In my infamous simulation of the twin paradox, i assume a near instantaneous acceleration, with the results being almost identical with the "real" thing sub some extremely small values which you could not see with the naked eye anyway. But i might want to take the simulation a step further, to give the exact value for slower accelerations. For the near instantaneous case, as an example. When two observers A and B are e local to each other, and B accelerates to 0.6c near instantaneously, then an event which B measured to be at x=1ls, t=0s pre-acceleration, post acceleration he will measure to be almost exactly at. x' = γ(x-vt) = 1.25ls t' = γ(t-vx/c2) = - 0.75s Increasing the acceleration, we would get arbitrary close to those two values. Now i could of course do the acceleration in small steps, accelerating to a given v, then let some time pass. The more steps, the closer i would get to the real value. But that would be too intensive computational and would not really be "perfect". Again i would have to work with just an approximation. Is there any way to get the exact values for x' and t' for let's say the case of the accelerating observer B measuring the acceleration towards 0.6c to be taking 1 second on his clock? Is it calculable without having to do it in "steps" where you could get arbitrary close to the results, but never get the exact results?