The Brass Tacks of the Twin Paradox

Back Story

Remember Einstein’s Twin Paradox? Someone goes walkabout at near light speed and winds up younger than their twin who stayed at home. It seems like a conundrum wrapped in an enigma because the the experts insist that moving clocks tick more slowly than one another. It's patent nonsense of course because you can't be both older and younger than your twin when you're standing face-to-face at the family reunion. So what's up with that? Are they pulling our proverbial legs? Here's the geometry in question:

Conventional Wisdom

You can find several explanations in the literature and on YouTube (see links below for example.) The general consensus is that the symmetry is broken by the traveller's antics at the waypoint. The about face at that location is supposed to be the give-away. That's how we know whose clock is running slow. But naysayers with an ounce of common sense are quick to point out that we're being presumptuous. How can we know who's actually doing the about face? Maybe the Earth and the waypoint are zigzagging through spacetime and the traveller is standing still.

Fair enough, but it's easy to sort that out because you inevitably feel a force when you change your speed and/or heading. We can interview the participants about their experience after the fact or equip them with decent accelerometers. Problem solved. The symmetry is broken by acceleration at the waypoint. It's not a very satisfying explanation though because acceleration doesn't factor into a Lorentz transformation. It's all about velocity. Acceleration is just a means to an end.

Barking up the Wrong Tree

Einstein tried to make it better by invoking general relativity: Diaglogue about Objections to the Theory of Relativity. It's an interesting application of his equivalence principle, but it doesn't actually shed any light on the subject because the pesudo-gravity effect goes away as soon as you cut your engines and we can make the burn time arbitrarily short compared to the time spent at cruising speed by simply taking a longer trip. So GR is a distraction. Let's take that off the table first.

In fact, the return trip itself is a red herring for the problem at hand because your antics on arrival cannot retroactively change the time you spent getting there. Stop for tea, head for home, or do a fly-by and carry on. It just doesn't matter because the arrival event is right there in your vacation footage for all to see. It all boils down to constant cruising speed.

Faster or Slower Time?

The events in question for the outbound leg are as follows:

The traveller goes from A to D. The bystander goes from B to D. The chart on the right is scaled down for easier comparison. One interpretation of this result is, the traveller's clock runs slower than the bystander's clock because the A'D' time interval is shorter than the BD one, even after the right-hand chart is scaled up. That's because the bystander gets a head start from the traveller's point of view. One could argue that the traveller's clock actually runs faster because the B'D' time interval is longer than the BD one after the right-hand chart has been scaled up properly.

That's how an engineer would look at it. A Lorentz transformation has two additional parts compared to a Galilean one. First there's a temporal translation (t - vx/c^2) in addition to the spatial one (x - vt). Then there's a scaling factor, which dilates the space and time axes symmetrically. The temporal translation affects the amount by which the traveller ages in transit, but it doesn't change the scale of the time axis. (You can shift the origin from A to B if that makes it any easier to understand.)

Symmetry

However you choose to skin that cat, it does seem lopsided because there's a clear distinction between traveller and bystander. That's unexpected because there's no such distinction in a Galilean transformation. But here's what happens if you start your analysis from the other point of view:

Events A, C, and D are the same as before so the transformation is demonstrably symmetric. Event E is new, but it's not shenanigans. We could have included it in the previous analysis or included event B in this one. The traveller gets a head start in this case so the bystander ages less. The upshot is, the answer to the question of whose clock is running slow depends on when the other guys start counting or equivalently, how far they have to go from your point of view because the initial separation also depends on perceptions of simultaneity.

I would add that race committees for sailing regattas are well aware of this problem. They don't have to worry about scaling factors, but they do have to contend with the simultaneity problem. They spend a lot of time getting the start line perpendicular to the wind before the race starts. Otherwise there's a tactical advantage at one end and that can lead to a traffic jam if the sailors detect the error.

Even so, there's still an epistemological ambiguity because the roles of bystander and traveller are dealer's choice. We could call them Fred and Barney with no loss of generality. The important thing is, events can only be simultaneous in one reference frame. If you want to compare clock rates, you have to decide whether the other guy gets a head start or a late start in the other reference frame. It's all about initial conditions.

The Cosmic Rest Frame

One last thing. The cosmic rest frame is unique amongst all others because the universe has an event horizon. You know you've got it wrong if a transformed event occurs before the dawn of creation. The universe also has no net momentum and you can't go any slower than that. Cosmic time is therefore the fastest time there is so it's the best context for comparing clock rates. It eliminates the epistemological ambiguity.

Resources

Here are some good resources for further study on our collective state of confusion on this subject:

• A similar argument: Implications_of_an_Absolute Simultaneity Theory for Cosmology and Universe Acceleration
• Conventional wisdom: https://en.m.wikipedia.org/wiki/Twin_paradox
• Old school relativity: https://youtu.be/bJMYoj4hHqU
• Einstein himself: https://einsteinpapers.press.princeton.edu/vol7-trans/82
• Minute Physics spacetime globe: https://youtu.be/Rh0pYtQG5wI
• An online version of that: https://www.desmos.com/calculator/pc7azsxteh
• Kevin Brown on the Twin Paradox: The Inertia of Twins
• Viascience on the Twin Paradox: https://youtu.be/kN_d7eknfYk
• Fermilab's Don Lincoln on the Twin Paradox:  https://youtu.be/noaGNuQCW8A
• Dialect on the incompleteness of Twin Paradox solutions: https://youtu.be/FGoAZKyI6ZY
• Brian Greene on Special Relativity: https://youtu.be/XFV2feKDK9E

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