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Book: Gravitational Waves-Vol 1 by Maggiore; pg 147, note 48

The book says that the components of angular momentum $\vec{L}$ are unchanged under parity (reversing the orientation of the axes).

Now, whether I think of $\vec{L}$ as

  1. $\vec{r}\times \vec{p}$ using the right-hand rule (as taught in introductory physics) OR

  2. $\epsilon^{i}_{jk} r^{j} p^{k} $

(where $\epsilon^{i}_{jk}$ is the Levi-Civita tensor), I find that the sign of components of $L^i$ flips. Why is there this discrepancy between my conclusion and the book?

PS: When the axes orientation is reversed, a right-handed coordinate system becomes a left-handed one and the sign of components of the Levi-Civita tensor flips, thus changing the sign of $\epsilon^{i}_{jk} r^{j} p^{k} $.

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  • $\begingroup$ The sign of the Levi-Civita symbol does not change under parity. $\endgroup$ – Buzz Feb 16 at 21:10
  • $\begingroup$ @Buzz Pg 82 of Sean Carroll's book says that the Levi-Civita symbol does change sign under right-handed to left-handed coordinate system transformation. When I reverse the orientation of the axes, I change from right to left-handedness. $\endgroup$ – Sashwat Tanay Feb 17 at 18:10
  • $\begingroup$ I don't remember what Sean says in his book. However, there is an issue of passive versus active transformations that you are probably misunderstanding. $\endgroup$ – Buzz Feb 18 at 1:17
  • $\begingroup$ I am trying to work in the passive sense. $\endgroup$ – Sashwat Tanay Feb 18 at 3:34
  • $\begingroup$ There is no physics in passive transformations, just a relabeling of the coordinates. You cannot learn anything new by looking at the passive transformations. $\endgroup$ – Buzz Feb 18 at 3:57

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