Consider equation (2.7.8) page 42 in the book Gravitation and Cosmology by Weinberg(adsbygoogle = window.adsbygoogle || []).push({});

F'^{αβ}= Λ^{α}_{γ}Λ^{β}_{δ}F^{γδ}

Now consider the time reversal Lorenz transformation

Λ^{μ}_{ν}= 0 if μ ≠ ν, 1 if μ = ν = 1..3 and -1 if μ = ν = 0

then

F'^{00}= 0

F'^{0i}= -F^{0i}

F'^{ij}= F^{ij}

Using equation (2.7.5) of the same book this is equivalent to

E'_{i}= -E_{i}

B'_{i}= B_{i}

but the result should be

E'_{i}= E_{i}

B'_{i}= -B_{i}

since time reversal reverse the sign of the electric charge current.

It look like for time reversal the equation (2.7.8) needs a minus sign.

How to justify this exception?

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# A Electromagnetic tensor and time reversal

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