# Sign of components of $\vec{L}$ seems to change under parity; book says otherwise

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

• The sign of the Levi-Civita symbol does not change under parity. – Buzz Feb 16 at 21:10
• @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. – Sashwat Tanay Feb 17 at 18:10
• 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. – Buzz Feb 18 at 1:17
• I am trying to work in the passive sense. – Sashwat Tanay Feb 18 at 3:34
• There is no physics in passive transformations, just a relabeling of the coordinates. You cannot learn anything new by looking at the passive transformations. – Buzz Feb 18 at 3:57