Spontaneous Emissions

[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.

[Original draft 2024-11-15. After publishing my original series of posts analyzing the twins paradox, I realized I had more to say. This and the next post were written a few months later, but not published. A recent memory about an old textbook brought me back to this theme.]

I published a series of posts about the twins paradox and got bemused responses from those who looked at it.  It is a popular puzzle among physics nerds, but most of us are not bothered by such impossible scenarios.

I did however, get a reply from someone who made an effort to resolve it with me, yet came up short.  We both agreed that the paradox was broken by the situations not being symmetric (one twin experienced acceleration; the other did not).  There was no paradox, yet there was still a puzzling discrepancy.  Why would one twin’s path through spacetime result in a different time experience?  What was the cause of it?

It didn’t seem to matter whether the acceleration (to change directions and return home) happened gradually or instantly; there was always a net proper-time difference.  So the details of the acceleration didn’t seem to matter.  Acceleration would not explain the time discrepancy.

I considered some other scenarios.  Here’s one I found amusing. Recall that the twins Amelia and Ben take different life paths: Amelia stays home while Ben travels to an exoplanet three light-years away, and then returns to find that Amelia has aged during his absence. But what if Ben stays on the exoplanet, and Amelia travels to join him there.  Guess what?  When they reunite, they are the same age!