# Electromagnetic field tensor: invariance of inner product

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 7; Problem 7.7.

We’ve worked out earlier the scalar quantity ${F_{ij}F^{ij}=2\left(B^{2}-E^{2}\right)}$. We can check this for the case of a Lorentz transformation in flat space-time. We found there that

 $\displaystyle E_{x}^{\prime}$ $\displaystyle =$ $\displaystyle E_{x}\ \ \ \ \ (1)$ $\displaystyle E_{y}^{\prime}$ $\displaystyle =$ $\displaystyle \gamma E_{y}-\gamma\beta B_{z}\ \ \ \ \ (2)$ $\displaystyle E_{z}^{\prime}$ $\displaystyle =$ $\displaystyle \gamma E_{z}+\gamma\beta B_{y}\ \ \ \ \ (3)$ $\displaystyle B_{x}^{\prime}$ $\displaystyle =$ $\displaystyle B_{x}\ \ \ \ \ (4)$ $\displaystyle B_{y}^{\prime}$ $\displaystyle =$ $\displaystyle -\gamma\beta E_{y}+\gamma B_{z}\ \ \ \ \ (5)$ $\displaystyle B_{z}^{\prime}$ $\displaystyle =$ $\displaystyle -\gamma\beta E_{z}-\gamma B_{y} \ \ \ \ \ (6)$

Calculating the invariant in the new system we get

 $\displaystyle B^{\prime2}-E^{\prime2}$ $\displaystyle =$ $\displaystyle B_{x}^{\prime2}+B_{y}^{\prime2}+B_{z}^{\prime2}+E_{x}^{\prime2}+E_{y}^{\prime2}+E_{z}^{\prime2}\ \ \ \ \ (7)$ $\displaystyle$ $\displaystyle =$ $\displaystyle B_{x}^{2}+\left(-\gamma\beta E_{y}+\gamma B_{z}\right)^{2}+\left(-\gamma\beta E_{z}-\gamma B_{y}\right)^{2}-\ \ \ \ \ (8)$ $\displaystyle$ $\displaystyle$ $\displaystyle E_{x}^{2}-\left(\gamma E_{y}-\gamma\beta B_{z}\right)^{2}-\left(\gamma E_{z}+\gamma\beta B_{y}\right)^{2}\ \ \ \ \ (9)$ $\displaystyle$ $\displaystyle =$ $\displaystyle B_{x}^{2}+\left(B_{y}^{2}+B_{z}^{2}\right)\gamma^{2}\left(1-\beta^{2}\right)-E_{x}^{2}-\left(E_{y}^{2}+E_{z}^{2}\right)\gamma^{2}\left(1-\beta^{2}\right)\ \ \ \ \ (10)$ $\displaystyle$ $\displaystyle =$ $\displaystyle B^{2}-E^{2} \ \ \ \ \ (11)$

since ${\gamma=1/\sqrt{1-\beta^{2}}}$. All the cross terms involving the product of a component of ${E}$ and one of ${B}$ cancel out between lines 2/3 and 4.

## 2 thoughts on “Electromagnetic field tensor: invariance of inner product”

1. gwrowe Post author

I had already worked out earlier that ${F^{\mu\nu}F_{\mu\nu}=2\left(B^{2}-E^{2}\right)}$, so Moore’s problem 7.7 in effect duplicates his problems 4.8 and 4.9. Here I was just verifying that ${B^{2}-E^{2}}$ is actually invariant under a Lorentz transformation.