Pseudotensor

In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g., a proper rotation), but additionally changes sign under an orientation reversing coordinate transformation (e.g., an improper rotation, which is a transformation that can be expressed as a proper rotation followed by reflection). In this sense, it is a generalization of a pseudovector.

There is a second meaning for pseudotensor, restricted to general relativity; tensors obey strict transformation laws, but pseudotensors are not so constrained. Consequently, the form of a pseudotensor will, in general, change as the frame of reference is altered. An equation containing pseudotensors which holds in one frame will not necessarily hold in a different frame; this makes pseudotensors of limited relevance because equations in which they appear are not invariant in form.

Two quite different mathematical objects are called a pseudotensor in different contexts.

The first context is essentially a tensor multiplied by an extra sign factor, such that the pseudotensor changes sign under reflections when a normal tensor does not. According to one definition, a pseudotensor P of the type (p, q) is a geometric object whose components in an arbitrary basis are enumerated by (p + q) indices and obey the transformation rule

under a change of basis.

Here ${\displaystyle {\hat {P}}_{\,j_{1}\ldots j_{p}}^{i_{1}\ldots i_{q}},P_{l_{1}\ldots l_{p}}^{k_{1}\ldots k_{q}}}$ are the components of the pseudotensor in the new and old bases, respectively, ${\displaystyle A^{i_{q}}{}_{k_{q}}}$ is the transition matrix for the contravariant indices, ${\displaystyle B^{l_{p}}{}_{j_{p}}}$ is the transition matrix for the covariant indices, and ${\displaystyle (-1)^{A}=\mathrm {sign} (\det(A^{i_{q}}{}_{k_{q}}))=\pm {1}}$. This transformation rule differs from the rule for an ordinary tensor only by the presence of the factor (−1)^A.

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