The twin paradox is a thought experiment in special relativity that involves two twins, one of whom remains on Earth while the other travels at high speed to a distant star and back. The paradox arises because, according to special relativity, time dilation occurs when an object moves relative to another. This means that the traveling twin experiences less time passing than the stationary twin.
In the scenario you described, where one twin departs slowly to Alpha Centauri and later another twin leaves at a faster speed and catches up to the first twin near Alpha Centauri, the situation can be resolved using special relativity alone. The resolution lies in the fact that the second twin experiences a different reference frame during their journey.
When the second twin accelerates to catch up with the first twin, they enter a different inertial reference frame. Special relativity tells us that when an object accelerates, different rules apply. During the acceleration phase, the second twin's reference frame changes, and they experience time dilation just like the first twin did during their journey.
As a result, when the second twin catches up with the first twin near Alpha Centauri and they exchange photos, they will find that the elapsed time on their clocks is different. However, both twins can agree on the sequence of events and their respective experiences using the principles of special relativity.
In summary, special relativity provides a consistent framework to explain the twin paradox. By considering the acceleration phase when the second twin catches up with the first, the apparent contradiction is resolved, and the paradox is no longer an issue.