Feynman's Lectures, Volume 2, says that the electromagnetic force is invariant in any reference frame, and the magnetic force in one frame becomes the electric field in another.
And Wikipedia says:
That is, the magnetic field is simply the electric field, as seen in a moving coordinate system.
Can we then say that the magnetic field is just a modified "relativistic" version of the electric field?
Answer
In above Figure-02 an inertial system S′ is translated with respect to the inertial system S with constant velocity
υ=(υ1,υ2,υ3)=(υn1,υn2,υn3)=υn,υ∈(−c,c)
a matrix representing the vectorial projection on the direction n.
The electromagnetic field is an entity and this is more clear if we take a look at its transformation. So, for the Lorentz transformation (02), the vectors E and B are transformed as follows E′=γE−(γ−1)(E⋅n)n+γ(υ×B)B′=γB−(γ−1)(B⋅n)n−γc2(υ×E)
Now, if in (09) we replace E′,B′,u′ by their expressions (07a),(07b) and (10) respectively, then we end up with the following relation between the force 3-vectors
f′=f+(γ−1)(n⋅f)n−γυ(f⋅uc2)γ(1−υ⋅uc2)
That's why in the early years of Special Relativity transformation (11) was believed to be valid for any force at least of the same type as the Lorentz force (more exactly for any force that doesn't change the rest mass of the particle).
Following the same path by which we construct from (10) the velocity 4-vector U U=(γuu,γuc)
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