Tensor density


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, 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. A distinction is made among tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight W 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 vectors weighted by some coefficients such as
where 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 and the vector will remain 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 given a linear change of basis. In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation. In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation. A prototypical example is a matrix representing the cross product on. The representation is given by in the standard basis by
which, when expanded is just the original expression but multiplied by the determinant of, which is also. 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. 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 the domain of integration.

Definition

Some authors classify tensor densities into the two types called 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.
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. 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.

Tensor and pseudotensor densities

For example, a mixed rank-two tensor density of weight W transforms as:
where is the rank-two tensor density in the coordinate system, is the transformed tensor density in the coordinate system; and we use the Jacobian determinant. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when W is an integer.
We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-two pseudotensor density of weight W transforms as
where sgn is a function that returns +1 when its argument is positive or −1 when its argument is negative.

Even and odd tensor densities

The transformations for even and odd tensor densities have the benefit of being well defined even when W 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.
When W is an even integer the above formula for an tensor density can be rewritten as
Similarly, when W is an odd integer the formula for an tensor density can be rewritten as

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 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 +1.

Algebraic properties

  1. A linear combination of tensor densities of the same type and weight is again a tensor density of that type and weight.
  2. A product of two tensor densities of any types and with weights and is a tensor density of weight.
  3. :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.
  4. The contraction of indices on a tensor density with weight again yields a tensor density of weight.
  5. Using and one sees that raising and lowering indices using the metric tensor leaves the weight unchanged.

    Matrix inversion and matrix determinant of tensor densities

If is a non-singular matrix and a rank-two tensor density of weight W with covariant indices then its matrix inverse will be a rank-two tensor density of weight −W with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant.
If is a rank-two tensor density of weight W with covariant indices then the matrix determinant will have weight, where N is the number of space-time dimensions. If is a rank-two tensor density of weight W with contravariant indices then the matrix determinant will have weight. The matrix determinant will have weight NW.

General relativity

Relation of Jacobian determinant and metric tensor

Any non-singular ordinary tensor transforms as
where the right-hand side can be viewed as the product of three matrices. Taking the determinant of both sides of the equation, dividing both sides by, and taking their square root gives
When the tensor T is the metric tensor,, and is a locally inertial coordinate system where diag, the Minkowski metric, then −1 and so
where is the determinant of the metric tensor.

Use of metric tensor to manipulate tensor densities

Consequently, an even tensor density,, of weight W, can be written in the form
where is an ordinary tensor. In a locally inertial coordinate system, where, it will be the case that and will be represented with the same numbers.
When using the metric connection, the covariant derivative of an even tensor density is defined as
For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely
to the expression that would be appropriate for the covariant derivative of an ordinary tensor.
Equivalently, the product rule is obeyed
where, for the metric connection, the covariant derivative of any function of is always zero,

Examples

The expression is a scalar density. By the convention of this article it has a weight of +1.
The density of electric current is a contravariant vector density of weight +1. It is often written as or, where and the differential form are absolute tensors, and where is the Levi-Civita symbol; see below.
The density of Lorentz force 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 authentic tensor density of weight −1 or a rank-N contravariant authentic tensor density of weight +1. Notice that the Levi-Civita symbol does not obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that
but in general relativity, where is always negative, this is never equal to.
The determinant of the metric tensor,
is an authentic scalar density of weight +2.