Why is $E^i = E_i$ instead of $E^i = - E_i$?

The fundamental reason would be that the electric and magnetic fields, $E$ and $B$ *do not form four vectors*. Rather they are *three-dimensional vectors* without a fourth component as explained here. The transformation between co- and contra- forms is identity transformation. This is due to different vectorial nature of the respective fields. The electric field is a **polar vector** (or true vector) because it changes the sign if coordinates are reversed, $\mathbf{r} \rightarrow \mathbf{-r}$. In contrast, the magnetic field given by

$$\mathbf{B}(\mathbf{r}) = \frac{1}{c} \int \dfrac{(\mathbf{r}-\mathbf{r'})\times \mathbf{J}(\mathbf{r})}{|\mathbf{r}-\mathbf{r'}|^3} \text dV'$$

remains unchanged against coordinate inversion since both $(r - r')$ and current density $J(r)$ change sign. The magnetic field is a **pseudo-vector** (or axial vector).

A detailed calculation of how it turns out to be the same in Minkowski space-time metric regardless of its form is shown here.

The signs of the components of the electromagnetic tensor $F^{\mu \nu}$ and $F_{\mu \nu}$ depend on the metric convention. However, the mixed tensor $F^\mu{}_\nu$ is independent of such a choice. Considering $c = 1$, we have

$$F^\mu{}_\nu=\left(\begin{array}{cccc}0&E_x&E_y&E_z\\E_x&0&B_z&-B_y\\E_y&-B_z&0&B_x\\E_z&B_y&-B_x&0\end{array}\right)$$

Here $\mu,\nu~\in~\{0,1,2,3\}$ and $i~\in~\{1,2,3\}$.

Defining $\eta_{\mu \nu} = \text{diag}(-1,+1,+1,+1) = \eta^{\mu \nu}$, we know $F_{\mu \nu} = \eta_{\mu \rho}F^\rho{}_{\nu}$ so we obtain $E_i = -F_{0i}$

Since, $F^{\mu \nu} = \eta^{\mu \rho}\eta^{\nu \lambda}F_{\rho \lambda}$, for $i \ne 0$ we obtain $$E^i = F^{0i} = \eta^{00}\eta^{i i}F_{0 i} = -F_{0i} = E_i$$

Now, it is fairly straight-forward to prove that if we use $\eta_{\mu \nu} = \text{diag}(+1,-1,-1,-1)$, the result will be $$E^i = -F^{0i} = - \eta^{00}\eta^{i i}F_{0i} = F_{0i} = E_i $$

In both the cases it turns out that $E_i = E^i$.

This post imported from StackExchange Physics at 2020-11-09 19:26 (UTC), posted by SE-user Abhay Hegde