I have been watching a series of lectures on general relativity by Neil Turok and I have run into a problem.
In one of the lectures, the professor writes the momentum 4-vector as a contraction of the electromagnetic stress-energy tensor with the epsilon tensor inside an integral. He first writes it in the following way:
$p^\gamma = \int_Vd^3xT^{\gamma 0}$
and then claims that this is proportional to the following
$\int_V T^{\gamma\mu}\epsilon_{\mu\nu\alpha\beta}dx^\nu dx^\alpha dx^\beta$
being proportional with only a factor of 1/6th since there are 6 permutations for which $\mu$ = 0
I dont understand this. I would think that by the antisymmetric nature of $\epsilon$, there would be 6 parts summed over, where 3 are the exact negative of the other 3, resulting in a final value of 0. I dont think the order of the dx's should matter. Can someone explain if I'm wrong in my reasoning?