I am confused as to how Gauss's law can hold for moving charges. When discussing how to use Gauss's law to calculate the charge of a moving object, my textbook says: "At the instant the moving charge passes the center of the sphere, the force on each test charge is measured, and the average of the force magnitudes is used to compute Q." So we are using a surface in the "stationary" or "lab" frame, to calculate the flux integral. By taking for granted that the surface integral is the same whether the object is moving or not, and then using conservation of charge, the book then derives the relativistic transformations of the electric field. But If the perpendicular component of the electric field of a moving charge at any point is greater, and the parallel component is unchanged, shouldn't E be greater almost everywhere? That would seem to make the flux through a given surface greater for a moving charge.