In an inertial frame, consider that a particle's position and the time measured by a clock in this frame are respectively, ##(t,x)##.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose there's another frame, moving with constant speed ##v_R## with respect to the frame described above.

The particle acceleration is given in the first frame by ##d^2 x / dt^2##. In the second frame I would expect it would be given by ##d^2 x' / dt'^2##. I know the relation between the primed and unprimed coordinates: $$t' = \gamma (t - v_R x) \\ x' = \gamma (x - v_R t)$$ But in expressing ##d^2 x' / dt'^2## by chain rule, etc... in terms of ##(t,x)## I got an ugly expression with several terms. But when I evaluate the vector quantity ##(d^2 t' / dt^2, d^2 x' / dt^2)## I get a beautiful expression for ##a'##, namely ##a' = \gamma a##, aside from the fact that that vector is Lorentz invariant.

What bothers me, however, is that in the last expression I used ##t##, instead of ##t'##, and it seems senseless to use the time ##t## measured from the first frame to get a quantity measured in the second frame.

If the particle was not relativistically moving, then it would be easy: ##a' = d^2 x' / dt'^2 = d^2 x' / dt^2 = a##. What can I do?

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# I Acceleration of a relativistic particle

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