Tensor densities

(FB/Math/BM – Question by Vasudev Shyam)

Why is the completely antisymmetric unit tensor of fourth rank εiklm considered a pseudotensor?

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εiklm is a tensor density but I think ”pseudotensor” denotes the same thing. Also known as the Levi-Civita density after a mathematician.
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εiklm is defined by ε1234 = 1 and complete antisymmetry in every coordinate s…ystem in a 4-dimensional riemannian space. Now, under a coordinate transformation x → x’, a 4-th rank (contravariant) tensor Aiklm in such a space transforms as

(1) A’iklm = Σ [pqrs] ∂x’i/∂xp·∂x’k/∂xq·∂x’l/∂xr·∂x’m/∂xs·Apqrs

and √g, where g is the determinant g of the metric tensor, transforms as

(2) √g’ = J-1·√g

where J is the Jacobian (aka functional determinant) of the coordinate transformation. If A is a tensor then √g·A is called a tensor density, and it transforms as

(3) √g’·A’iklm = J·√g· Σ [pqrs] ∂x’i/∂xp·∂x’k/∂xq·∂x’l/∂xr·∂x’m/∂xs·Apqrs

if A is 4-th rank (contravariant) and similarly in other cases.

With A = ε the right hand side of (1) becomes

Σ [pqrs] ∂x’i/∂xp·∂x’k/∂xq·∂x’l/∂xr·∂x’m/∂xs·εpqrs =

= ∂x’i/∂x1·∂x’k/∂x2·∂x’l/∂x3·∂x’m/∂x4∂x’i/∂x1·∂x’k/∂x2·∂x’l/∂x3·∂x’m/∂x4·1 + (perm)

if iklm are different, 0 else. There will be 24 terms with signs + or – because of the antisymmetry of ε, and only terms with different pqrs will be non-zero. This sum is equal to the Jacobian J of the transformation, so we get J·εiklm. But ε^iklm is defined to have the same values in all coordinate systems, i e ε’^iklm = ε^iklm so it das *not* transform as a tensor. Instead

Σ [pqrs] ∂x’i/∂xp·∂x’k/∂xq·∂x’l/∂xr·∂x’m/∂xs·εiklm/√g =

= J·εiklm/√g = J·ε’iklm/(√g’/J^(-1)) = ε’iklm/√g’

so εiklm/√g is a tensor and thus εiklm a tensor density.

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The name ”tensor density” has to do with integration. The ”volume element” dⁿx is not invariant but dⁿx’ = J·dⁿx (cf variable change in multiple integrals) and thus √g’·dⁿx’ = J-1·√g·J·dⁿx = √g·dⁿx so √g·d…ⁿx is invariant.

This implies that if A(x) is a tensor field then ∫ A(x) √g·dⁿx but *not*
∫ A(x) dⁿx is a tensor so to get a tensor by integration the integrand should be (tensor field)·√g, which appropriately is called a tensor *density*; it is a density with respect to the ”coordinate volume”.

tensor density = tensor·√g
(tensor density)/√g = tensor

In particular, in General Relativity, usuallully g < 0 so √(-g) is used instead of √g, and also n = 4. So if A(x) is a tensor (field) then A(x)·√(-g) is a tensor density under space-time coordinate transformations.

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