The Twin Dilemma

Imagine you have a pair of twins. On their birthday, the twins are drugged and then wake up in identical cells. Only one twin is in a cell on the surface of Earth and the other is in a cell on a spaceship accelerating at 1g. How does each twin work out whether they are on surface of the Earth or on the starship?

The answer, of course, is when the twin on the starship experiences a period of weightless when the starship is flipped for the return. That's what breaks the symmetry. Ler's say the twins are put in the cells on their birthday and are allowed to send a birthday greeting. The twin on Earth sends a message one year after being placed in the cell.

Now, I can't find a damned a damned calculator for what happens for a statship accelering at 1g, The one I thought was doing whay I thought it was was actually calculating the times for missions involving deaccelerating and possibly even a round trip. So I suppose I am going to have work it all out for scratch (at some point). I remember doing the calculation for an accelerating starship. It's a nice calculation and I remember finding it in a book later, so I did so correctly and, indeed, we find a FoAK article on the Hyperbolic motion (relativity). The point is that during the outbound leg, whether the starship is accelerating or travelling at a constant speed (which makes the calculation easier, but obviously misses the point about one twin being on Earth), the twins receive the messages at exactly the same ages. In the accelerating case, they must be increasingly spread out. But it is not until turn around that symmetry is broken.

This is one of those weird relativity things that is relevant to my notion of the extended now. There is no universal now. Although we might say that a picture of Andromeda shows it as it was 2 million years ago, in another sense, what we see is the shared now. This is something that I need to thing about a lot more and I wish we had had someone teach it us at university. We did do special relativity, of course, but only, I think, a couple of tutorials. I recall Ken making a complete howler about the relativistic Doppler shift and wondering how he could possibly have made that error, but he did get a double first and I didn't.

The thing is even if we drug the twin again in the starship at turnover and rearrange the cell, on the return leg, the starship will cross all of the messages the twin on Earth sent in a rush. THat almost makes it feel as though the Earth twin is put through an aging weapon, but, of course, that's not really right. Both twins have lead their perfectly normal lives, it is just on the return can the starship twin see this. If threy kept on accelerating in the same direction forever they would never know. 

It did strike me though that there is one way for the twins to tell which is on Earth and which on the starship. Coroilis effect is one, of course, Could you measure that in the cell? And, on a spinning spaceship, things would be weird, but there's no need to spin a torchship. On Earth though, the gravitational field strength is lower at the ceiling than at the floor. I don't how east it is to measure that with a pendulum, but it is not a completely minite amount over a few metres. At 64 km, there's about a 2% reduction or 2 parts in 100. That corresponds to 2 parts in 100,000 at 64 m or 2 parts in a million at 6.4 metres. If you had a time source accurate to the microsecond, you could potentially measure the difference in the period of a pendulum if a cell with a high enough ceiling. Of course, it would be easier if the cell were more of an office block.But a starship the acceleration would be the same at all heights. So there is a difference between the gravity field on Earth and one on an accelerating spaceship.

Comments

[John B]

If you had true prisoners, without tech, I don't think they'd know. It would be easy to drug them during turnover.

If you had scientist prisoners with full tech, they could

1) test for micro earthquakes or the rumble from the drive

2) Observe variation in penetrating radiation, which might be coming from the drive, or space, and might have diurnal variation etc

3) thrust and gravity would differ in angle off-axis differently, and more so than with height

4) Atmospheric composition different in a life support system.

[Paul M. Cray]

Yes, if you have enough technical knowledge and equipment, there is a lot you can to determine where you are. For instance, a Foucault's pendulum will tell you if you are on a rotating planet. An accelerating starship will fairly soon be travelling at a fair clip. If you have a convenient supply of dry cleaning fluid, you can make yourself a neutrino detector. If you are travelling at close to the speed of light, you will detect a lot more neutrino bursts from supernovae that you would if you were at rest with respect to the fixed stars.