Stochastic Scribbles
Random musings in a variety of subjects, from science to religion.

# Twin paradox: the other twin

In the twin paradox, we have two twins where one remains on Earth and the other travels to a star and back near the speed of light. Because of time dilation, time for the traveling twin passes slowly from the reference frame of the twin on Earth, and vice versa. If for both twins, time for the other twin passes slowly, then both twins should be older than the other, which is nonsensical.

The standard resolution in popular treatments of the twin paradox note that the traveling twin does not stay in a single inertial reference frame, so that what they see does not count. So less time ends up passing for the traveling twin when the twins meet back up. This is correct. It is also an explanation that I have always found highly unsatisfactory.

There are two reasons I have been unhappy with it.

• If the traveling twin coasts along with no change in speed for most of the journey to the star and back, time for the twin on Earth should still flow slowly in the traveling twin’s reference frame while they are coasting. The popular explanation glosses over how the traveling twin ends up younger despite this.

• It does not give us a way to figure out what will be seen by the other twin who travels to the star and back.

The secret to resolving the paradox from the other twin’s point of view is that when they switch reference frames, there is a time skip in the time passing for the twin on Earth. The “now” for the twin on Earth in the traveling twin’s reference frame when they arrive at the star is much earlier than the “now” for the twin on Earth when the traveling twin starts coasting back to Earth. From the twin’s reference frame, time flows slowly on Earth, it goes through time quickly when it turns around, and it slows down again on the way back. The amount of time skipped is more than the amount of time slowed, so the twin on Earth ends up being older than the traveling twin when they meet again.

However, this is not what the traveling twin actually sees, because light from the twin on Earth does not reach the traveling twin instantly. If one works things out, what the traveling twin actually sees is that when the twin is traveling to the star, it looks like time for the twin on Earth passes very slowly, but it looks like it starts flowing much quicker once the traveling twin turns around at the star and travels back to Earth.

For anyone who thought that the other twin was ignored too much in explanations of the twin paradox, this is what happens from their perspective. I also have a more in-depth article discussing the twin paradox using spacetime diagrams elsewhere.