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Could you explain me a few things I still don't get about special relativity ?

I've been tought a few years ago that special relativity wasn't good for handling accelerations... well I've been reading books and a few topics here, and I'm now convinced it is not true... for exactly the same reason galilean mechanics can handle accelerations as seen from inertial frames...

ok....

My question is... As SR can handle accelerations (forces), I can calculate what is, in an inertial reference frame, the trajectory of a relativistic particle in an electromagnetic field. To do this, I have to write the relativistic equation of the dynamic :

[tex]\frac{d\mathbf{P}}{dt} = \mathbf{F}[/tex]

where [tex]\mathbf{P}[/tex] and [tex]\mathbf{F}[/tex] are the 4-vector energy-impulsion and 4-force respectively.

[tex]\mathbf{F}[/tex] in this problem, being the electromagnetic Lorentz-force.

It is often said "SR is good at handling accelerations, as long as there is no gravity", that I don't understand !

Why couldn't I replace the electromagnetic field in the previous problem, by a gravitationnal field (newtonian GM/r^2) ? Why couldn't I solve the dynamic of a relativistic particle in the gravitationnal field of the sun for example ?

I've been told many things which have not convinced me. Among them :

- the reason would be that electromagnetic interaction is possible in SR because it propagates at a finite speed whereas newtonian universal gravitation violates the principle of relativity by propagating at a infinite speed. I accept these facts, but I can't see why it would explain that one can use lorentz force but not gravitational force in SR. (after all there are probably some cases where the body has a relativistic velocity but the propagation time of the gravity is negligeable ?)

- the reason would also be that electric charge is invariant in respect to the inertial transformation, whereas the mass is not. I don't even understand that... to me, the mass is also conserved ... no ? It is just the inertial factor (gamma times the mass) that changes. How could this explain that gravitational forces are prohibited in SR ?

Further related question... if electromagnetic forces are tolerated, how can a electromagnetic two body problem can be formulated ? If I take an electron and a proton, moving at some relative speed, they will feel attraction due to coulomb force, and their trajectory will be somehow changed. From an inertial reference frame, their motion can, according to me, be described via the relativistic equation of the dynamic... where the force is the coulomb force.

What I don't understand in what I've just said, is that with coulomb force, there is no time in my equations, so any change in the electron motion will immediatly be felt by the proton... just like in Newtonian physics (and newtonian gravitational force) thus violating the principle of relativity... exactly as if I would have written the gravitationnal two body problem with the newton universal gravitation law..

Where is my mistake ?

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# Few concepts that I miss in SR

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