Schrödinger's Cormorant

A2.5 Nefertiti and Cormorant Tackle the Twin Paradox

If two twins are each moving relative to the other, each will measure that less time has passed for the other, due to time dilation. But can both be right?

Nefertiti has arrived at the pub, to find that Cormorant and his buddies are discussing Star Wars. At first she sighs but then spots an opportunity. She can divert the conversation to get them all thinking about a classic special relativity conundrum. Here’s what she sets out for them to contemplate:

Luke and Leia are (as, of course, they know) twins. Leia climbs aboard a spaceship and goes off, travelling at high speed, into space, while Luke remains at home. Being well versed in special relativity, Luke knows that time will pass more slowly on board the space ship, because of its high velocity in his frame of reference, than it does for himself, remaining still at home. When the spaceship finally returns to Earth, Leia will therefore have lived through less time than he has, so they will be twins no longer: Leia will now be younger than Luke.

Which is weird but it’s not the really weird part. Leia, on board the spaceship, is the static one in her frame of reference while Luke, back on the home planet, is moving really fast relative to her. So Leia’s understanding is that Luke is the one who’ll experience less time, and so she agrees that they’ll no longer be twins when they meet again but she knows that it will be Luke who’ll be the younger one.

How can they both be right? It doesn’t make sense: it’s a paradox. Except …. Hendrik and Uncle Albert are back to shake their heads sagely ….. that perhaps it isn’t really.

“I suspect that there’s been a false assumption made somewhere” says Nefertiti. “While I figure out what it is, let’s do the experiment.” Cormorant is, as ever, enthusiastic.

They synchronise watches. Cormorant commandeers a spacecraft and heads towards Mars, which is 90 million km away. Cormorant is famed for his harsh acceleration, so we’ll assume that the time taken for him to reach his cruising speed is negligible. This cruising speed is 50% of the speed light. Cormorant bounces (more-or-less) elastically off Mars and flies back again at the same speed. Nefertiti remains in the pub to await his return. When he arrives, Cormorant has a few big bruises, sustained in his collison with Mars, and Nefertiti has an expectant look on her face. She has figured it out. Before they compare their watches, she insists on explaining …..

Have you too spotted the flaw in the reasoning? If not, here’s a helpful reminder: the time dilation equation that is the foundation for the apparently paradoxical predictions relates the time that passes in two inertial frames of reference.

Got it yet? If not, here’s a less subtle clue. The equation relates two inertial frames of reference.

Inertial means remaining in a state of rest or uniform motion – in other words, not subject to any external forces. Now that seems like a reasonable description of Nefertiti’s reference frame as she has been sitting in the pub. But as for Cormorant, he has that unsightly bump on his head that says his frame of reference has been far from inertial. We can’t just ignore the fact that his flight involved a 180o reversal of direction at the halfway point. If we ignore the brief periods of acceleration that he’s been through, that means he’s been in two different inertial frames while he was away from the pub. That’s not something we should be ignoring.

“One damn minute” interjects Cormorant. “From my perspective it was you who changed direction, not me. So doesn’t it still work both ways?”

“I don’t have a headache” points out Nefertiti. “That force that changed the direction definitely acted on you, not me. So this time there’s no equality: it’s your reference frame that changed.”

Well that’s not entirely clear, but let’s not argue for now. Instead we’ll try to pick it all apart, with the aid of some maths.

The complication this time is that the frame of reference where Cormorant is stationary changes when he bounces off Mars. We will have to concern ourselves with three frames of reference, so let’s define our abbreviations: we have Nefertiti’s (N), Cormorant’s on his outward journey (C1) and on his return journey (C2). So, for example, xC2(N) means Nefertiti's position (x-coordinate) in Cormorant's frame of reference during his return journey.

We can reduce the wear and tear on your calculator by expressing velocities as fractions of the speed light and using the light minute as our unit of distance (for the x-coordinate).

We’re going to approach this two ways – from Nefertiti’s perspective and then from Cormorant’s. Here’s a table to fill in as we work through each step:

Nefertiti's point of view:

Question A2 10

These calculations should have confirmed, for you, Nefertiti’s expectation that she will have aged more than Cormorant has when they are finally reunited on Earth.

Cormorant’s point of view:

This is a bit more complicated because the time Cormorant actually experiences (his proper time) is the time measured in the frame of reference where he is stationary and that frame is going to switch when he bounces off Mars. Let's first consider that time he himself will experience:

Question A2 11

OK, so Cormorant and Nefertiti agree, albeit through different deductive routes, about the amount of time Cormorant will experience on the mission. But what about the time that Nefertiti will spend, awaiting Cormorant’s return. She measured it to be 20.0 minutes, but what will it be from Cormorant’s perspective? Since he changes his frame of reference at the halfway point, we'll approach this one leg at a time:

Question A2-12

At this halfway stage, when Cormorant has just reached Mars, we have a result which is, qualitatively at least, exactly what we anticipated. Nefertiti, static in her reference frame, observes that less time has passed for Cormorant. Meanwhile, however, Cormorant has observed that less time has passed for Nefertiti. However they are not in the same place in space at this point, so this is not necessarily a paradox.

We need to work out what happens in Cormorant's reference frame on the return leg. This will complete our calculations and we'll find out who has actually aged more when they are reunited.

This is the tricky bit: up until now, Cormorant has been stationary in frame C1, so his spatial coordinate, xC1(C) = 0. Now, though, he has reversed his direction, so xC1(C) will start to change. In order to keep track of this movement, we need to figure out what his velocity now is, in C1. That will allow us to figure out how long it will take him to get home and that will - in turn, after a bit of relativistic correction - give us our answer:

Question A2-13

If you worked through what happens, in this leg, you should end up with this final table.

Looking at the numbers we can see that the final outcome is, remarkably, agreement. Whether you approach it from Nefertiti’s perspective or from Cormorant’s, you reach the same conclusion: when the two meet up again at the end of the mission, Nefertiti has aged by a full 2.68 minutes more than Cormorant.

So perhaps, after all, that there is no twin paradox. The key was properly considering the effects of a change in direction – and hence in the static frame of reference – of one of the protagonists. Working through all the relativistic corrections carefully, we have found that – regardless of whose perspective we approach it from – the timings work out the same at the end, once they are back together at the same location.

The key asymmetry in this was our decision that it was definitely Cormorant and not Nefertiti who was moving. We know that Cormorant underwent all the forces that accelerated him this way and that (he can show you the bruises to prove it) while Nefertiti experienced none of these. So it might seem entirely reasonable to make the assumption that he was moving and Nefertiti was not and therefore that her static reference frame remained constant whereas his changed when he bounced off Mars. The trouble is that there is nothing in the theory of special relativity to say that the effect of forces should be taken into account like this. The whole essence of the theory is that all motion is relative, so if Cormorant was accelerating in his motion relative to Nefertiti then Nefertiti was accelerating, in an equal and opposite sort of a way, in her motion relative to Cormorant.

We seem, in this problem, at least, to have resolved the twin paradox: the shared conclusion was that whoever had actually been moving (and had therefore changed their static reference frame at the midpoint) would experience less time by the end of the journey. This may seem to have wrapped things up satisfactorily but the problem is that in doing this, we have thrown away a fundamental tenet of Special Relativity: if all motion is relative then the idea that one is actually moving and the other is not would be meaningless. Perhaps this means that there is, after all, such a thing as absolute motion. Or perhaps not. This seems like something that needs more advanced study. Nefertiti is intrigued. Cormorant needs a drink. For us, this is as far as we can go with the theory we have, so on this slightly tantalising note, that’s where we must leave them.

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