Why is the Lorentz force always perpendicular to velocity?

[JiuBeixin]

TL;DR

Exploring the relativistic origin of Lorentz force. I'm deriving how a transverse force (from length contraction) and a longitudinal force (from relativistic momentum compensation) combine to stay perpendicular to velocity.

I represent the conductor as two parallel lines: stationary protons and electrons moving at velocity $v$. A test object with a positive charge also moves at velocity $v$ parallel to the electrons.
In frame $S'$ (the rest frame of the object), the object experiences a repulsive force (the origin of the Lorentz force) due to the length contraction of the moving proton line.

If I introduce an additional vertical velocity $u$ to the object in the frame $S$ the rest frame of the conductor), the entire conductor appears to move upward in frame $S'$. Since length contraction only occurs in the direction of relative motion, this vertical shift seemingly shouldn't create any additional charge density imbalance, so there isn't any extra force.

Yet, someone told me that because of a kind of Transformation of Forces, the force in frame $S$ would be vertical, but WHT and HOW it transform?

PS: I am a high school student and not a native English speaker, so please excuse any grammar issuee and awkward phrase :)​

 

[A.T.]

If I introduce an additional vertical velocity u to the object in the frame S the rest frame of the conductor), the entire conductor appears to move upward in frame S′.

Not just upward, but to the right (like it was moving before) and upward. So you have to apply the Lorentz transformation along that diagonal direction.

Since length contraction only occurs in the direction of relative motion, this vertical shift seemingly shouldn't create any additional charge density imbalance, so there isn't any extra force.

The Lorentz transformation is non-linear, so you cannot decompose it into independent dimensions like with Galilean transformations.