Duration of an accelerated movement in special relativity

Question

Consider a point $P$ which does the following (one dimensional) movement:

1. Uniform acceleration $a_0$ (proper aceleration) from $t=0$ to $t=T$ (time measured in lab's frame).

2. Uniform deceleration $a_0$ from $t=T$ to $t=3T$.

3. Uniform acceleration $a_0$ from $t=3T$ to $t=4T$.

I want to know how much time has passed in $P$'s frame. That is, what is $ \Delta\tau$?

Since the movement starts with no velocity, we can easily calculate its four-velocity: $$u^\mu=c(\cosh(a_0\tau/c),\sinh(a_0\tau/c),0,0).$$

As $u^0=\gamma c$, we conclude that $\gamma=\cosh(a_0\tau/c)$ and thus the duration of the first part can be calculated as $$T=\int_0^{\Delta\tau_i} \cosh\left(\frac{a_0}{c}\tau\right)\mathrm{d}\tau\quad\implies\quad \Delta\tau_i=\frac{c}{a_0}\mathrm{arcsinh }\left(\frac{a_0}{c}T\right).$$

However, for the other parts the initial velocity is not zero and then the function I get when calculating $\Delta t=\int\gamma\:\mathrm{d}\tau$ is not easily invertible.

Is there a better way to do this?

Answer 1

From your setup, there is symmetry to exploit. If you break up the T-to-3T trip in half, you can see [by drawing a spacetime diagram] that you have four congruent legs to your trip. You just calculated the duration of the first leg. So, by symmetry, multiply that by 4 to get the total duration.

update2:

In this second update, I have shown three uniformly accelerated observers as with dashed worldlines, which are hyperbolas on a spacetime diagram. Since the magnitudes of their proper-accelerations are equal, these hyperbolas are congruent--related by reflections in space, reflections in time, and translations on this spacetime diagram.
[If we were doing Galilean relativity, these would be congruent parabolas.]

This round-trip starting and ending at rest in this frame is essentially the splicing of four congruent portions.

[As a simpler case, suppose the traveler departed with a nonzero outgoing speed and returned at the same speed.... essentially the magenta observer. Is it clear that if that trip were split into halves, those halves have equal elapsed times.]

Note that the 0-to-T leg is the mirror image in space of the 2T-to-3T leg.
Note that the T-to-2T leg is the mirror image in space of the 3T-to-4T leg.

The 0-to-T leg is the "time-reverse" of the 3T-to-4T leg.
The T-to-2T leg is the "time-reverse" of the 2T-to-3T leg.

update3:

Consider the first leg of the trip... uniform-acceleration $a_0$ to the right.

Suppose instead the traveler went in the opposite direction in space.
After the same time $\Delta t_1$, is it clear that the traveler would have the opposite velocity (same speed, opposite direction) as the original?
This space-reflected segment is a translation of the leg that starts [back] at the turnaround event.

Similarly, if you ran time in reverse [retrodiction, if you will], then at earlier time $-\Delta t_1$, is it clear that the traveler would have had the opposite velocity (same speed, opposite direction) as the original?
This time-reflected segment is a translation of the leg that returns at the reunion event.

Answer 2

You correctly wrote the expression for the four-velocity and $\gamma$ but it'd be much easier to use straight forward the time coordinate (Rindler) transform for a uniformly accelerated frame (without integral calculus and 4-velocities) as $$t=\frac{c}{a_0}\mathrm{asinh }\left(\frac{a_0}{c}\tau\right)$$ where $\tau$ is proper time in (in P's frame) and $t$ is time in "lab frame" or at rest frame. So, using this general transform for the first part $$t_1=\frac{c}{a_0}\mathrm{asinh }\left(\frac{a_0}{c}T\right)$$ which is the same as you obtained. The next part can be derived by the use of the same time transform but with the negative $a_0$ and for the duration is $2T=3T-T$, so $$t_2=\frac{c}{-a_0}\mathrm{asinh }\left(\frac{-a_0}{c}2T\right)=\frac{c}{a_0}\mathrm{asinh }\left(\frac{a_0}{c}2T\right)$$ Then obviously $t_2 \neq 2t_1$. The third part is given by delta $T=4T-3T$, so $$t_3=\frac{c}{a_0}\mathrm{asinh }\left(\frac{a_0}{c}T\right)$$