compute the covariant and contravariant Levi-Civita pseudo-tensors
Levi_Civita(detg, dim, cov_LC, con_LC)
determinant of the covariant metric tensor components
dimension of the space
output parameter for the covariant Levi_Civita pseudo tensor
output parameter for the contravariant Levi_Civita pseudo tensor
Important: The tensor package has been deprecated. Use the superseding command Physics[LeviCivita] instead.
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.
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).
dim must be an integer greater than 1.
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).
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 −detg is used in computing the covariant components and a factor of −1−detg is used in computing the contravariant components.
Indexing Function: the results are completely anti-symmetric; their component arrays use Maple's antisymmetric indexing function.
Compute the Levi-Civita pseudo-tensor in the Schwarzschild geometry of Relativity:
detg ≔ −r4⁢sin⁡θ2
cov_LC and con_LC are totally antisymmetric:
The Levi-Civita components for the Poincare half-plane:
detg ≔ 1v4
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