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 Exercise [20.18], page 491 
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Joined: 12 Jul 2010, 07:44
Posts: 154
Post Exercise [20.18], page 491
I attempted this exercise (using the components of F as given on page 443), and my result for the Lagrangian was \frac{1}{2}\left(\|\boldsymbol{B}\|^2-\|\boldsymbol{E}\|^2\right), which differs from Penrose's description by a factor of 4:

...this quantity being \frac{1}{8} of the difference between the squared lengths of the electric and magnetic field vectors, in 3-dimensional terms.

So either I've made some basic error (but I can't see where!), or else there's an error in Penrose's description, or in the Lagrangian he supplies (\mathcal{L}=\frac{1}{4}F_{ab}F^{ab}), or some discrepancy in the scale of F as used here vs. as given on page 443.

Note that the electromagnetic Lagrangian is given on both page 442 and 443, but the sign changes between these two versions!

Note that rescaling the Lagrangian has no effect on the equations of motion in any case, so if this is an error, it actually isn't a very significant one.

19 May 2012, 10:08
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