In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. 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 of the Jacobian determinant of the coordinate transition function or its absolute value. A tensor density with a single index is called a vector density are called tensor capacity. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.

Motivation

In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself. An example would be decomposing a vector into a sum of basis is a vector in 3-dimensional Euclidean space 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 remains fixed in physical space. 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

is completely infeasible). 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.

Definition

Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article. Other authors classify them differently, into the types called even tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd.

Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-reversing coordinate transformations. Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-preserving coordinate transformations.

, the determinant of the metric tensor expressed with covariant indices. With this choice, classical densities, like charge density, will be represented by tensor densities of weight +1. Some authors use a sign convention for weights that is the negation of that presented here.

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.

Tensor and pseudotensor densities

transforms as:

1. )

coordinate system; and we use the Jacobian determinant is an integer. (However, see even and odd tensor densities below.)

transforms as

1. )

where is a function that returns +1 when its argument is positive or −1 when its argument is negative.

Even and odd tensor densities

is not an integer. Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2.

An even tensor density transforms as follows. Although the formula works for any real valued weight , the name arises because the transformation is equivalent to the transformation of an (authentic) tensor density transform when its weight is even.

1. )

Similarly, an odd tensor density transforms as follows. Although the formula works for any real valued weight , the name arises because the transformation is equivalent to the transformation of an (authentic) tensor density transform when its weight is odd.

1. )

Weights of zero and one

A tensor density of any type that has weight zero is also called an absolute tensor. An authentic tensor density of weight zero, which is also an even tensor density of weight zero, is also called an ordinary tensor.

If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is .

Algebraic properties

1. A linear combination (also known as a weighted sum is again a tensor density of that type and weight.

2. Furthermore, a product of authentic tensor densities and pseudotensor densities will be an authentic tensor density when an even number of the factors are pseudotensor densities; it will be a pseudotensor density when an odd number of the factors are pseudotensor densities. Similarly, a product of even tensor densities and odd tensor densities will be an even tensor density when an even number of the factors are odd tensor densities; it will be an odd tensor density when an odd number of the factors are odd tensor densities.

3. Raising and lowering indices using the metric tensor (which is authentic, even, and of weight 0) leaves the weight unchanged, as can be proved by combining (2) and (3).

Matrix inversion and matrix determinant of tensor densities

with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant.

General relativity

Relation of Jacobian determinant and metric tensor

is the metric tensordiag(−1,+1,+1,+1), the Minkowski metric

Use of metric tensor to manipulate tensor densities

will be represented with the same numbers.

When using the metric connection (Levi-Civita connection), the covariant derivative

to the expression that would be appropriate for the covariant derivative of an ordinary tensor.

Examples

is a scalar density. By the convention of this article it has a weight of +1.

and the differential form are absolute tensors is the Levi-Civita symbol; see below.

The density of Lorentz force divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.

-dimensional space-time, the Levi-Civita symbol contravariant (odd) authentic tensor density of weight +1 covariant (odd) authentic tensor density of weight −1 Notice that the Levi-Civita symbol (so regarded) does

The determinant is an (even) authentic scalar density of weight +2, being the contraction of the product of 2 (odd) authentic tensor densities of weight +1 and four (even) authentic tensor densities of weight 0.

See also

Notes

References

• .