We have a tensor associated with a magnetic field in special relativity defined as $B_{jk} = \partial _j A_k - \partial _k A_j$ and two related equations: $B^i = \frac {1}{2} \epsilon ^{ijk} B_{jk}$ $\iff$ $B_{jk}= \epsilon _{jkl} B^{l}$, where $A$ is a vector field and $\epsilon$ is a Levi-Civita symbol. Even though I have a subtle knowledge about tensor algebra, but nonconsistent, I don't understand why is in the second equation that $1/2$. After I plugged the third equation into the second I got:

$B^i = \frac {1}{2} \epsilon ^{ijk}\epsilon _{jkl} B^{l} = \frac {1}{2}B^{l}\epsilon ^{i}_{l} $ ?


1 Answer 1


$$\varepsilon^{ijk}\varepsilon_{jkl} = 2\delta^i_l$$

So, you need $\frac{1}{2}$ to get rid of that $2$ factor.

Also if you are working in 4D (which I assume you do, because SR), then what you call magnetic field tensor ($B_{jk}$) is in fact electromagnetic field tensor, which contains both electric and magnetic fields, and it's customary to denote it $F_{jk}$.

  • $\begingroup$ I did not know this identity exists, thank you. $\endgroup$
    – user148787
    Commented Mar 4, 2018 at 14:55
  • $\begingroup$ @Leif, you could've derived it performing element-wise multiplication. $\endgroup$
    – Tajimura
    Commented Mar 4, 2018 at 15:13

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