In Berkeley Physics Course (Volume II) on Electricity and Magnetism: https://www.scribd.com/doc/128728926/Electricity-and-Magnetism-Berkeley-Physics-Course-Purcell ....Purcell discusses the invariance of charge (Section 5.4), the electric field measured in different frames of reference (Section 5.5), and the field of point charge moving at constant velocity (Section 5.6). What I find peculiar is that toward the end of Section 5.5, he states that if one observer were to view a charge as stationary in one frame, and moving relative to second observer in another frame, the longitudinal component of the electric field is the same according to both observers. However, in section 5.6, he starts with one observer where the charge is already moving in one frame, while a relatively moving second observer sees a different longitudinal component of this charge's electric field. To see what I mean, look at figure 5.13. Also, upon looking at equation 12 in Section 5.6, it appears that the transverse electric field scales with the Lorentz factor, while the longitudinal electric field (where sin(theta) = 0) scales inversely to the square of the Lorentz factor. What's going on here? Anyway, it would appear to me that if one were to have increased transverse components to the electric field of a moving charge upon switching to a different observer, then invariance of charge would require that longitudinal components of the electric field would have to be less in this other frame so that the total surface integral of the electric field on a Gaussian surface enclosing the charge remains unaffected.