My question is that when we want to find the Lorentz force acted on a particle moving in an electric and magnetic field , the equation is invariant in any two inertial relativistic frames. Why is that so ? And only the electric and magnetic fields transform ?


The force f also transforms. In one frame you see f,E,v, and B. If you boost yourself wrt this frame, you will see f ',E',v',and B'. In the new frame the primed quantities satisfy the Lorentz force equation just like the unprimed quantities did in the original frame. The covariant form of the Lorentz force equation makes obvious how everything transforms:

$$ \frac{dp^{\alpha}}{d\tau}=qF^{\alpha \beta}U_{\beta} $$

The force on the charge q are the 3 spatial components of $\frac{dp^{\alpha}}{d\tau}$. The EM field tensor $F^{\alpha \beta}$ has the components of E and B in it. The 4-velocity $U^{\beta}=(\gamma,\gamma v_x, \gamma v_y, \gamma v_z)$. Please see "Relativistic form of the Lorentz force" down the page in https://en.wikipedia.org/wiki/Lorentz_force if you would like more details on all these covariant objects and how the vector form of the Lorentz force equation falls out.

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    $\begingroup$ Can you please suggest me a reference from where you studied about tensors and 4-vectors $\endgroup$ – Hazel Apr 10 '16 at 9:49

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