Why does a travelling twin age slower but not get shorter?

[atyy]

Incidentally, if we restrict ourselves to inertial worldlines, where we don't need the clock hypothesis, then although they are not exact analogues, we do have time dilation and length contraction as weirdy somewhat counterparts.

 

[Rasalhague]

I was thinking that maybe ageing cannot even be defined, if the accelerated twin is killed by the acceleration. If he's snapped in half, then is he longer or shorter? OK, this is morbid, sorry , I'm just doing this off the top of my head.

Uh oh, I think the top of my head just snapped off. Is he more than the sum of his parts? Does it make any difference if we accelerate the evil twin? What if he's an amoeba?

 

[atyy]

What if he's an amoeba?

:rofl: Amoebas are immortal.

 

[atyy]

How would you measure the height of the twins if one always stood straight and the other always stood bent over at the waist? If you perform an analogous measure of the lifespan of the twins you will get an analogous result. I don't think there is much difference conceptually between time and space in this scenario.

In the classic twin "paradox" proper time (arc length) along the trajectory of the traveling twin is shorter than the proper time along the trajectory of the stay-at-home twin. In twin the home-twin's rest frame, the traveling twin's spatial coordinates vary over time. This makes a shape on a spacetime diagram with one straight edge (home-twin's position, parallel to the time axis of home-twin's rest frame), and one curvilinear edge (traveling twin's worldline--path through spacetime--having both time and space components in anyone frame).

Can we find an equivalent geometric scenario in which the roles of time and space are reversed, and would it have any physical meaning? Here's my attempt. It sounds a bit nonsensical to me, which leads me to suspect the answers to these questions are possibly YES there is an analogous geometric scenario, but NO it doesn't have an obvious physical significance.

Here goes... I suppose the spatial analog of the stay-at-home twin would have no duration in time in his rest frame, so he'd be an instantaneous twin in that frame. But he'd have some spatial extent. He'd extend in a straight line in some direction. And the spatial equivalent of the traveling twin, I suppose, we could think of as extending in an arc from one end of the instantaneous twin to the other, a bit like this: http://www.crystalinks.com/geb.gif (sky goddess Nut = spatial analog of traveling twin; Earth god Geb = spatial analog of stay-at-home twin), except that the gap between them in the middle isn't a spatial gap but due only to a difference in time. (And the tangent line to Nut's body never, at any point, makes as great an angle from the x-axis as that of the a lightlike worldline, just as the tangent line to the traveling twin's worldline in the traditional twin "paradox" never makes as great an angle from the t axis, of any frame, as a lightlike worldline.) I guess that would mean that Nut's toes and fingers exist simuntaneously in Geb's rest frame, but Nut's middle only comes into existence after Geb has disappeared. (So, every bit as frustrating as the myth.) After Geb disappeared, we'd see Nut's ankles and wrists, then they'd be replaced by her calves and forearms, and so on till we saw her middle. If we integrated all the infinitesimal spacelike intervals along her, I suppose, the result would be less than the distance along Geb's body. But this isn't really the length of a physical object, as she doesn't all exist at the same time.

Thinking about this a bit more, and taking the cue from the above comments, I guess the twin paradox basically asks for the invariant length of two timelike curves - one geodesic and one not. So the exact analogue would be the invariant length of two spacelike curves - one geodesic and one not. The invariant length is the same formula defined using the metric in both cases, except with a minus sign on one of them to get a real answer. The invariant length is not defined for curves that switch from timelike to spacelike.

 

[pliu123123]

Coordinate time and space are "meaningless", they are just addresses for events in spacetime. Proper time is an invariant and potentially meaningful. The twin paradox is not about time dilation which is a coordinate effect and "meaningless", but about proper time. However, to make proper time meaningful requires the "clock hypothesis" which defines an ideal clock as one that reads proper time. Is true ageing proper time? No. Ageing is affected by acceleration. For a quick turn around, the accelerated twin's lifespan would be affected - as would his length! The twin paradox is just a fanciful way to illustrate proper time and the clock hypothesis using ideal point twins.

i argee that relativity illustrate proper time, and other term that can be similar to "proper time" would be "natural laws" or "physical laws" .

 

[DrGreg]

Thinking about this a bit more, and taking the cue from the above comments, I guess the twin paradox basically asks for the invariant length of two timelike curves - one geodesic and one not. So the exact analogue would be the invariant length of two spacelike curves - one geodesic and one not. The invariant length is the same formula defined using the metric in both cases, except with a minus sign on one of them to get a real answer. The invariant length is not defined for curves that switch from timelike to spacelike.

If one of the twins travels from Earth to Alpha Centauri and back (say), then the traveling twin reckons the round trip distance, Earth to Alpha Centauri in the outgoing frame, plus Alpha Centauri to Earth in the return frame, is shorter than the stay-at-home's reckoning of Earth to Alpha Centauri plus Alpha Centauri to Earth, both in the Earth's frame. Draw a spacetime diagram showing all those lengths as spacelike intervals, and what you get is pretty much what atyy said.