Editing Pseudotensor (section)

{{Short description|Type of physical quantity}}
{{Use American English|date=March 2019}}

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 (mathematics)|reflection]]. This is a generalization of a ''[[pseudovector]]''. To evaluate a tensor or pseudotensor sign, it has to be [[Tensor contraction|contracted]] with some vectors, as many as its [[Tensor (intrinsic definition)#Tensor rank|rank]] is, belonging to the space where the rotation is made while keeping the tensor coordinates unaffected (differently from what one does in the case of a base change). Under improper rotation a pseudotensor and a proper tensor of the same rank will have different sign which depends on the rank being [[Parity (mathematics)|even or odd]]. Sometimes inversion of the axes is used as an example of an improper rotation to see the behaviour of a pseudotensor, but it works only if vector space dimensions is odd otherwise inversion is a proper rotation without an additional reflection.

There is a second meaning for '''pseudotensor''' (and likewise for ''pseudovector''), restricted to [[general relativity]]. Tensors obey strict transformation laws, but pseudotensors in this sense 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 [[Covariance and contravariance of vectors|invariant]] in form.

Mathematical developments in the 1980s have allowed pseudotensors to be understood as [[Section (fiber bundle)|sections]] of [[jet bundle]]s.