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(This answer was originally posted for a newer question, asked on 2018 Nov 10, that was later marked as a duplicate question and linked to this one, so I relocated my answer here.)

There are two different kinds of situation that have both been described by the name "twin paradox". One is symmetric, and one is not.

First situation

Consider two objects that meet each other twice. Each object can record the proper time between these two meetings according to its own internal clock. If $\tau_A$ is the elapsed proper time between meetings according to object $A$, and $\tau_B$ is the elapsed proper time between meetings according to object $B$, then they can have $\tau_A=\tau_B$, but typically they will have $\tau_A\neq \tau_B$. In the typical case $\tau_A\neq \tau_B$, the situation is not symmetric. One of the two objects will age less than the other one, and both objects will agree about which one of them has aged less. For example, in flat (Minkowski) spacetime, suppose that:

  • Object $A$ remains in free-fall (that is, weightless) between the two meetings.

  • Object $B$ undergoes constant acceleration (in the sense that it has constant weight) between the two meetings.

In this case, object $B$ ages less between meetings than object $A$ does, and both objects agree about this. This situation is not symmetric.

Second situation

Now consider two objects flying past each other with constant velocities. They do not meet twice; they just keep on going after passing each other once. Each object has its own internal clock, and each object is able to observe (see) the other object's internal clock. In this situation, both of the following statements are true:

  • Object $A$ sees object $B$'s clock running more slowly than its own clock.

  • Object $B$ sees object $A$'s clock running more slowly than its own clock.

This is symmetric. The two objects are behaving symmetrically, so their observations of each other's clocks are necessarily also symmetric.

The second situation is more complicated, because in order for each object to observe the other object's clock, some kind of signal must travel from each object to the other. For example, each object could continually broadcast the time according to its own internal clock, using some kind of radio signal for the broadcast. Most importantly, this situation involves more than just the two objects; it also involves the radio signals that travel from one object to the other. This is why the second situation is more complicated.

Both of the situations described above, the first one and the second one, are described by special relativity using precisley the same principles. The principles are the same, but the situations are different. The second situation is symmetric, and the first one is not.


Appendix

For convenience, this appendix summarizes how the "principles" mentioned in the last paragraph can be expressed mathematically. In flat spacetime (which is the arena of special relativity), we can choose a coordinate system $t,x,y,z$ in which the proper-time increment $d\tau$ is given by $$ d\tau^2 = dt^2 - \frac{dx^2+dy^2+dz^2}{c^2} \tag{1} $$ whre $c$ is the vacuum speed of light and $dt,dx,dy,dz$ are the coordinate increments along any infinitesimal piece of the object's worldline. Equation (1) makes sense only when the right-hand side is non-negative, which is another principle: the worldline of a physical object must be such that the right-hand side of (1) is non-negative. Another principle gives a recipe for converting the proper-time equation into an equation that describes the motion of freely-falling objects. Applied to equation (1), this recipe says that the worldline of a freely-falling object is such that $x,y,z$ are all proportional to $t$. For a freely-falling massless entity, like a pulse of light, the worldline is such that the right-hand side of (1) is zero. This is consistent with calling the constant $c$ the "speed of light." Using these principles, we can analyze both of the types of situation that were described above. The principles about the motion of massless freely-falling entities are used, for example, to determine how light or radio signals propagate from one object to the other in the second type of scenario.

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