
 ${0}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.12)

Example 3.

The contraction of two bivector soldering forms on their tensor indices can be expressed in terms of the metric and the permutation tensor

.

We check this identity using the solder form from Example 1.  First we calculate the left-hand side.

 > $\mathrm{S3}≔\mathrm{BivectorSolderForm}\left(\mathrm{\sigma },"spinor",\mathrm{indextype}=\left["cov","cov","cov","cov"\right]\right)$
 ${\mathrm{S3}}{:=}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.13)
 > $\mathrm{LHS}≔\mathrm{ContractIndices}\left(\mathrm{S1},\mathrm{S3},\left[\left[3,3\right],\left[4,4\right]\right]\right)$
 ${\mathrm{LHS}}{:=}{-}{2}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{2}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{2}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{2}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{2}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{2}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{2}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dy}}{+}{2}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{2}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}$ (2.14)

To calculate the right-hand side we first construct the tensor product of the metric tensor with itself.

 > $G≔g\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}g$
 ${G}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dt}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dt}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dz}}{-}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dt}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.15)

We re-arrange the indices of G to obtain the first two terms on the right-hand side.

 > $\mathrm{RHS1}≔\mathrm{RearrangeIndices}\left(G,\left[\left[2,3\right]\right]\right)$
 ${\mathrm{RHS1}}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.16)
 > $\mathrm{RHS2}≔\mathrm{RearrangeIndices}\left(\mathrm{RHS1},\left[\left[1,2\right]\right]\right)$
 ${\mathrm{RHS2}}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.17)

We construct the epsilon tensor using the commands MetricDensity and PermutationSymbol.

 > $E≔\mathrm{MetricDensity}\left(g,1,\mathrm{detmetric}=-1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PermutationSymbol}\left("cov_bas"\right)$
 ${E}{:=}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.18)

Evaluate the right-hand side of the identity and check that it agrees with the left-hand side.

 > $\mathrm{RHS}≔\mathrm{evalDG}\left(2\left(\mathrm{RHS1}-\mathrm{RHS2}-IE\right)\right)$
 ${\mathrm{RHS}}{:=}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{2}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{2}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{2}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{2}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{2}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{2}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{2}{}{I}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{2}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{2}{}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{2}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{2}{}{I}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.19)
 > $\mathrm{LHS}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{RHS}$
 ${0}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}$ (2.20)

Example 4.

The bivector solder form is anti-self-dual, that is,

${S}_{\mathrm{ij}}^{\mathrm{AB}}=-\frac{i}{2}{\mathrm{ε}}_{\mathrm{ijhk}}{S}^{\mathrm{hkAB}}.$

We check this identity using the solder form from Example 1.  The left-hand side is just the solder form S1 from Example 1.

 > $\mathrm{LHS}≔\mathrm{S1}$
 ${\mathrm{LHS}}{:=}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}$ (2.21)

To evaluate the right-hand side we begin with the contravariant form of the bivector solder form.

 > $\mathrm{S4}≔\mathrm{BivectorSolderForm}\left(\mathrm{\sigma },"spinor",\mathrm{indextype}=\left["con","con","con","con"\right]\right)$
 ${\mathrm{S4}}{:=}{-}{\mathrm{D_t}}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{D_t}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{D_t}}{}{\mathrm{D_y}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{D_t}}{}{\mathrm{D_y}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{\mathrm{D_t}}{}{\mathrm{D_z}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{\mathrm{D_t}}{}{\mathrm{D_z}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{\mathrm{D_x}}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{D_x}}{}{\mathrm{D_t}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{D_x}}{}{\mathrm{D_z}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_t}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{D_z}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{\mathrm{D_z}}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{D_z}}{}{\mathrm{D_t}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{D_z}}{}{\mathrm{D_y}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}$ (2.22)

Construct the epsilon tensor and contract with S4 and to obtain the left-hand side.

 > $E≔\mathrm{MetricDensity}\left(g,1,\mathrm{detmetric}=-1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PermutationSymbol}\left("cov_bas"\right)$
 ${E}{:=}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{-}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{+}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{dt}}{-}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.23)
 > $\mathrm{RHS}≔\left(-\frac{I}{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ContractIndices}\left(E,\mathrm{S4},\left[\left[3,1\right],\left[4,2\right]\right]\right)$
 ${\mathrm{RHS}}{:=}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dt}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dx}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{dy}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{+}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}$ (2.24)
 > $\mathrm{LHS}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{RHS}$
 ${0}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}$ (2.25)