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On Zwiebach's "A First Course in String Theory" textbook, there is a problem 8.2 that derives the invariance of electric charge given a few assumptions: 1. Conservation of Maxwell 4-current $j = (c\rho, j^x, j^y, j^z)$: $\partial_{\alpha} j^{\alpha} = 0$ 2. Correctness of Lorentz Transformations. 3.charge densities and currents vanish at infinity. Here is how far I have gotten:

There are two inertial frames S and S'. S' moves with speed v relative to frame S. The charge as measured by observers in S and S' can be written as: $$ Q = \int \rho (t=0,\vec x) d^3x$$ $$ Q' = \int \rho'(t' = 0, \vec x') d^3x'$$ After applying Lorentz transformations, we can express all the primed variables in terms of the unprimed variables: $$ Q' = \int d^3x (\rho - \frac{v}{c^2} j^x) |_{t = \frac{vx}{c^2}, x,y,z}$$ Now we see that when v = 0, $Q' = Q$ as expected. Now we take derivative with respect to v. If we get zero, then the proof is done. So I got: $$ \frac{dQ'}{dv} = \int d^3x (\frac{\partial \rho}{\partial t} \frac{dt}{dv} - \frac{1}{c^2} j^x - \frac{v}{c^2} \frac{\partial j^x}{\partial t} \frac{dt}{dv}) $$ After a lot of algebra, (and using the conservation of 4-current), I simplified the expression to: $$ \frac{dQ'}{dv} = - \frac{1}{c^2} \int d^3x \nabla \cdot \vec J$$ where $\vec J = x \cdot[j^x,j^y,j^z]$. Using Gauss's law, this is: $$ \frac{dQ'}{dv} = - \frac{1}{c^2} \int \vec J \cdot d\vec A$$ where the integration is taken over a closed surface at infinity.

The goal is to show that the integral above actually goes to zero. But I don't see how the third assumption of charge densities and currents vanishing at infinity can help with the final step. Since $\vec J = x [j^x,j^y,j^z]$, I think the current densities have to vanish faster than $\frac{1}{distance^3}$ in order for the integral to vanish. But there is no guarantee for that in the problem statement. Can someone take a look at Zweibach's book and help? Thanks!!

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  • $\begingroup$ Without having looked at the problem or your derivation in detail - it is quite canonic in my experience to just assume sufficiently fast vanishing. $\endgroup$ – Sanya Jun 28 '16 at 12:16
  • $\begingroup$ Usually yes. But infinities are pretty tricky... For example, if you want to evaluate the integral $\int \vec E \cdot d\vec A$ at a spherical surface at infinity, the value you get is simply proportional to the total charge in the world by Gauss's law! We cannot simply assume that this total charge is zero in that case. $\endgroup$ – Zhengyan Shi Jun 29 '16 at 15:14

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