Sequential Lorentz transformation on an accelerating frame of reference? A rocket takes off from Earth and increases its speed by 0.001 c every second. When the speed reaches 0.99 c, it decreases its speed by 0.001 c every second until its speed returns to 0.
Then, it turns around and increases its speed by 0.001 c every second back towards the Earth. When it reaches 0.99 c in this homeward direction, it decreases its speed by 0.001 c and stops exactly when it reaches Earth.
By applying Lorentz transformation every second to switch into the rocket's frame,
$$x'=\gamma(x-vt)$$
$$t'=\gamma\left(t - \frac{vx}{c^2}\right)$$

the Earth clock is found to have ticked significantly more (3958) than the rocket clock (3136).
My questions are:

*

*Can this "discrete integration" method be used to find the Lorentz transform of an accelerating frame of reference, by making t extremely small (eg. 0.00000000001 second) and setting a different v for each small time interval, and applying the Lorentz transform piecewise?


*It does not make practical sense that the rocket can accelerate to 0.99 c in 2000 seconds. But this is nowhere encoded into Lorentz transformation. Which part of special relativity prohibits accelerating beyond the speed of light?
 A: 
By applying Lorentz transformation every second to switch into the rocket's frame

What you are switching into is not the rocket's frame. The rocket is non-inertial so its frame is also non-inertial. What you are switching to is called the momentarily co-moving inertial frame (MCIF). It is important to distinguish the MCIF from the rocket's frame since the MCIF is inertial and the rocket's frame is not.

Can this "discrete integration" method be used to find the Lorentz transform of an accelerating frame of reference, by making t extremely small (eg. 0.00000000001 second) and setting a different v for each small time interval, and applying the Lorentz transform piecewise?

No. There is no such thing as the Lorentz transform of an accelerating frame of reference. The Lorentz transform is the transform between inertial frames and an accelerating frame is not inertial.
You can, of course, construct a non-inertial frame for the accelerating rocket by using the simultaneity defined by each MCIF. However, that runs into some well-known problems, so other methods of constructing non-inertial frames are typically preferable (e.g. radar coordinates, or Fermi coordinates)

It does not make practical sense that the rocket can accelerate to 0.99 c in 2000 seconds. But this is nowhere encoded into Lorentz transformation. Which part of special relativity prohibits accelerating beyond the speed of light?

It is actually encoded into the Lorentz transform. The composition of $n$ Lorentz transforms, each of velocity $\delta \beta \ c$ is not the same as one Lorentz transform of velocity $v= \beta \ c = n \ \delta \beta \ c$. The actual formula is $$\beta=\sqrt{1-\frac{4 \left( 1-\delta \beta^2 \right)^n}{\left( \left( 1-\delta \beta \right)^n + \left( 1+ \delta \beta\right)^n \right)^2}}$$

This is probably more easily understood from the velocity addition formula. No matter how many times you add a small velocity you never get a total velocity that exceeds $c$
