Tensor density

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 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. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.

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.

Some authors use a sign convention for weights that is the negation of that presented here. In this article we have chosen the convention that assigns a weight of -2 rather than +2 to the determinant of the metric tensor expressed with covariant indices.

Tensor and pseudotensor densities

For example, a mixed rank-two (authentic) tensor density of weight W transforms as:

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

where ${\bar {\mathfrak {T}}}$ is the rank-two tensor density in the ${\bar {x}}$ coordinate system, ${\mathfrak {T}}$ is the transformed tensor density in the ${x}$ 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

${\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,,$     (pseudotensor density of (integer) weight W)

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

${\mathfrak {T}}_{\beta }^{\alpha }=\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.$     (even tensor density of weight W)

Similarly, when W is an odd integer the formula for an (authentic) tensor density can be rewritten as

${\mathfrak {T}}_{\beta }^{\alpha }=\operatorname {sgn} \left(\det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right)\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ^{W}\,{\frac {\partial {x}^{\alpha }}{\partial {\bar {x}}^{\delta }}}\,{\frac {\partial {\bar {x}}^{\epsilon }}{\partial {x}^{\beta }}}\,{\bar {\mathfrak {T}}}_{\epsilon }^{\delta }\,.$     (odd tensor density of weight W)

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.

Multiplication of tensor densities

A product of tensor densities of any types will have a weight equal to the sum of the weights of the factors. 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.

Matrix inversion and matrix determinant of tensor densities

If ${\mathfrak {T}}_{\alpha \beta }$ 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 ${\mathfrak {T}}_{\alpha \beta }$ is a rank-two tensor density of weight W with covariant indices then the matrix determinant $\det {\mathfrak {T}}_{\alpha \beta }$ will have weight NW + 2, where N is the number of space-time dimensions. The matrix determinant $\det {\mathfrak {T}}^{\alpha \beta }$ will have weight NW − 2. The matrix determinant $\det {\mathfrak {T}}_{~\beta }^{\alpha }$ will have weight NW.

General relativity

Relation of Jacobian determinant and metric tensor

A non-singular ordinary tensor $T_{\mu \nu }$ transforms as

$T_{\mu \nu }={\frac {\partial {\bar {x}}^{\kappa }}{\partial {x}^{\mu }}}{\bar {T}}_{\kappa \lambda }{\frac {\partial {\bar {x}}^{\lambda }}{\partial {x}^{\nu }}}\,,$

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 $\det({\bar {T}}_{\kappa \lambda })$, and taking their square root gives

$\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ={\sqrt {\frac {\det({T}_{\mu \nu })}{\det({\bar {T}}_{\kappa \lambda })}}}\,.$

When the tensor T is the metric tensor, ${g}_{\kappa \lambda }$, and ${\bar {x}}^{\iota }$ is a locally inertial coordinate system where ${\bar {g}}_{\kappa \lambda }=\eta _{\kappa \lambda }=$diag(−1,+1,+1,+1), the Minkowski metric, then $\det({\bar {g}}_{\kappa \lambda })=\det(\eta _{\kappa \lambda })=$ −1 and so

$\left\vert \det {\left[{\frac {\partial {\bar {x}}^{\iota }}{\partial {x}^{\gamma }}}\right]}\right\vert ={\sqrt {-{g}}}\,,$

where ${g}=\det({g}_{\mu \nu })$ is the determinant of the metric tensor ${g}_{\mu \nu }$.

Relation between tensors and tensor densities

Given a type $(n,m)$ tensor density of weight $w$ one can make it into a 'ordinary' ('normal') tensor by multiplying it by $\left\vert g\right\vert ^{\frac {w}{2}}$. So, $T_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}=\left\vert g\right\vert ^{\frac {w}{2}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$, where ${\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ is the tensor density and $T_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ is the corresponding tensor.

To see why this is so consider the transformation of the metric tensor:

$g_{\mu ^{\prime }\nu ^{\prime }}={\frac {\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}}{\frac {\partial x^{\nu }}{\partial x^{\nu ^{\prime }}}}g_{\mu \nu }$.

Taking the determinant of both sides and letting $g\equiv {\text{det}}(g_{\mu \nu })$ we get:

$g^{\prime }={\text{det}}\left({\frac {\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}}{\frac {\partial x^{\nu }}{\partial x^{\nu ^{\prime }}}}g_{\mu \nu }\right)={\text{det}}\left({\frac {\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}}\right){\text{det}}\left({\frac {\partial x^{\nu }}{\partial x^{\nu ^{\prime }}}}\right)g={\text{det}}\left({\frac {\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}}\right)^{2}g$.

Taking the (positive) square root and rearranging we get:

${\sqrt {\left\vert g\right\vert }}=\left\vert {\frac {\partial x^{\mu }}{\partial x^{\mu ^{\prime }}}}\right\vert ^{-1}{\sqrt {\left\vert g^{\prime }\right\vert }}$ .

Thus ${\sqrt {\left\vert g\right\vert }}$ is a tensor density of weight $-1$.

Consider now, the expression $\left({\sqrt {\left\vert g\right\vert }}\right)^{p}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ where ${\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ is a tensor density of weight $w$ and $p$ is a real constant to be solved. We want the exprssion to be a ordinary tensor (i.e. weight is $0$). Since ${\sqrt {\left\vert g\right\vert }}$ is a tensor density of weight $-1$, we have a total weight of $-p+w$. Thus for $\left({\sqrt {\left\vert g\right\vert }}\right)^{p}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ to be a ordinary tensor we must have $-p+w=0$. Thus $p=w$!

So if ${\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ is a tensor density of weight $w$; then, $T_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}=\left\vert g\right\vert ^{\frac {w}{2}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$ is a good, ordinary $(n,m)$ type tensor.

Examples

See also: Maxwell's equations in curved spacetime

The expression ${\sqrt {-g}}$ is a scalar density. By the convention of this article it has a weight of -1, though, other conventions exist. The density of electric current ${\mathfrak {J}}^{\mu }$ (e.g., ${\mathfrak {J}}^{2}$ is the amount of electric charge crossing the 3-volume element $dx^{3}\,dx^{4}\,dx^{1}$ 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 $J^{\mu }={\mathfrak {J}}^{\mu }{\sqrt {-g}}$, where $J^{\mu }\,$ is an absolute tensor.

The density of Lorentz force ${\mathfrak {f}}_{\mu }$ (i.e., the linear momentum transferred from the electromagnetic field to matter within a 4-volume element $dx^{1}\,dx^{2}\,dx^{3}\,dx^{4}$ 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 (ε_α1…αN) or a rank-N contravariant (odd) authentic tensor density of weight +1 (ε^α₁…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

$\varepsilon ^{\alpha \beta \gamma \delta }\,g_{\alpha \kappa }\,g_{\beta \lambda }\,g_{\gamma \mu }g_{\delta \nu }\,=\,\varepsilon _{\kappa \lambda \mu \nu }\,g\,,$

but in general relativity, where $g$ is always negative, this is never equal to $\varepsilon _{\kappa \lambda \mu \nu }$.

The determinant of the metric tensor,

$g=\det \left(g_{\rho \sigma }\right)={\frac {1}{4!}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }\,,$

is an (even) authentic scalar density of weight -2.

See also

• relative scalar
• Pseudotensor
• Noether's theorem
• Variational principle
• Conservation law
• Action (physics)

References

• Spivak, Michael (1999), A comprehensive introduction to differential geometry, Vol I (3rd ed.), p. 134.
• Kuptsov, L.P. (2001) [1994], "Tensor density", Encyclopedia of Mathematics, EMS Press.
• Charles Misner; Kip S Thorne & John Archibald Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. p. 501ff. ISBN 0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link)