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Tensor[NullVector] - construct a null vector from a solder form and a rank 1 spinor
Calling Sequences
NullVector(sigma phi)
NullVector(sigma phi  psi)
Parameters
sigma - a spin-tensor defining a solder form on a 4-dimensional spacetime
phi, psi - rank 1 spinors
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Description
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With two arguments, the NullVector command returns the real vector with components ![X^i = sigma^(diff(iAA(x), x))*phi[A(x)]*conjugate(phi[diff(A(x), x)])](/support/helpjp/helpview.aspx?si=5662/file05899/math61.png)
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• With three arguments, the NullVector command returns the (complex) vector with components ![X^(i )= sigma^(iAA')phi[A]psi[A' ].](/support/helpjp/helpview.aspx?si=5662/file05899/math66.png)
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NullVector(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-NullVector.
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Examples
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Example 1.
First create the spinor bundle  with spacetime coordinates ![[t, x, y, z]](/support/helpjp/helpview.aspx?si=5662/file05899/math87.png) and fiber coordinates ![[z, z^2, w, w^2]](/support/helpjp/helpview.aspx?si=5662/file05899/math89.png) .
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![DGsetup([t, x, y, z], [z1, z2, w1, w2], M)](/support/helpjp/helpview.aspx?si=5662/file05899/math94.png)
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| (2.1) |
Define a spacetime metric  on  with signature ![[1, -1, -1, -1]](/support/helpjp/helpview.aspx?si=5662/file05899/math107.png) .
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 1], [[2, 2], -1], [[3, 3], -1], [[4, 4], -1]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math115.png)
| (2.2) |
Define an orthonormal tetrad  on  with respect to the metric Use the command SolderForm to create a solder form  .
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![F := [D_t, D_x, D_y, D_z]](/support/helpjp/helpview.aspx?si=5662/file05899/math135.png)
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![[_DG([["vector", M, []], [[[1], 1]]]), _DG([["vector", M, []], [[[2], 1]]]), _DG([["vector", M, []], [[[3], 1]]]), _DG([["vector", M, []], [[[4], 1]]])]](/support/helpjp/helpview.aspx?si=5662/file05899/math138.png)
| (2.3) |
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![_DG([["tensor", M, [["cov_bas", "con_vrt", "con_vrt"], []]], [[[1, 5, 7], (1/2)*2^(1/2)], [[1, 6, 8], (1/2)*2^(1/2)], [[2, 5, 8], (1/2)*2^(1/2)], [[2, 6, 7], (1/2)*2^(1/2)], [[3, 5, 8], -((1/2)*I)*2^(1/2)], [[3, 6, 7], ((1/2)*I)*2^(1/2)], [[4, 5, 7], (1/2)*2^(1/2)], [[4, 6, 8], -(1/2)*2^(1/2)]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math145.png)
| (2.4) |
Define rank 1 spinors ![phi[1], phi[2]](/support/helpjp/helpview.aspx?si=5662/file05899/math150.png) and ![phi[3].](/support/helpjp/helpview.aspx?si=5662/file05899/math152.png)
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![_DG([["vector", M, []], [[[5], 1]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math159.png)
| (2.5) |
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![_DG([["vector", M, []], [[[5], a], [[6], b]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math166.png)
| (2.6) |
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![_DG([["vector", M, []], [[[8], 1]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math173.png)
| (2.7) |
Use the command NullVector to find the corrresponding null vectors  .
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![_DG([["vector", M, []], [[[1], (1/2)*2^(1/2)], [[4], (1/2)*2^(1/2)]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math188.png)
| (2.8) |
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![Y := `assuming`([NullVector(sigma, phi2)], [a::real, b::real])](/support/helpjp/helpview.aspx?si=5662/file05899/math192.png)
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![_DG([["vector", M, []], [[[1], (1/2)*2^(1/2)*b^2+(1/2)*2^(1/2)*a^2], [[2], 2^(1/2)*a*b], [[4], -(1/2)*2^(1/2)*b^2+(1/2)*2^(1/2)*a^2]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math195.png)
| (2.9) |
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![_DG([["vector", M, []], [[[2], (1/2)*2^(1/2)], [[3], ((1/2)*I)*2^(1/2)]]])](/support/helpjp/helpview.aspx?si=5662/file05899/math202.png)
| (2.10) |
We can use the command TensorInnerProduct to check that the vectors  are indeed null vectors.
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| (2.11) |
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| (2.12) |
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| (2.13) |
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