Examples
See also: Maxwell's equations in curved spacetimeThe expression 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 (e.g., is the amount of electric charge crossing the 3-volume element 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, where is an absolute tensor.
The density of Lorentz force (i.e., the linear momentum transferred from the electromagnetic field to matter within a 4-volume element 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 (εα1…αN) or a rank-N contravariant (odd) authentic tensor density of weight +1 (εα1…αN). 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
but in general relativity, where is always negative, this is never equal to .
The determinant of the metric tensor,
is an (even) authentic scalar density of weight +2.
Read more about this topic: Tensor Density
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