I am reading Kleppner.(Lorentz transformations) He said,we take the most general transformation relating the coordinates of a given event in the two systems to be of the form $$x'=Ax +Bt, y'=y, z'=z, t'=Cx +Dt,$$ and then he found out the constants considering four cases in which we know a priori how an event appears in the two systems.
but why the transformations are linear? He said ,a nonlinear transformation would predict acceleration in one system even if the velocity were constant in the other. But i think thats what happen when we consider Lorentz force(without electric field) if $v$(velocity of a charged particle) is zero in one inertial frame then there is no Lorentz force on the particle(hence the particle has no acceleration in that frame). but this may not be the case in other inertial frames(where the velocity of the charged particle is not zero)? whats wrong here?
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