# What is pseudo-tensor?

What is the pseudo-tensor in relativity? How do we transform tensor and pseudo-tensor under parity?

• When you say 'the' pseudotensor, do you mean a specific one? There's more than one, you know. Aug 4 '14 at 12:15
• So, Can we write for a vector as a tensor $V^a$, $P(V^a)=V^a$ for $a=0,1,2,3$, I think we can not. Aug 4 '14 at 13:22
• More on pseudo-tensors: physics.stackexchange.com/q/32159/2451 Aug 4 '14 at 14:51

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.
• Can you give me an example? Aug 4 '14 at 12:28
• @user55944 the determinant of the metric, $g$
– Danu
Aug 4 '14 at 12:31
• Yes. Aug 4 '14 at 12:31
• ok!, What about $g_{ab}$, $R_{ab}$, $\epsilon_{abcd}$? How we kow whether they are tensor or pseudo tensor? Aug 4 '14 at 12:42
• The fully antisymmetric one, like $\epsilon$, is the best example of the pseudotensor. Unless something else is explicitly written using such an epsilon and other things that are ordinary tensors, it is probably an ordinary tensor, like g and R. So you de facto count the number of epsilon factors in the most appropriate "definition" of the tensor. If the number is odd, then it is a pseudotensor, even is tensor. Aug 4 '14 at 12:53

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.