[Originally drafted 2024-11-18]
The annoying thing about the twins paradox is that even though we can see that one of the twins did not remain in an inertial frame the whole time, it seems like a minor difference, one that shouldn’t matter. Just why is it that we can’t treat the outbound and inbound segments of Ben’s trip as just two sections of a single trip taken at the same velocity, and then invoke relativity to claim that it was Amelia traveling away and then back? At least that’s how it looked to Ben. He was in a non-inertial frame of reference for only a short time, which if he could withstand high enough acceleration could approach zero. Not much aging difference would have accrued during that brief transition.
The answer is that space is involved. The most helpful thought I’ve encountered is to consider that we are moving through spacetime at lightspeed. This means we are moving through time at lightspeed, but only when we are not moving through space. If we are moving through space, the motion through time has to slow down, because the combined velocity through both space and time cannot exceed c, the speed of light.
If you are not moving through space, your velocity through time is c. If you are moving through space at velocity c, then your motion through time is zero. The rules for the tradeoff are given by the equations of relativity which specify the time dilation as one moves at some velocity through space. It turns out that even as we move around in our daily activities on Earth, we are undergoing small time dilations. But they are so miniscule, that we never notice. When Ben makes a trip to the grocery store, he is a little bit younger than Amelia when he returns.
This still presents the issue of “who is moving?” On Earth we can declare a reference location and mark our positions in space. In the universe at large, who is to say who is moving and who is staying in place?
Well, there does not seem to be an absolute reference point in the universe. So it could be that Ben’s view from his spaceship is that the Earth and stars were moving past him at 0.6c, and then moving toward him at 0.6c, although there was a break in between where he underwent a transition to the second half of the experience.
Ben can’t claim to be on a straight inertial worldline the whole time. We can take one segment, or the other to be the inertial frame of reference, but they can’t both be combined to make a single straight worldline.
We can make any leg of the “paradox triangle” (Amelia’s stay on Earth, Ben’s outbound and return trips) as the inertial frame of reference. We can then calculate what the proper time experiences are for each leg with respect to the chosen reference frame. They all show that Ben’s biologic aging is less than Amelia’s.
Ben is traveling at the same speed on his return trip, but he is also traversing the same space, in reverse. His time dilation (from Amelia’s perspective) occurs whether he continued on, or returned. By returning to a frame of reference that had been inertial the whole time, the effect of Ben’s travel, which traded off his velocity through time with his velocity through space, became unexpectedly apparent.
So to conclude: Ben’s and Amelia’s inertial experiences are not symmetric. Therefore, there is no paradox. The apparent paradox is from assuming that separate inertial travel segments and their time dilations combine linearly. They do not. A full spacetime plot reveals that any one inertial segment can be used as a reference frame, and the proper-time experiences of Ben and Amelia are always consistent.
Traveling through space slows down your travel through time. And yet… we are left with the question of just why can’t we linearly add the separate inertial travel segments? And how do we interpret the timelines where Ben, in his return trip reference frame, appears to have left Earth years before he actually did?
There is still more to understand.