Tensor density

In physics, the term of the tensor of Hermann Weyl was introduced to the " difference between quantity and intensity, as far as he has physical meaning " to grasp " the tensors are the intensity, the tensor-densities the quantity sizes ." After Weyl tensor one assigns a coordinate system, a tensor field such that it is multiplied with a change of coordinates to the absolute value of the Jacobian. A tensor of zero level is therefore a scalar density whose integral according to the transform set provides an invariant.

More generally we define by multiplying by a power of the magnitude of the Jacobian of a weighted tensor. The weight is the exponent in this potency. ( By contrast, uses the term Weyl tensor ( density) with weight in a different meaning: The weight is the exponent in the power of the calibration ratio is multiplied by the at a rescaling of the metric. ). A different definition uses the Funktionaldeterminate instead of its amount. For straight weight, both definitions coincide. For odd weight the terms and tensor Pseudotensordichte be reversed, because Pseudotensoren or Pseudotensordichten be multiplied by the sign of the Jacobian. Hereinafter the first definition is used. ( Another variant differs in the sign of the weight. )

Definition

A tensor density of weight assigns coordinates to a tensor, which under a coordinate change the relationship

Applies. The tensor components with respect to the coordinates are. Then the following transformation law is the change of coordinates:

Examples

A tensor density with weight zero is an ordinary tensor field.

It is the value of the determinant of the matrix components of the metric tensor (or more generally of a doubly covariant tensor ). Then because of the product set for a scalar determinant density from the weight 2 and a density of the weight of the first scalar is a tensor, it is a tensor on the weight. Conversely, any tensor density of weight written as such a product by putting.

An example of a weight of -1 Pseudotensordichte the Levi -Civita tensor.