Two protons move parallel to each other with speed v1 and v2 net force between them is

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Force, like all vectors, is not a Lorentz invariant. It is therefore not surprising that the force between two protons depends on which frame of reference is considered.

The calculation is simple to do by calculating the electric and magnetic fields observed in the frame in which the protons are seen to be moving. Both fields appear different in the moving frame. See https://en.m.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity for the relevant transformation equations.

You will find the repulsive force between the protons is still there, but reduced by a factor of $\gamma = (1 -v^2/c^2)^{-1/2}$.

Details:

In the stationary frame of the protons then there is just the Coulomb repulsion between them given by $$F_{\rm rest} = e\vec{E}_{\rm rest} = \frac{e^2}{4\pi \epsilon_0 r^2}\hat{r} ,$$ where $\vec{E}_{\rm rest}$ is the E-field of a stationary proton.

In the lab frame, the electric field in the direction between the two protons is increased by the Lorentz factor to $\vec{E}_{\rm lab} =\gamma \vec{E}_{\rm rest}$, where $\gamma = (1-v^2/c^2)^{-1/2}$ and where $\gamma \geq 1$. At the same time, there is a magnetic field caused by the motion of the protons and this contributes a force $e \vec{v} \times \vec{B}_{\rm lab}$, where $\vec{B}_{\rm lab}$ is the B-field measured in the lab frame.

The lab B-field is found using the appropriate transform as $$ \vec{B}_{\rm lab} = -\frac{\gamma}{c^2} \vec{v} \times \vec{E}_{\rm rest}$$

Thus the force between the protons on the lab frame is $$F_{\rm lab} = e(\vec{E}_{\rm lab} + \vec{v}\times \vec{B}_{\rm lab}) = e (\gamma \vec{E}_{\rm rest} - \frac{\gamma v^2}{c^2} \vec{E}_{\rm rest}) = \frac{e \vec{E}_{\rm rest}}{\gamma} = \frac{F_{\rm rest}}{\gamma}. $$ This is exactly as required by the rules for transforming forces under special relativity. The force acting between the two protons is smaller in the lab frame and approaches zero as the protons become more and more relativistic.

The same thing would be true for two uncharged particles connected by a spring.