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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 a 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 a the transformation matrix is  where  is the determinant of the metric tensor and is < 0. Hence its appearance as - g.
Consequently a tensor density,  , of weight W, is of the form:
where  is a tensor.
The covariant derivative of a tensor density is defined as:
Or equivalently the product rule is obeyed:
and the covariant derivative of the g, the determinant of the metric tensor, is always zero:
See also
External links
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