I am learning special relativity and still not sure how to correctly apply the time dilation formula. Take for instance the following example:
A spaceship leaves earth and travels to Alpha Centauri 4.3 ly away with the speed $v = 0.8c$. Seen from earth, the trip takes $\Delta{t} \approx5.3 $ years. Now with the time dilation formula we can see that the austronaut experienced the time $$\Delta{t'} = \frac{\Delta t}{\gamma} \approx 3.12 \ \text{years} $$
Now what I'm tempted to do is to say that time dilation is symmetric and so, from the perspective of the austronaut, the time elapsed on earth is
$$\Delta{t'} = \frac{3.12 \ \text{years}}{\gamma} \approx 1.82 \ \text{years} $$
which is obviously nonsense, however I can't say why this reasoning is flawed.
Best Answer
Let's say a clock on Earth reads $t = 0$ when the spaceship takes off. According to an observer on Earth, the clock reads $t = 5.3\text{ years}$ when the ship reaches Alpha Centauri.
According to an observer on the ship, the same clock reads $t = 0$ when the spaceship takes off, and $t = \text{ 1.82 years}$ when the ship reaches Alpha Centauri, as you have calculated.
There is no contradiction here. What's happening is that the observers don't agree on which event (the clock showing $\text{5.3 years}$ or $\text{1.82 years}$) happens at the same time as the event of the spaceship reaching Alpha Centauri.
If two events happen simultaneously according to observer 1, observer 2 who is in motion with respect to observer 1 will in general conclude that the two events are not simultaneous. There is no concept of absolute simultaneity for spatially separated events in special relativity.