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.

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.^[1]

Tensor and pseudotensor densities

For example, a mixed rank-two (authentic) tensor density of weight W transforms as:^[2]^[3]

{\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} \,,

((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}}^\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} \,,

(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} \,.

(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}}^\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} \,.

(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.

Algebraic properties

A linear combination of tensor densities of the same type and weight $W$ is again a tensor density of weight $W$ .
A product of two tensor densities of any types and with weights $W 1$ and $W 2$ is a tensor density of weight $W 1 + W 2$ .
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 $W$ .^[4]
Using (2) and (3) one sees that raising and lowering indices using the metric tensor (weight 0) leaves the weight unchanged.^[5]

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. If ${\mathfrak{T}}^{\alpha\beta}$ is a rank-two tensor density of weight W with contravariant indices then the matrix determinant $\det {\mathfrak{T}}^{\alpha\beta}$ will have weight $NW - 2$ . The matrix determinant $\det {\mathfrak{T}}^{\alpha}_{~\beta}$ will have weight NW.

General relativity

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}

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Relation of Jacobian determinant and metric tensor

Any 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

The expression $\sqrt{-g}$ is a scalar density. By the convention of this article it has 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.

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…α_N}) 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.

References

↑ E.g. Weinberg 1972 pp 98. The chosen convention involves in the formulae below the Jacobian determinant of the inverse transition $x \to x$ , while the opposite convention considers the forward transition $x \to x$ resulting in a flip of sign of the weight.
↑ M.R. Spiegel, S. Lipcshutz, D. Spellman (2009). Vector Analysis (2nd ed.). New York: Schaum's Outline Series. p. 198. ISBN 978-0-07-161545-7.
↑ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). p. 1417. ISBN 0-07-051400-3.
↑ Weinberg 1972 p 100.
↑ Weinberg 1972 p 100.

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 978-1-55608-010-4 .
Charles Misner; Kip S Thorne & John Archibald Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. p. 501ff. ISBN 0-7167-0344-0.
Weinberg, Steven (1972), Gravitation and Cosmology, John Wiley & sons, Inc, ISBN 0-471-92567-5

Tensors

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