Do not rush to write formulas
The equations of SRT are of utmost relevance when discussing the predictions of SRT. The definitive statement of what SRT predicts is the Lorentz transformation. If something is not predicted by the Lorentz transformation, it is not predicted by SRT.
Answer the question: What are we going to have: the double time-dilation or the time-dilation+acceleration?
The second. The traveller's time returns to the normal rate. That is the only prediction SRT makes because it is the only prediction the Lorentz transformation makes.
If you state that the traveler will return the normal time, then explain us: why you replaced second time-dilation to time-acceleration?
Simple. The way time dilation is usually described, the time dilation factor in a reference frame is given by $$\gamma(v) \,=\, 1 / \sqrt{1 \,-\, v^{2}/c^{2}}$$ [sup]*[/sup]. If $$v = 0$$ then $$\gamma(0) \,=\, 1$$.
You are asking why the time dilation factor is not $$\gamma(v)^{2}$$. The answer is that there is no way to derive this result in SRT. Time dilation factors do not multiply in SRT. As I explained, if observer B is time dilated by a factor $$\gamma$$ compared with observer A, and observer C is time dilated by a factor $$\gamma'$$ compared with observer B, then in general C is not time dilated by the factor $$\gamma \gamma'$$ compared with A. That logic only works if simultaneity is absolute, which it is not in SRT.
You are acting as if the full relation between time coordinates in SRT was given by $$t' \,=\, t/\gamma$$. In SRT it is not. The full relation is $$t' \,=\, \gamma(t \,-\, \frac{v}{c^{2}} x)$$. The $$- vx/c^{2}$$ term describes relativity of simultaneity and has the effect that time dilation factors do not multiply in SRT. You find this when you multiply successive Lorentz transformations together, as I suggested you do in [POST=2976278]post #499[/POST]. As I said, the Lorentz transformation is the definitive statement on what SRT predicts, and if you get a result that contradicts the Lorentz transformation, then it is not predicted by SRT.
[sup]*[/sup]The way time dilation is usually described can be a bit confusing, because the relation between time coordinates is the opposite of the time dilation factor: $$t' \,=\, t / \gamma$$. Time dilation means that the time of the moving observer increases more slowly than time in the reference frame at "rest".
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