Tensor Density - Definition

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.