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 (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. A distinction is made among (authentic) 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 formula_2 is a vector in 3-dimensional Euclidean space, formula_3 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 formula_4 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 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). 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 (area of spanned parallelogram) on formula_5. The representation is given by in the standard basis by 
If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to formula_7 where formula_8 is some 2 by 2 matrix of real numbers. Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be:
which, when expanded is just the original expression but multiplied by the determinant of formula_10, which is also formula_11. 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 formula_11, rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is formula_13 matrix multiplications, which for large formula_14 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.
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.
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.
For example, a mixed rank-two (authentic) tensor density of weight "W" transforms as:
where formula_16 is the rank-two tensor density in the formula_17 coordinate system, formula_18 is the transformed tensor density in the formula_19 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. (However, see even and odd tensor densities below.)
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 (authentic) tensor density can be rewritten as
Similarly, when "W" is an odd integer the formula for an (authentic) 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 (even) 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.
Matrix inversion and matrix determinant of tensor densities.
If formula_23 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 formula_23 is a rank-two tensor density of weight "W" with covariant indices then the matrix determinant formula_25 will have weight , where "N" is the number of space-time dimensions. If formula_26 is a rank-two tensor density of weight "W" with contravariant indices then the matrix determinant formula_27 will have weight . The matrix determinant formula_28 will have weight "NW".
General relativity.
Relation of Jacobian determinant and metric tensor.
Any non-singular ordinary tensor formula_29 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 (using that the determinant of a matrix product is the product of the determinants), dividing both sides by formula_31, and taking their square root gives
When the tensor "T" is the metric tensor, formula_33, and formula_34 is a locally inertial coordinate system where formula_35diag(−1,+1,+1,+1), the Minkowski metric, then formula_36−1 and so
where formula_38 is the determinant of the metric tensor formula_39.
Use of metric tensor to manipulate tensor densities.
Consequently, an even tensor density, formula_40, of weight "W", can be written in the form
where formula_42 is an ordinary tensor. In a locally inertial coordinate system, where formula_43, it will be the case that formula_40 and formula_42 will be represented with the same numbers.
When using the metric connection (Levi-Civita 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 formula_49 is always zero,
Examples.
The expression formula_51 is a scalar density. By the convention of this article it has a weight of +1. 
The density of electric current formula_52 (e.g., formula_53 is the amount of electric charge crossing the 3-volume element formula_54 divided by that element — do not use the metric in this calculation) is a contravariant vector density of weight +1. It is often written as formula_55 or formula_56, where formula_57 and the differential form formula_58 are absolute tensors, and where formula_59 is the Levi-Civita symbol; see below.
The density of Lorentz force formula_60 ("i.e.", the linear momentum transferred from the electromagnetic field to matter within a 4-volume element formula_61 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⋯α"N") or a rank-"N" contravariant (odd) authentic tensor density of weight +1 (εα1⋯α"N"). Notice that the Levi-Civita symbol (so regarded) 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 formula_63 is always negative, this is never equal to formula_64.
The determinant of the metric tensor,
is an (even) authentic scalar density of weight +2.