Tensor density

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In differential geometry, 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.

For example, a mixed rank-2 tensor density of weight W transforms as:

$\bar{\mathfrak{A}}^{\alpha}_{\beta} = \det{\left[\frac{\partial x^{\iota}}{\partial \bar{x}^{\gamma}}\right]} ^{W} \, \frac{\partial \bar{x}^{\alpha}}{\partial x^{\delta}} \, \frac{\partial x^{\epsilon}}{\partial \bar{x}^{\beta}} \, \mathfrak{A}^{\delta}_{\epsilon}$

where $\mathfrak{A}$ is the rank-2 tensor density in the x coordinate system, $\bar{\mathfrak{A}}$ is the transformed tensor density in the $\bar{x}$ coordinate system; and we use the Jacobian determinant.

A tensor density of weight zero is an ordinary tensor.

A distinction is made between odd tensor densities, in which (as here) the term attributable to the determinant may be negative, and even tensor densities which have a power of the absolute value of the determinant, or an even power of it, in the transformation rule.

[edit] General relativity

If one transforms from a (locally inertial) coordinate system where the metric is the Minkowski metric, diag (-1, +1, +1, +1), to an arbitrary coordinate system, the absolute value of the Jacobian determinant will be equal to the square-root of the negative of the determinant of the (covariant) metric, i.e. $\sqrt{- g}$ where $g = \, \det \, [g_{\mu \nu}]\,$ is the determinant of the metric tensor, which is negative. Because the determinant is negative, its sign must be reversed before taking the square-root.

$\left\vert \det{\left[\frac{\partial x^{\iota}}{\partial \bar{x}^{\gamma}}\right]} \right\vert = \sqrt{-\bar{g}} \,, \, \text{if} \, g_{\alpha \beta} = \eta_{\alpha \beta} \,.$

Consequently, an even tensor density, $\mathfrak{p}^{\mu ...}$ , of weight W, is of the form

$\mathfrak{p}^{\mu ...} = \sqrt{(-g)}^W p^{\mu ...} \,$

where $p^{\mu ...} \,$ is a tensor which everywhere has the same values as the tensor density in a locally inertial coordinate system.

The covariant derivative of an even tensor density is defined as

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

In practice, this means that you add another term to the covariant derivative, namely

$- W \, \Gamma^{\delta}_{\delta \alpha} \, \mathfrak{p}^{\mu ...} \,.$

Or equivalently, the product rule is obeyed

$(\mathfrak{p}^{\mu ...} \mathfrak{q}^{\nu ...})_{; \alpha} = (\mathfrak{p}^{\mu ...}_{; \alpha}) \mathfrak{q}^{\nu ...} + \mathfrak{p}^{\mu ...} (\mathfrak{q}^{\nu ...}_{; \alpha}) \,$