Meaning and significance of the Levi-Civita symbol

Question

I am recently reading Sean Carroll's Spacetime and Geometry: An introduction to General relativity. I am much of a beginner but am really curios to learn about GR. In the first chapter, after introducing the concept of tensors and giving examples such as the Minkowski metric and inverse metric he introduced another type of tensor, a (0,4) tensor which he called the Levi-Civita symbol, defined as follows,

$$ \epsilon_{\mu\nu\rho\sigma}= \begin{cases} +1&\text{if}\,{\mu\nu\rho\sigma} \rm \, is\,an\,even\,permutation\,of\,0123 \\ -1&\text{if}\, {\mu\nu\rho\sigma}\rm \,is\,an\,odd\,permutation\,of\,0123\\ 0&\text{otherwise} \end{cases} $$

I do not understand what is meant here by a permutation or even what 0123 exactly refer to here. I tried re-reading the subsequent paragraphs but it did not make anything clear.

Any help would be appreciated, and i would prefer if someone could explain in terms that don't require very heavy mathematics.

Answer 1

It might be good to first study the analogue in three dimensional space. $$\epsilon_{abc}= \begin{cases} +1&\text{if}\,{abc} \rm \, is\,an\,even\,permutation\,of\,123 \\ -1&\text{if}\, {abc}\rm \,is\,an\,odd\,permutation\,of\,123\\ 0&\text{otherwise} \end{cases}$$ Then try it out with $$ \hat e_1 =\hat x \qquad \hat e_2 =\hat y \qquad \hat e_3 =\hat z $$ on $$\hat e_i \times \hat e_j=\epsilon_{ijk} \hat e_k$$ [implied summation over repeated indices... don't worry about raising and lowering indices.]

For further practice, you can try to prove the BAC-CAB formula involving cross-products.

Answer 2

The $0123$ refer to the indices of $\epsilon$, so it’s saying that $\epsilon_{0123}=1$. And permutations are just the reordering of the numbers. Examples are {$0,2,1,3$}, {$0,1,3,2$}. So each time two numbers are switched that’s a permutation. Even permutations would be doing 2, 4, 6... permutations, while odd would be 1,3,5... Finally, the Levi-Civita symbol is 0 for any repetition of indices.

Some examples of evaluating the Levi-Civita symbol are: $$\epsilon_{0213}=-1$$ $$\epsilon_{0231}=1$$ $$\epsilon_{1213}=0$$

Answer 3

There are several ways to calculate the parity of a permutation. One way is to count whether you need to swap two numbers at a time an even number of times (+1) or an odd number of times (-1) in order to reach another permutation from a starting permutation.

123 to 321 needs one (or three or five or ...) swaps (transpositions) so the parity of 321 with respect to 123 is odd (-1).