Tensor density

pseudo-tensor

A geometric object described in a coordinate system $ x = ( x ^ {1}, \dots, x ^ {n} ) $ by $ n ^ {p+q} $ components $ a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $, $ 1 \leq i _ \nu , j _ \mu \leq n $, transforming under a change of coordinates $ x \mapsto y = ( y ^ {1} \dots y ^ {n} ) $ according to the formula

$$ a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } = \ \Delta ^ {- \kappa } a _ {\beta _ {1} \dots \beta _ {q} } ^ {\alpha _ {1} \dots \alpha _ {p} } \frac{\partial y ^ {i _ {1} } }{\partial x ^ {\alpha _ {1} } } \dots \frac{\partial y ^ {i _ {p} } }{\partial x ^ {\alpha _ {p} } } \cdot $$

$$ \cdot \frac{\partial x ^ {\beta _ {1} } }{\partial y ^ {j _ {1} } } \dots \frac{\partial x ^ {\beta _ {q} } }{\partial y ^ {j _ {q} } } , $$

where $ \Delta = \mathop{\rm det} ( \partial y ^ {i} / \partial x _ {k} ) $. The number $ \kappa $ is called the weight of the tensor density. When $ \kappa = 0 $, the tensor density is a tensor (cf. Tensor on a vector space). Concepts such as type, valency, covariance, contravariance, etc. are introduced similar to the corresponding tensor concepts. Tensor densities of types $ ( 1, 0) $ and $ ( 0, 1) $ are called vector densities. Tensor densities of type $ ( 0, 0) $ are called scalar densities.

Comments

A tensor density as defined above is also called a relative tensor. One distinguishes between odd relative tensors of weight $ k $, which transform as above, and even relative tensors, which transform according to the same formula except that $ \Delta $ is replaced by its absolute value $ | \Delta | $. In [a2] an even tensor density is simply called a "tensor density" and an odd one is called a tensor $ \Delta $-density.

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , I , Publish or Perish (1970) pp. 437ff
[a2]	J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 12 (Translated from German)