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)

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