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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 1, 1, 1,1

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

 

Description

Examples

See Also

Description

• 

The solder form 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 σiAA'. (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 1,1,1,1. A tetrad of vectors X1,X2,X3,X4 is an orthonormal frame with respect to the metric if gXa,Xb=0 for ab, gX1,X1=1, g(Xa, Xa) = -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 P1,P2,P3,P4, given below in Example 1. The matrix elements of the Pa can be viewed as components of a Hermitian spinor, PaAA'. Let X1,X2,X3,X4 be an orthonormal frame with respect to a metric g and let ω1,ω2,ω3,ω4 be the dual co-frame (see DualBasis). Then the associated solder form is

σiAA'=22ωaPaAA'     (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 σiAA' 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

withDifferentialGeometry:withTensor:

 

Example 1.

First create a vector bundle over M with base coordinates t,x,y,z and fiber coordinates z1, z2, w1, w2. It is understood that w1,w2 are complex conjugates of z1,z2.

DGsetupt,x,y,z,z1,z2,w1,w2,M

frame name: M

(2.1)

 

Define a spacetime metric g on M with signature 1,1,1,1.

M > 

gevalDGdt&tdtdx&tdxdy&tdydz&tdz

g:=dtdtdxdxdydydzdz

(2.2)

 

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

M > 

FD_t,D_x,D_y,D_z

F:=D_t,D_x,D_y,D_z

(2.3)
M > 

GRQueryF,g,OrthonormalTetrad

true

(2.4)

 

Calculate the solder form σ from the frame F.

M > 

σSolderFormF

σ:=122dtD_z1D_w1+122dtD_z2D_w2+122dxD_z1D_w2+122dxD_z2D_w112I2dyD_z1D_w2+12I2dyD_z2D_w1+122dzD_z1D_w1122dzD_z2D_w2

(2.5)

 

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

M > 

P1,P2,P3,P4Matrix1,0,0,1,Matrix0,1,1,0,Matrix0,I,I,0,Matrix1,0,0,1

 

Define the corresponding rank 2 Hermitian spinors.

M > 

S1evalDGD_z1&tD_w1+D_z2&tD_w2

S1:=D_z1D_w1+D_z2D_w2

(2.6)
M > 

S2evalDGD_z1&tD_w2+D_z2&tD_w1

S2:=D_z1D_w2+D_z2D_w1

(2.7)
M > 

S3evalDGID_z1&tD_w2+ID_z2&tD_w1

S3:=ID_z1D_w2+ID_z2D_w1

(2.8)
M > 

S4evalDGD_z1&tD_w1D_z2&tD_w2

S4:=D_z1D_w1D_z2D_w2

(2.9)

 

Define the dual coframe to F.

M > 

ωdt,dx,dy,dz

ω:=dt,dx,dy,dz

(2.10)
M > 

σ0evalDGsqrt22addωi&tensorSi,i=1..4

σ0:=122dtD_z1D_w1+122dtD_z2D_w2+122dxD_z1D_w2+122dxD_z2D_w112I2dyD_z1D_w2+12I2dyD_z2D_w1+122dzD_z1D_w1122dzD_z2D_w2

(2.11)

 

This coincides with σ.

M > 

evalDGσσ0

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:

σiAA'σjAA'=gij

 

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

 

σjAA'σjBB'=εABεA'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 > 

sigmaCovSolderFormF,indextype=cov,cov,cov

sigmaCov:=122dtdz1dw1+122dtdz2dw2122dxdz1dw2122dxdz2dw112I2dydz1dw2+12I2dydz2dw1122dzdz1dw1+122dzdz2dw2

(2.13)

 

Note that this coincides with the result of using RaiseLowerSpinorIndices to lower the spinor indices of σ using the epsilon spinor.

M > 

sigmaCov&minusRaiseLowerSpinorIndicesσ,2,3

0dtdz1dz1

(2.14)

 

The contraction of σ and sigmaCov over their spinor indices gives the metric g.

M > 

ContractIndicesσ,sigmaCov,2,2,3,3

dtdtdxdxdydydzdz

(2.15)

 

The same result can be obtained using SpinorInnerProduct.

M > 

SpinorInnerProductσ,σ

dtdtdxdxdydydzdz

(2.16)

 

To check the second identity calculate the contravariant form of σ.

M > 

sigmaConSolderFormF,indextype=con,con,con

sigmaCon:=122D_tD_z1D_w1+122D_tD_z2D_w2122D_xD_z1D_w2122D_xD_z2D_w1+12I2D_yD_z1D_w212I2D_yD_z2D_w1122D_zD_z1D_w1+122D_zD_z2D_w2

(2.17)

 

Note that this coincides with the result of using RaiseLowerIndices to raise the tensor index of σ using the inverse of the metric g.

M > 

sigmaCon&minusRaiseLowerIndicesInverseMetricg,σ,1

0D_tD_z1D_z1

(2.18)

 

The contraction of σ and sigmaCon over their tensor indices gives a product of epsilon spinors (EpsilonSpinor).

M > 

E1ContractIndicesσ,sigmaCon,1,1

E1:=D_z1D_w1D_z2D_w2D_z1D_w2D_z2D_w1D_z2D_w1D_z1D_w2+D_z2D_w2D_z1D_w1

(2.19)

 

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

M > 

E2RearrangeIndicesE1,2,3

E2:=D_z1D_z2D_w1D_w2D_z1D_z2D_w2D_w1D_z2D_z1D_w1D_w2+D_z2D_z1D_w2D_w1

(2.20)
M > 

evalDGE2EpsilonSpinorcon,spinor&tEpsilonSpinorcon,barspinor

0D_z1D_z1D_z1D_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 t,x,y,z and fiber coordinates z1, z2, w1, w2.

M > 

DGsetupt,x,y,z,z1,z2,w1,w2,M

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 1,1,1,1 used by the spinor formalism in DifferentialGeometry .)

M > 

gevalDGdt&tdt+expxdt&sdzdx&tdxdy&tdy+12exp2xdz&tdz

g:=dtdt+12ⅇxdtdzdxdxdydy+12ⅇxdzdt+12ⅇ2xdzdz

(2.23)

 

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

M > 

FDGGramSchmidtD_t,D_x,D_y,D_z,g,signature=1,1,1,1assumingx::real

F:=D_t,D_x,D_y,ID_t2IⅇxD_z

(2.24)

 

Use SolderForm to compute the solder form σ from the orthonormal frame F.

M > 

σSolderFormF

σ:=122dtD_z1D_w1+122dtD_z2D_w2+122dxD_z1D_w2+122dxD_z2D_w112I2dyD_z1D_w2+12I2dyD_z2D_w1+14Iⅇx2+14ⅇx2dzD_z1D_w1+14Iⅇx2+14ⅇx2dzD_z2D_w2

(2.25)

 

Example 4.

For any metric of Lorentz signature 1,1,1,1, a compatible solder form can be constructed.

M > 

DGsetupu,v,x,y,z1,z2,w1,w2,N

frame name: N

(2.26)

 

Define a spacetime metric g3.

N > 

g3evalDGx4du&sdvy2dx&tdxv2dy&tdy

g3:=12x4dudv+12x4dvduy2dxdxv2dydy

(2.27)

 

Use the command DGGramSchmidt to find an orthonormal frame.

N > 

F3DGGramSchmidtD_u&comma;D_v&comma;D_x&comma;D_y&comma;g3&comma;signature=1&comma;1&comma;1&comma;1assuming0<y,0<v,x::real

F3:=D_ux2&plus;D_vx2&comma;D_ux2D_vx2&comma;D_xy&comma;D_yv

(2.28)

 

Calculate the solder form from F3.

N > 

&sigma;3SolderFormF3

&sigma;3:=14x22duD_z1D_w1&plus;14x22duD_z1D_w2&plus;14x22duD_z2D_w1&plus;14x22duD_z2D_w2&plus;14x22dvD_z1D_w114x22dvD_z1D_w214x22dvD_z2D_w1&plus;14x22dvD_z2D_w212Iy2dxD_z1D_w2&plus;12Iy2dxD_z2D_w1&plus;12v2dyD_z1D_w112v2dyD_z2D_w2

(2.29)

 

Use SpinorInnerProduct to check that &sigma;3 is compatible with the metric g3.

N > 

SpinorInnerProduct&sigma;3&comma;&sigma;3

12x4dudv&plus;12x4dvduy2dxdxv2dydy

(2.30)

See Also

DifferentialGeometry, Tensor, BivectorSolderForm, convert/DGspinor, convert/DGtensor, DGGramSchmidt, DualBasis, EpsilonSpinor, GRQuery, NullTetrad, OrthonormalTetrad, RaiseLowerIndices, RaiseLowerSpinorIndices, RicciSpinor,  SpinConnection, SpinorInnerProduct, WeylSpinor