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

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  • $\begingroup$ When you say 'the' pseudotensor, do you mean a specific one? There's more than one, you know. $\endgroup$ Aug 4 '14 at 12:15
  • $\begingroup$ 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. $\endgroup$
    – user55944
    Aug 4 '14 at 13:22
  • $\begingroup$ More on pseudo-tensors: physics.stackexchange.com/q/32159/2451 $\endgroup$
    – Qmechanic
    Aug 4 '14 at 14:51
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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.
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  • $\begingroup$ Can you give me an example? $\endgroup$
    – user55944
    Aug 4 '14 at 12:28
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    $\begingroup$ @user55944 the determinant of the metric, $g$ $\endgroup$
    – Danu
    Aug 4 '14 at 12:31
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    $\begingroup$ Yes. $\endgroup$ Aug 4 '14 at 12:31
  • $\begingroup$ ok!, What about $g_{ab}$, $R_{ab}$, $\epsilon_{abcd}$? How we kow whether they are tensor or pseudo tensor? $\endgroup$
    – user55944
    Aug 4 '14 at 12:42
  • $\begingroup$ 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. $\endgroup$ Aug 4 '14 at 12:53
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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.

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