- Tensor density
A tensor density transforms as a
tensor , except that it is additionally multiplied or "weighted" by a power of theJacobian determinant.For example, a rank-3 tensor density of weight W transforms as::A_{ijk}^prime =egin{vmatrix} alpha end{vmatrix}^Wcdot alpha_i^l cdot alpha_j^m cdot alpha_k^n cdot A_{lmn}where "A" is the rank-3 tensor density, "A′" is the transformed tensor density, and "α" is the transformation matrix. Here egin{vmatrix} alpha end{vmatrix} is the Jacobian determinant.
A tensor density of weight zero is an ordinary tensor.
A distinction is made between "odd" tensor densities, in which (as here) the term attributable to the determinant may be negative, and "even" tensor densities which have a power of the
absolute value of the determinant, or an even power of it, in the transformation rule.General relativity
In general relativity "α" the transformation matrix is sqrt -g where g = , det , (g_{mu u}), is the
determinant of the metric tensor and is < 0. Hence its appearance as g .Consequently a tensor density, mathfrak{g}^{mu ...}, of weight W, is of the form:
:mathfrak{g}^{mu ...}=sqrt{(-g)^W}g^{mu ...}
where g^{mu ...}, is a tensor.
The
covariant derivative of a tensor density is defined as::mathfrak{g}^{mu ...};_lambda=sqrt{(-g)^W}(g^{mu ...});_lambda
Or equivalently the product rule is obeyed::mathfrak{g}^{mu ...}mathfrak{h}^{ u ...});_lambda=(mathfrak{g}^{mu ...};_lambda) mathfrak{h}^{ u ...} + mathfrak{g}^{ u ...} (mathfrak{h}^{mu ...};_lambda)
and the covariant derivative of the g, the determinant of the metric tensor, is always zero:
g;_lambda=0 ,
ee also
*
Pseudotensor
*Noether's theorem
*Variational principle
*Conservation law
*Action (physics) External links
* [http://mathworld.wolfram.com/TensorDensity.html Mathworld description for pseudotensor] .
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