A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is additionally multiplied or weighted by a power W of the Jacobian determinant of the coordinate transition function or its absolute value.
are the usual standard basis vectors in Euclidean space.
This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations.
However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to change the basis while the vector
More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation.
Physicists will often call this representation of a geometric object a tensor if it transforms under a sequence of linear maps given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids).
A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on
which, when expanded is just the original expression but multiplied by the determinant of
In fact this representation could be thought of as a two index tensor transformation, but instead, it is computationally easier to think of the tensor transformation rule as multiplication by
rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is
Objects which transform in this way are called tensor densities because they arise naturally when considering problems regarding areas and volumes, and so are frequently used in integration.
Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-reversing coordinate transformations.
In this article we have chosen the convention that assigns a weight of +2 to
, the determinant of the metric tensor expressed with covariant indices.
Some authors use a sign convention for weights that is the negation of that presented here.
[4] In contrast to the meaning used in this article, in general relativity "pseudotensor" sometimes means an object that does not transform like a tensor or relative tensor of any weight.
For example, a mixed rank-two (authentic) tensor density of weight
Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when
The transformations for even and odd tensor densities have the benefit of being well defined even when
is an even integer the above formula for an (authentic) tensor density can be rewritten as Similarly, when
is a non-singular matrix and a rank-two tensor density of weight
with covariant indices then its matrix inverse will be a rank-two tensor density of weight −
For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely
where, for the metric connection, the covariant derivative of any function of
is the amount of electric charge crossing the 3-volume element
divided by that element — do not use the metric in this calculation) is a contravariant vector density of weight +1.
(that is, the linear momentum transferred from the electromagnetic field to matter within a 4-volume element
divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.
In N-dimensional space-time, the Levi-Civita symbol may be regarded as either a rank-N covariant (odd) authentic tensor density of weight −1 (εα1⋯αN) or a rank-N contravariant (odd) authentic tensor density of weight +1 (εα1⋯αN).
Notice that the Levi-Civita symbol (so regarded) does not obey the usual convention for raising or lowering of indices with the metric tensor.