How can I resolve this version of the twin paradox in special relativity?

Question

I have a version of the twin paradox which I am completely stumped by. There is a similar question on the forum but this particular version is unanswered. I really hope someone a lot better at physics than me is able to solve it!

Imagine two twins (Max and Tony) which are accelerated at birth in two opposite directions equally (at exactly the same rate of acceleration) for a long time, and then decelerated (at the exact same rate) until they are at rest with respect to each other far, far apart. Since they both accelerated the same amount, they are now the same age (say 20) in the same inertial reference frame.

Then, they accelerate towards each other until their relative speed is .99c, at which point they stop accelerating and are both travelling at constant velocity in an inertial (non-accelerating) reference frame (special relativity now applies).

At .99c, let's say they pass each other after say 30 years (or we can just stipulate a number of years x). The time dilation factor for .99c is about 1/7, so, when they pass, Max will see Tony at age 20+(30/7) = 24.3 years, and Max will see himself (Max) at age 20+30 = 50 years.

However, by the symmetry of special relativity the same goes for Tony: when they pass, each twin will see the other as 50 - 24.3 or about 26 years younger (or just (6/7x) years younger where x is the number of years travelled at .99c).

Finally, suppose they both accelerate (by decelerating, accelerating and decelerating equally and oppositely) into the same reference frame and come to rest together. Since they accelerate equally, they should age equally (say by y years) to each other. All other acceleration throughout their lives has been equal so no aging differences should have occurred at any point other than when they are travelling at constant speed of 0.99c (when special relativity applies). However, this means that both twins are now both 24.3+y and 50+y years at the same time by the symmetry of special relativity. How can this possibly be?

I’m sure there is a very simple explanation for this but I have no idea how to resolve it. I would expect that both twins are exactly the same age but how do you get round the fact that under special relativity each twin can absolutely legitimately claim the other is younger?

Answer 1

they stop accelerating and are both travelling at constant velocity in an inertial (non-accelerating) reference frame (special relativity now applies).

Special relativity applies to the whole problem, including the accelerating parts, since there's no gravity.

Max will see Tony at age 20+(30/7) = 24.3 years, and Max will see himself (Max) at age 20+30 = 50 years.

That's incorrect. By symmetry they are the same age when they meet, and they'll see each other as being the same age.

Note that when Max is at his farthest from Earth at age 20, he doesn't see Tony at age 20. He sees an infant Tony not long after Tony leaves Earth, because light from most of Tony's trip hasn't had time to reach him yet. While Max travels back toward Earth, he sees Tony age to 20, accelerate toward him, and then very quickly age to 50. The blueshift factor for this last part is $(1+.99)/(1-.99) = 199$, so Tony appears to age 30 years in about 2 months. That means he appears to age from infancy to age 20 in about 29 years and 10 months.

When Max is at his farthest from Earth at age 20, if you draw a spacetime plane perpendicular to his worldline and take the intersection of that plane with Tony's worldline, the intersection will be at the symmetric point where Tony is 20. While people are very fond of drawing these so-called "planes of simultaneity", it's important to understand that they have no physical significance whatsoever, and won't help you understand the nature of special relativity. Except in simple cases, they aren't even helpful in doing calculations. Nevertheless, you can analyze this problem using Max's planes of simultaneity if you really want to. In this case, the resolution of the paradox is that when Max accelerates back toward Earth (an acceleration is a rotation in spacetime), the plane of simultaneity also rotates, sweeping over a large part of Tony's worldline (roughly 26 years of Tony's proper time) "during" the acceleration. Once Max is traveling at his constant speed of $.99c$, the plane moves along Tony's worldline $1/\sqrt{1-.99^2}\approx 7$ times slower than Max's proper time, so Tony ages roughly another $30/7$ years "during" Max's trip back in this unphysical sense of "during". In total Tony ages exactly 30 years "during" Max's acceleration and trip back.

Everything in the last two paragraphs is also true if you swap the names Max and Tony.