I just studied SR and I understand how velocity affects time and length; how do they evolve during the process of reaching a certain velocity?

  • $\begingroup$ For constant acceleration, see the relativistic rocket. Those formulae aren't hard to derive, if you know some calculus. $\endgroup$ – PM 2Ring May 6 at 4:01
  • $\begingroup$ What if acceleration isn't constant? $\endgroup$ – Matteo Scandola May 6 at 20:01
  • $\begingroup$ Then you don't get simple formulae, but you can still work out what goes on by integration, as G. Smith explains. $\endgroup$ – PM 2Ring May 6 at 20:07

At any instant, the time dilation and the length contraction of a moving object just depend on its instantaneous velocity relative to the observer, not on its acceleration.

If the particle is accelerating and you want to know, for example, the time that elapses on “its clock” over a finite interval, then you have to integrate over each infinitesimal interval of time that elapses for it. In doing so, you will take its acceleration into account because you have to use its varying instantaneous velocity.

  • $\begingroup$ Could you be more specific? Maybe with equations? $\endgroup$ – Matteo Scandola May 6 at 7:26
  • $\begingroup$ For example, the elapsed proper time for a moving object is $$\Delta\tau=\int d\tau=\int\sqrt{1-\frac{v(t)^2}{c^2}}dt.$$ $\endgroup$ – G. Smith May 6 at 15:45
  • $\begingroup$ I don't understand how can you simply apply SR in a non inertial reference frame. $\endgroup$ – Matteo Scandola May 6 at 19:57
  • $\begingroup$ @Matteo That's a good question. Strictly speaking, you can't apply SR in a non-inertial frame. But what you can do is to use a sequence of comoving inertial frames. That is, at each instant, you use an inertial frame that has the same origin and velocity as the frame of the accelerating body. See en.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime) $\endgroup$ – PM 2Ring May 6 at 20:13
  • $\begingroup$ @MatteoScandola Also see en.wikipedia.org/wiki/Time_dilation#Clock_hypothesis $\endgroup$ – G. Smith May 7 at 20:39

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