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 tensor density with a single index is called a vector density. 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.^[1][2][3] 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 ${\displaystyle {\vec {v}}}$ is a vector in 3-dimensional Euclidean space, ${\displaystyle c_{i}\in \mathbb {R} ^{1}{\text{ and }}{\vec {e}}_{i}}$ 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 ${\displaystyle {\vec {v}}}$ 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 ${\displaystyle \mathbb {R} ^{2}.}$ 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 ${\textstyle {\begin{bmatrix}u'_{1}&u'_{2}\end{bmatrix}}^{\textsf {T}}=A{\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}}^{\textsf {T}}}$ where ${\displaystyle A}$ 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 ${\displaystyle A^{-1},}$ which is also ${\textstyle {\frac {1}{\det A}}.}$ 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 ${\textstyle {\frac {1}{\det A}},}$ rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is ${\displaystyle n,n\times n}$ matrix multiplications, which for large ${\displaystyle n}$ 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

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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 ${\displaystyle g=\det \left(g_{\rho \sigma }\right)}$, 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.^[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.

Tensor and pseudotensor densities

For example, a mixed rank-two (authentic) tensor density of weight ${\displaystyle W}$ transforms as:^[5][6]

${\displaystyle {\mathfrak {T}}_{\beta }^{\alpha }=\left(\det {\left}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,}$     ((authentic) tensor density of (integer) weight W)

where ${\displaystyle {\bar {\mathfrak {T}}}}$ is the rank-two tensor density in the ${\displaystyle \stackrel{¯<!--\; ¯\; -->}{x}}$Zdroj:https://en.wikipedia.org?pojem=Tensor_density
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