1
$\begingroup$

In a curved spacetime with a general Minkowskian metric $g_{\mu\nu}$, I understand the difference between the Levi-Civita symbol $\pmb{\in}^{\mu\nu\rho\sigma}$ (which is a tensor-density of weight $W=+1$) with the reference $\pmb{\in}_{0123}=1$, and Levi-Civita tensor $$\varepsilon^{\mu\nu\rho\sigma} = \frac{1}{\sqrt{-g}}\pmb{\in}^{\mu\nu\rho\sigma}$$ with the reference $\varepsilon_{0123}=\sqrt{-g}$ and weight $W=0$. However, I don't understand which of them should I use in order to construct the Hodge dual. For example, consider the electromagnetic tensor $$F_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu,$$ then I have two options $$F^{*\mu\nu}=\frac12 \pmb{\in}^{\mu\nu\rho\sigma}F_{\rho\sigma}\tag{1} = \pmb{\in}^{\mu\nu\rho\sigma}\partial_\rho A_\sigma,\label{tag1}$$ $$H^{*\mu\nu}=\frac12 \varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}\tag{2} = \varepsilon^{\mu\nu\rho\sigma}\partial_\rho A_\sigma\label{tag2}.$$

I personally like option \eqref{tag1} because it implies $$\partial_\nu F^{*\mu\nu} = 0, $$ and then automatically (because for any anti-symmetric tensor-density and torsion-free connection, one has $\partial_\nu F^{*\mu\nu} = \nabla_\nu F^{*\mu\nu}$) we have the covariant conservation law $$\nabla_\nu F^{*\mu\nu} = 0.\tag{1a}$$ However, the second option \eqref{tag2} is used everywhere in the literature, which is not conserved in the ordinary sense: $$\partial_\nu H^{*\mu\nu} =-H^{*\mu\nu}\Gamma^\lambda_{\ \nu\lambda},$$ but, nevertheless, it is conserved covariantly $$\nabla_\nu H^{*\mu\nu} = 0.\tag{2a}$$

So, is there any controversy in using the Levi-Civita symbol $\pmb{\in}^{\mu\nu\rho\sigma}$ in constructing the Hodge dual instead of the conventionally used Levi-Civita tensor $\varepsilon^{\mu\nu\rho\sigma}$?

$\endgroup$
  • 1
    $\begingroup$ For one thing, your Hodge dual depends on the coordinates used. $\endgroup$ – knzhou Jun 4 at 16:00
  • 2
    $\begingroup$ You must use the Levi-Civita tensor if you want to have a proper tensor equation. $\endgroup$ – mike stone Jun 4 at 16:27
  • 1
    $\begingroup$ $\epsilon$ and $\varepsilon$ are a really bad choice for different variables because they look almost identical in some fonts, especially at small sizes. I had to look at the source of this post to see the symbols are supposed to be distinct. $\endgroup$ – ACuriousMind Jun 4 at 17:04
  • $\begingroup$ More on Levi-Civita symbol & tensor. $\endgroup$ – Qmechanic Jun 4 at 19:02
  • $\begingroup$ @mike stone, I agree that in order to have Hodge dual being a proper tensor one has to use the standard definition with Levi-Civita tensor. However, the resulting PDEs are covariant in both cases, (1a) and (2a). And from this standpoint, I cannot see fundamental contradictions so far. $\endgroup$ – peshenator Jun 5 at 11:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.