tensor[Levi_Civita] - compute the covariant and contravariant Levi-Civita pseudo-tensors
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Calling Sequence
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Levi_Civita(detg, dim, cov_LC, con_LC)
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Parameters
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detg
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determinant of the covariant metric tensor components
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dim
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dimension of the space
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cov_LC
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output parameter for the covariant Levi_Civita pseudo tensor
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con_LC
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output parameter for the contravariant Levi_Civita pseudo tensor
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Description
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The function Levi_Civita(detg, dim, cov_LC, con_LC) computes the Levi-Civita pseudo-tensor in the dimension dim using the metric determinant detg. The covariant Levi-Civita tensor is output via the parameter cov_LC and the contravariant Levi-Civita tensor is output via the parameter con_LC. The return value is NULL.
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detg must be an algebraic type. It can be computed from the covariant metric tensor using tensor[invert]. Because the square root of detg is used in computing the components of the results, it is assumed that detg is positive (except in the case where dim=4, where it is assumed to be negative; see below).
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dim must be an integer greater than 1.
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cov_LC and con_LC must be unassigned names to be used as output parameters for the results. Recall that the Levi-Civita pseudo-tensor is equal to the permutation symbol multiplied by a factor involving the square root of detg. cov_LC is the covariant permutation symbol multiplied by square root of detg and con_LC is the contravariant permutation symbol multiplied by the reciprocal of the square root of detg (except in the case where dim=4; see below).
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In the case where the dimension is 4, it is assumed that the geometry is for Relativity applications, in which case, detg is assumed to be negative. Thus, a factor of is used in computing the covariant components and a factor of is used in computing the contravariant components.
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Indexing Function: the results are completely anti-symmetric; their component arrays use Maple's antisymmetric indexing function.
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Examples
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Compute the Levi-Civita pseudo-tensor in the Schwarzschild geometry of Relativity:
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cov_LC and con_LC are totally antisymmetric:
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The Levi-Civita components for the Poincare half-plane:
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