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 transform under the somewhat pathological 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.^{[citation needed]}

Tensor and pseudotensor densities

For example, a mixed rank-2 (authentic) tensor density of weight W transforms as

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

where $\bar{\mathfrak{T}}$ is the order-2 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-2 pseudotensor density of weight W transforms as

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

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

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

This formula has the advantage of being well defined even when W is an arbitrary real number.

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

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

This formula has the advantage of being well defined even when W is an arbitrary real number.

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.

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 ${g}_{\mu\nu}$.

Use of metric tensor to manipulate tensor densities

Consequently, an even tensor density, $\mathfrak{T}^{\mu \dots}_{\nu \dots}$, of weight W, can be written in the form

$\mathfrak{T}^{\mu \dots}_{\nu \dots} = \sqrt{-g}\;^W T^{\mu \dots}_{\nu \dots} \,,$

where $T^{\mu \dots}_{\nu \dots} \,$ is an ordinary tensor. In a locally inertial coordinate system, where $g_{\kappa\lambda} = \eta_{\kappa\lambda}$, it will be the case that $\mathfrak{T}^{\mu \dots}_{\nu \dots}$ and $T^{\mu \dots}_{\nu \dots} \,$ 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

$\mathfrak{T}^{\mu \dots}_{\nu \dots ; \alpha} = \sqrt{-g}\;^W T^{\mu \dots}_{\nu \dots ; \alpha} = \sqrt{-g}\;^W (\sqrt{-g}\;^{-W} \mathfrak{T}^{\mu \dots}_{\nu \dots})_{;\alpha} \,.$

For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely

$- W \, \Gamma^{\delta}_{\delta \alpha} \, \mathfrak{T}^{\mu \dots}_{\nu \dots} \,$

to the expression which would be appropriate for the covariant derivative of an ordinary tensor.

Equivalently, the product rule is obeyed

$(\mathfrak{T}^{\mu \dots}_{\nu \dots} \mathfrak{S}^{\sigma \dots}_{\tau \dots})_{; \alpha} = (\mathfrak{T}^{\mu \dots}_{\nu \dots; \alpha}) \mathfrak{S}^{\sigma \dots}_{\tau \dots} + \mathfrak{T}^{\mu \dots}_{\nu \dots} (\mathfrak{S}^{\sigma \dots}_{\tau \dots; \alpha}) \,,$

where, for the metric connection, the covariant derivative of any function of $g_{\kappa\lambda}$ is always zero,

$\begin{align} g_{\kappa\lambda ; \alpha} & = 0 \\ (\sqrt{-g}\;^W)_{; \alpha} & = (\sqrt{-g}\;^W)_{, \alpha} - W \Gamma^{\delta}_{\delta \alpha} \sqrt{-g}\;^W = \frac W2 g^{\kappa\lambda} g_{\kappa\lambda,\alpha} \sqrt{-g}\;^W - W \Gamma^{\delta}_{\delta \alpha} \sqrt{-g}\;^W = 0 \,. \end{align}$

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, e.g., Weinberg uses a convention that gives it a weight of −1.

The density of electric current $\mathfrak{J}^\mu$ (e.g., $\mathfrak{J}^2$ is the amount of electric charge crossing the 3-volume element $d x^3 \, d x^4 \, d x^1$ 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 $\mathfrak{J}^\mu = J^\mu \sqrt{-g}$, 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 $d x^1 \, d x^2 \, d x^3 \, d x^4$ divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.

The determinant of the metric tensor, g = det(g_μν), is an (even) authentic scalar density of weight +2.

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 $\varepsilon_{\kappa\lambda\mu\nu}$.

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), "Tensor density", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=t/t092390 .
• Charles Misner; Kip S Thorne & John Archibald Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. p. 501ff. ISBN 0-7167-0344-0.