What is the pseudo-tensor in relativity? How do we transform tensor and pseudo-tensor under parity?
From the Wikipedia page on pseudotensors,
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).
The action of parity on a tensor or pseudotensor depends on the number of indices it has (i.e. its tensor rank):
- Tensors of odd rank (e.g. vectors) reverse sign under parity.
- Tensors of even rank (e.g. scalars, linear transformations, bivectors, metrics) retain their sign under parity.
- Pseudotensors of odd rank (e.g. pseudovectors) retain their sign under parity.
- Pseudotensors of even rank (e.g. pseudoscalars) reverse sign under parity.
The word "pseudotensor" is used in the sense that Emilio Pisanty mentioned, but it also has a completely different and fairly common meaning in general relativity: a multidimensional array of numbers indexed by spacetime coordinates that doesn't transform as a tensor. Energy pseudotensors are an example. Both of these meanings are mentioned in the Wikipedia article.