Errors on pp. 444-449

deant · 26 Feb 2012, 10:38

(1) On page 444, if it is to match the definition on the previous page, the caption to figure 19.1 should read:

Quote:

"...normalized so that $\varepsilon_{0123}=\epsilon^{0123}=1$ in a standard Minkowski frame, ..."

(Note it should be $\varepsilon$ rather than the $\epsilon$ in the first instance).

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(2) Also in this caption, it reads:

Quote:

"In the diagrams (left middle, lower two lines) this sign change is absorbed by an effective index reversal."

This is simply false: A quick check will show that reversing the order of 4 indices is an even permutation, so does not produce a sign change! In the figure, the raised and lowered versions of $\varepsilon_{abcd}$ and $\epsilon^{abcd}$ respectively (i.e. the big U-shaped diagrams) should have minus signs in front of them.

This sign change between $\varepsilon$ and $\epsilon$ is actually due to the Lorentzian signature of the metric, and would not occur in Euclidean 4-space, even though the diagrams for raising and lowering would be identical. (In Euclidean space $\varepsilon\equiv\epsilon$ ).

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(3) In two places on page 445, the second Maxwell equation is given as:

$\mathrm{d} {^*}\!F = 4\pi {^*}\!J$

Whereas it should actually be:

$\mathrm{d} {^*}\!F = \frac{4\pi}{3} {^*}\!J$

This accords with the coordinate expression for this equation in the last line of Fig. 19.1's caption, and also with the results of exercises [19.02] and [19.03].

Similarly the equation on p. 446 should read:

$\mathrm{d} {^+}\!F = -\frac{2\pi i}{3} {^*}\!J$

...and the equations on p. 448 and 449 need a similar factor of 1/3 correction.

NOTE that the other alternative is to take the above equations to be correct as they are, which would mean the component versions are wrong, and should have " $12\pi$ " whereever they now have " $4\pi$ ", e.g.

$\nabla_a F^{ab} = 12\pi J^b$

Since the difference depends only the scale of the units of electric charge, and since Penrose doesn't actually define this specifically, it's not possible to say which alternative is "right", given only what's in the book. The problem is that whichever one you choose, there's a factor of 1/3 introduced when expanding the exterior derivative operator out into coordinate terms, and it's been neglected here, leading to conflicting versions of the equations that can't be reconciled.