I am looking at the first three pages of this file (https://www.mtholyoke.edu/courses/tdray/phys310/electromag.pdf).
In the lab frame, there is an infinitely long wire stretching from left to right, consisting of positive and negative charges with equal linear charge density $\lambda_+ = \lambda_-$ so that the wire is overall charge-neutral, $\lambda = \lambda_+ - \lambda_- = 0$. There is a test charge $q$ at some distance $r$ from the wire moving at speed $v = c\tanh{\beta}$ to the right. The positive charges in the wire are moving at a speed $u = c\tanh{\alpha}$ to the right and the negative charges are moving at the same speed $u= c\tanh{\alpha}$ to the left, so that the wire has a net positive current to the right of $I = 2\lambda_+ u$, leading to a (tangential $\hat{\phi}$) magnetic field at the moving test charge and thus a magnetic force (radially attractive in $\hat{r}$).
Now boost to the rest frame of the test charge. The wire's positive charges are now moving at $u_+ = c\tanh({\alpha-\beta})$ and the negative charges at $u_- = c\tanh({\alpha+\beta})$. The paper is trying to show that the force in the rest frame of the test charge now appears to be due to an electric force, because the linear charge densities have changed due to Lorentz-length contraction from the new velocities, and so $\lambda = \lambda_+ -\lambda_-$ is now nonzero.
My question is this: why is the wire not electrically neutral in the rest frame of the test charge also? I thought charge was a relativistic invariant. If we boost to the rest frame of the test charge than the spacing for both the positive and negative charges should contract by $\gamma_v$. I can agree that the relativistic velocity addition works as stated, but why are we using that to contract the charge densities, rather than the original boost factor?
Thanks for any help.