Tensor density

A tensor density transforms as a tensor, except that it is additionally multiplied or "weighted" by a power of the Jacobian determinant.

For example, a rank-3 tensor density of weight W transforms as::where "A" is the rank-3 tensor density, "A′" is the transformed tensor density, and "α" is the transformation matrix. Here 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] .