Pseudo-tensor

A tensor considered up to multiplication by an arbitrary function (cf. Tensor on a vector space).

Comments

More precisely, a pseudo-tensor (also called relative tensor) is a quantity $ p _ {j _ {1} \dots j _ {n} } ^ {i _ {1} \dots i _ {m} } $ which under a coordinate change transforms as

$$ \overline{p}\; {} _ {j _ {1} \dots j _ {n} } ^ {i _ {1} \dots i _ {m} } = \ \tau ( \overline{x}\; ) p _ {k _ {1} \dots k _ {n} } ^ {l _ {1} \dots l _ {m} } \frac{\partial \overline{x}\; {} ^ {i _ {1} } }{\partial x ^ {l _ {1} } } \dots \frac{\partial \overline{x}\; {} ^ {i _ {m} } }{\partial x ^ {l _ {m} } } \cdot \frac{\partial x ^ {k _ {1} } }{\partial \overline{x}\; {} ^ {j _ {1} } } \dots \frac{\partial x ^ {k _ {n} } }{\partial \overline{x}\, {} ^ {j _ {n} } } , $$

where $ \tau $ is a scalar-valued function. Most frequently (in applications), the function $ \tau $ depends in a simple manner on the Jacobian determinant $ \Delta = \mathop{\rm det} ( {\partial \overline{x}\; {} ^ {i} } / {\partial x ^ {j} } ) $ of the coordinate transformation. In [a1] the following cases are distinguished:

i) $ \tau = \Delta ^ {-w} \overline \Delta \, {} ^ {- w ^ \prime } $, a tensor $ \Delta $-density of weight $ w $ and anti-weight $ w ^ \prime $;

ii) $ \tau = | \Delta | ^ {w} $, a tensor density of weight $ w $;

iii) $ \tau = \Delta / | \Delta | $, a $ W $- tensor.

Here $ \overline \Delta $ is the complex conjugate of $ \Delta $. A tensor density of weight zero is an ordinary tensor (cf. Tensor on a vector space).

In [a2] a tensor $ \Delta $-density of weight 1 and anti-weight 0 is called a tensor density and a tensor $ \Delta $-density of weight $ - 1 $ and anti-weight 0 a tensor capacity.

References

[a1]	J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 11ff (Translated from German)
[a2]	R. Sauer (ed.) I. Szabó (ed.) , Mathematische Hilfsmittel des Ingenieurs , III , Springer (1968) pp. Sect. G.II.6