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.

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