 DifferentialGeometry/Tensor/SolderForm - Maple Help

Tensor[SolderForm] - calculate the solder form from an orthonormal frame

Calling Sequences

SolderForm(OrthFr, indexlist)

Parameters

OrthFr    - a list of 4 vectors defining an orthonormal frame for a metric g with signature

indexlist - (optional) the keyword argument indextype = ind, where ind is a list of 3 index types "con" or "cov" Description

 • The solder form $\mathrm{σ}$ is a rank 3 spin-tensor which defines an isomorphism between vectors and Hermitian rank 2 spinors. The first index type is a covariant tensor index, the second index type is a contravariant spinor index, and the third index is a contravariant barred (primed) spinor index. Denote the components of the solder form by ${\mathrm{σ}}_{i}^{\mathrm{AA}'}.$ (The components of the solder form are often referred to as the Infeld-van der Waerden symbols.) To define the solder form, first recall the definition of an orthonormal frame. Let $M$ be a 4-dimensional manifold and let $g$ be a metric on $M$ with signature $\left[1,-1,-1,-1\right]$. A tetrad of vectors $\left({X}_{1},{X}_{2},{X}_{3},{X}_{4}\right)$ is an orthonormal frame with respect to the metric if $g\left({X}_{a},{X}_{b}\right)=0$ for $a\ne b$, $g\left({X}_{1},{X}_{1}\right)=1$, ) = -1, $a=2,3,4$. The command DGGramSchmidt can be used to create an orthonormal frame. The command GRQuery or TensorInnerProduct can used to check that a list of vectors constitutes an orthonormal frame for a given metric. Recall also the definition of the 4 Pauli spin matrices $\left({P}_{1},{P}_{2},{P}_{3},{P}_{4}\right)$, given below in Example 1. The matrix elements of the ${P}_{a}$ can be viewed as components of a Hermitian spinor, ${P}_{a}^{\mathrm{AA}'}.$ Let $\left({X}_{1},{X}_{2},{X}_{3},{X}_{4}\right)$ be an orthonormal frame with respect to a metric and let $\left({\mathrm{ω}}^{1},{\mathrm{ω}}^{2},{\mathrm{ω}}^{3},{\mathrm{ω}}^{4}\right)$ be the dual co-frame (see DualBasis). Then the associated solder form is

(sum on $a$).

 • The command SolderForm(OrthFr) calculates the solder form from the orthonormal frame OrthFr.
 • The keyword argument indexlist = ind allows the user to specify the index structure for the solder form. For example, withindexlist = ["con", "con", "con"], the contravariant form ${\mathrm{σ}}^{\mathrm{iAA}'}$ is returned.
 • The solder form satisfies a number of important identities.  These are given in Example 2.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SolderForm(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-SolderForm. Examples

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

Example 1.

First create a vector bundle over $M$ with base coordinates $\left(t,x,y,z\right)$ and fiber coordinates . It is understood that $\mathrm{w1},\mathrm{w2}$ are complex conjugates of $\mathrm{z1},\mathrm{z2}$.

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

Define a spacetime metric $g$ on $M$ with signature $\left(1,-1,-1,-1\right)$.

 M > $g≔\mathrm{evalDG}\left(\mathrm{dt}&t\mathrm{dt}-\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}-\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

Define an orthonormal frame on $M$ with respect to the metric $g$.  Verify the frame is orthonormal using the command GRQuery.

 M > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.3)
 M > $\mathrm{GRQuery}\left(F,g,"OrthonormalTetrad"\right)$
 ${\mathrm{true}}$ (2.4)

Calculate the solder form ${\mathrm{σ}}_{}$ from the frame $F$.

 M > $\mathrm{σ}≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.5)

Let us obtain this result directly from the definition. First we define the Pauli matrices.

 M > $\mathrm{P1},\mathrm{P2},\mathrm{P3},\mathrm{P4}≔\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[1,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,-I\right],\left[I,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right)$ Define the corresponding rank 2 Hermitian spinors.

 M > ${S}_{1}≔\mathrm{evalDG}\left(\mathrm{D_z1}&t\mathrm{D_w1}+\mathrm{D_z2}&t\mathrm{D_w2}\right)$
 ${{S}}_{{1}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.6)
 M > ${S}_{2}≔\mathrm{evalDG}\left(\mathrm{D_z1}&t\mathrm{D_w2}+\mathrm{D_z2}&t\mathrm{D_w1}\right)$
 ${{S}}_{{2}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}{\mathrm{D_z2}}{}{\mathrm{D_w1}}$ (2.7)
 M > ${S}_{3}≔\mathrm{evalDG}\left(-I\mathrm{D_z1}&t\mathrm{D_w2}+I\mathrm{D_z2}&t\mathrm{D_w1}\right)$
 ${{S}}_{{3}}{:=}{-}{I}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}{I}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}$ (2.8)
 M > ${S}_{4}≔\mathrm{evalDG}\left(\mathrm{D_z1}&t\mathrm{D_w1}-\mathrm{D_z2}&t\mathrm{D_w2}\right)$
 ${{S}}_{{4}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.9)

Define the dual coframe to $F$.

 M > $\mathrm{ω}≔\left[\mathrm{dt},\mathrm{dx},\mathrm{dy},\mathrm{dz}\right]$
 ${\mathrm{ω}}{:=}\left[{\mathrm{dt}}{,}{\mathrm{dx}}{,}{\mathrm{dy}}{,}{\mathrm{dz}}\right]$ (2.10)
 M > $\mathrm{σ0}≔\mathrm{evalDG}\left(\frac{\sqrt{2}\mathrm{add}\left(\left({\mathrm{ω}}_{i}\right)&tensor\left({S}_{i}\right),i=1..4\right)}{2}\right)$
 ${\mathrm{σ0}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.11)

This coincides with $\mathrm{σ}$.

 M > $\mathrm{evalDG}\left(\mathrm{σ}-\mathrm{σ0}\right)$
 ${0}$ (2.12)

Example 2.

The solder form satisfies two important identities. The first identity involves contracting a pair of solder forms over their spinor indices:

${\mathrm{σ}}_{i}^{\mathrm{AA}'}{\mathrm{σ}}_{\mathrm{jAA}'}={g}_{\mathrm{ij}}$

The second identity involves contracting a pair of solder forms over their tensor indices:

${\mathrm{σ}}_{j}^{\mathrm{AA}'}{\mathrm{σ}}^{\mathrm{jBB}'}={\mathrm{ε}}^{\mathrm{AB}}{\mathrm{ε}}^{A'B'}.$

Let us check the first identity using the solder form from Example 1.  First calculate the covariant form of the solder form, using the orthonormal frame of the previous example.

 M > $\mathrm{sigmaCov}≔\mathrm{SolderForm}\left(F,\mathrm{indextype}=\left["cov","cov","cov"\right]\right)$
 ${\mathrm{sigmaCov}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}$ (2.13)

Note that this coincides with the result of using RaiseLowerSpinorIndices to lower the spinor indices of ${\mathrm{σ}}_{}$ using the epsilon spinor.

 M > $\mathrm{sigmaCov}&minus\left(\mathrm{RaiseLowerSpinorIndices}\left(\mathrm{σ},\left[2,3\right]\right)\right)$
 ${0}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.14)

The contraction of $\mathrm{σ}$ and sigmaCov over their spinor indices gives the metric $g$.

 M > $\mathrm{ContractIndices}\left(\mathrm{σ},\mathrm{sigmaCov},\left[\left[2,2\right],\left[3,3\right]\right]\right)$
 ${\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.15)

The same result can be obtained using SpinorInnerProduct.

 M > $\mathrm{SpinorInnerProduct}\left(\mathrm{σ},\mathrm{σ}\right)$
 ${\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.16)

To check the second identity calculate the contravariant form of $\mathrm{σ}$.

 M > $\mathrm{sigmaCon}≔\mathrm{SolderForm}\left(F,\mathrm{indextype}=\left["con","con","con"\right]\right)$
 ${\mathrm{sigmaCon}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.17)

Note that this coincides with the result of using RaiseLowerIndices to raise the tensor index of $\mathrm{σ}$ using the inverse of the metric $g$.

 M > $\mathrm{sigmaCon}&minus\left(\mathrm{RaiseLowerIndices}\left(\mathrm{InverseMetric}\left(g\right),\mathrm{σ},\left[1\right]\right)\right)$
 ${0}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}$ (2.18)

The contraction of $\mathrm{σ}$ and sigmaCon over their tensor indices gives a product of epsilon spinors (EpsilonSpinor).

 M > $\mathrm{E1}≔\mathrm{ContractIndices}\left(\mathrm{σ},\mathrm{sigmaCon},\left[\left[1,1\right]\right]\right)$
 ${\mathrm{E1}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{-}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}$ (2.19)

Rearrange the indices so that the spinor indices are first, the barred spinor indices second.

 M > $\mathrm{E2}≔\mathrm{RearrangeIndices}\left(\mathrm{E1},\left[\left[2,3\right]\right]\right)$
 ${\mathrm{E2}}{:=}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{}{\mathrm{D_w2}}{-}{\mathrm{D_z1}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{}{\mathrm{D_w1}}{-}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{}{\mathrm{D_w2}}{+}{\mathrm{D_z2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{}{\mathrm{D_w1}}$ (2.20)
 M > $\mathrm{evalDG}\left(\mathrm{E2}-\left(\mathrm{EpsilonSpinor}\left("con","spinor"\right)\right)&t\left(\mathrm{EpsilonSpinor}\left("con","barspinor"\right)\right)\right)$
 ${0}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}$ (2.21)

Example 3.

Here we compute a solder form for the Gödel spacetime.  (See (12.26) in Stephani Kramer et al.) First create a vector bundle over $M$ with base coordinates $\left(t,x,y,z\right)$ and fiber coordinates .

 M > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.22)

Define the Gödel metric $g$ on $M$. (Note that we have adjusted the metric to conform to the signature convention used by the spinor formalism in DifferentialGeometry .)

 M > $g≔\mathrm{evalDG}\left(\mathrm{dt}&t\mathrm{dt}+{ⅇ}^{x}\mathrm{dt}&s\mathrm{dz}-\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}+\frac{1{ⅇ}^{2x}\mathrm{dz}&t\mathrm{dz}}{2}\right)$
 ${g}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.23)

Use DGGramSchmidt to calculate an orthonormal frame $F$ for the metric $g$.

 M > $F≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right],g,\mathrm{signature}=\left[1,-1,-1,-1\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}x::\mathrm{real}$
 ${F}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{I}{}{\mathrm{D_t}}{-}{2}{}{I}{}{{ⅇ}}^{{-}{x}}{}{\mathrm{D_z}}\right]$ (2.24)

Use SolderForm to compute the solder form $\mathrm{σ}$ from the orthonormal frame $F$.

 M > $\mathrm{σ}≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\left(\frac{{1}}{{4}}{}{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}{+}\frac{{1}}{{4}}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}\right){}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\left({-}\frac{{1}}{{4}}{}{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}{+}\frac{{1}}{{4}}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}\right){}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.25)

Example 4.

For any metric of Lorentz signature $\left[1,-1,-1,-1\right]$, a compatible solder form can be constructed.

 M > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],N\right)$
 ${\mathrm{frame name: N}}$ (2.26)

Define a spacetime metric $\mathrm{g3}$.

 N > $\mathrm{g3}≔\mathrm{evalDG}\left({x}^{4}\mathrm{du}&s\mathrm{dv}-{y}^{2}\mathrm{dx}&t\mathrm{dx}-{v}^{2}\mathrm{dy}&t\mathrm{dy}\right)$
 ${\mathrm{g3}}{:=}\frac{{1}}{{2}}{}{{x}}^{{4}}{}{\mathrm{du}}{}{\mathrm{dv}}{+}\frac{{1}}{{2}}{}{{x}}^{{4}}{}{\mathrm{dv}}{}{\mathrm{du}}{-}{{y}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{{v}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.27)

Use the command DGGramSchmidt to find an orthonormal frame.

 N > $\mathrm{F3}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D_u},\mathrm{D_v},\mathrm{D_x},\mathrm{D_y}\right],\mathrm{g3},\mathrm{signature}=\left[\left[1,-1\right],-1,-1\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0
 ${\mathrm{F3}}{:=}\left[\frac{{\mathrm{D_u}}}{{{x}}^{{2}}}{+}\frac{{\mathrm{D_v}}}{{{x}}^{{2}}}{,}\frac{{\mathrm{D_u}}}{{{x}}^{{2}}}{-}\frac{{\mathrm{D_v}}}{{{x}}^{{2}}}{,}\frac{{\mathrm{D_x}}}{{y}}{,}\frac{{\mathrm{D_y}}}{{v}}\right]$ (2.28)

Calculate the solder form from $\mathrm{F3}$.

 N > $\mathrm{σ3}≔\mathrm{SolderForm}\left(\mathrm{F3}\right)$
 ${\mathrm{σ3}}{:=}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{du}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{du}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{du}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{du}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{dv}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{dv}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{-}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{dv}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{4}}{}{{x}}^{{2}}{}\sqrt{{2}}{}{\mathrm{dv}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{-}\frac{{1}}{{2}}{}{I}{}{y}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}{y}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}{v}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{v}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.29)

Use SpinorInnerProduct to check that is compatible with the metric $\mathrm{g3}$.

 N > $\mathrm{SpinorInnerProduct}\left(\mathrm{σ3},\mathrm{σ3}\right)$
 $\frac{{1}}{{2}}{}{{x}}^{{4}}{}{\mathrm{du}}{}{\mathrm{dv}}{+}\frac{{1}}{{2}}{}{{x}}^{{4}}{}{\mathrm{dv}}{}{\mathrm{du}}{-}{{y}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{{v}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.30) See Also