In differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another (see classical treatment of tensors), 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.
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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.
Tensor and pseudotensor densities
For example, a mixed rank-two (authentic) tensor density of weight W transforms as:
where
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.
Algebraic properties
- A linear combination of tensor densities of the same type and weight W is again a tensor density of that type and weight.
- A product of two tensor densities of any types and with weights W1 and W2 is a tensor density of weight W1 + W2.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.
- The contraction of indices on a tensor density with weight W again yields a tensor density of weight W.
- Using (2) and (3) one sees that raising and lowering indices using the metric tensor (weight 0) leaves the weight unchanged.
Matrix inversion and matrix determinant of tensor densities
If
If
Relation of Jacobian determinant and metric tensor
Any non-singular ordinary tensor
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
When the tensor T is the metric tensor,
where
Use of metric tensor to manipulate tensor densities
Consequently, an even tensor density,
where
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 which 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
Examples
The expression
The density of electric current
The density of Lorentz force
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. That is, it is true that
but in general relativity, where
The determinant of the metric tensor,
is an (even) authentic scalar density of weight +2.