# Electromagnetic field tensor: invariance under Lorentz transformations

Required math: algebra

Required physics: special relativity

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 4; Problems 4.10.

The electromagnetic field tensor ${F^{ij}}$ transforms between inertial frames by using 2 Lorentz transformations (we’ll see why in a future post):

$\displaystyle F^{\prime ij}=\Lambda_{\;\; a}^{i}\Lambda_{\;\; b}^{j}F^{ab} \ \ \ \ \ (1)$

Using this, let’s see how the quantity ${\eta_{ia}\eta_{jb}F^{ij}F^{ab}}$ that we considered in the last post transforms. Since the metric tensor is invariant under Lorentz transformations

$\displaystyle \eta_{ia}\eta_{jb}F^{\prime ij}F^{\prime ab}=\eta_{ia}\eta_{jb}\Lambda_{\;\; c}^{i}\Lambda_{\;\; d}^{j}F^{cd}\Lambda_{\;\; e}^{a}\Lambda_{\;\; f}^{b}F^{ef} \ \ \ \ \ (2)$

We can now use

$\displaystyle \left(\Lambda^{-1}\right)_{\;\; k}^{a}\eta_{ab}=\eta_{kj}\Lambda_{\;\; b}^{j} \ \ \ \ \ (3)$

So we substitute in 2

 $\displaystyle \eta_{ia}\Lambda_{\;\; c}^{i}$ $\displaystyle =$ $\displaystyle \left(\Lambda^{-1}\right)_{\;\; a}^{i}\eta_{ic}\ \ \ \ \ (4)$ $\displaystyle \eta_{jb}\Lambda_{\;\; d}^{j}$ $\displaystyle =$ $\displaystyle \left(\Lambda^{-1}\right)_{\;\; b}^{j}\eta_{jd} \ \ \ \ \ (5)$
 $\displaystyle \eta_{ia}\eta_{jb}\Lambda_{\;\; c}^{i}\Lambda_{\;\; d}^{j}F^{cd}\Lambda_{\;\; e}^{a}\Lambda_{\;\; f}^{b}F^{ef}$ $\displaystyle =$ $\displaystyle \left(\Lambda^{-1}\right)_{\;\; a}^{i}\eta_{ic}\left(\Lambda^{-1}\right)_{\;\; b}^{j}\eta_{jd}F^{cd}\Lambda_{\;\; e}^{a}\Lambda_{\;\; f}^{b}F^{ef}\ \ \ \ \ (6)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(\Lambda^{-1}\right)_{\;\; a}^{i}\Lambda_{\;\; e}^{a}\eta_{ic}\left(\Lambda^{-1}\right)_{\;\; b}^{j}\Lambda_{\;\; f}^{b}\eta_{jd}F^{cd}F^{ef}\ \ \ \ \ (7)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \delta_{\;\; e}^{i}\eta_{ic}\delta_{\;\; f}^{j}\eta_{jd}F^{cd}F^{ef}\ \ \ \ \ (8)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \eta_{ec}\eta_{fd}F^{cd}F^{ef} \ \ \ \ \ (9)$

Thus the quantity ${\eta_{ia}\eta_{jb}F^{ij}F^{ab}}$ is invariant under Lorentz transformations. As we saw in the last post, this quantity is

$\displaystyle \eta_{ia}\eta_{jb}F^{ij}F^{ab}=2\left(B^{2}-E^{2}\right) \ \ \ \ \ (10)$