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

Well, if you ever figure that out, tell me so I can write a paper on it and win the Nobel prize.
No, I won't tell you. Want that prize all to myself.

You might have a look at The End of Time by Julian Barbour. Not that this will dispel any mysteries.

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 any one 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.

Hmm... I think I preferred it back in the hornets' nest.

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Does this not make proper time a dimension, too? or do dimensions in relativity comprise only intervals that can seem time-like or space-like to different observers? I'm puzzled by this, but perhaps it is just a semantic distinction made by professionals.

It's also interesting that you distinguish "proper" and "coordinate" times from "real" time. I still have the feeling that if it flows, it's time --- and if not, it's not. What other hypothesis do were you referring to?
Real time is just a (confusing!) synonym for coordinate time.

If an object has a constant velocity over some interval of proper time, we can identify this proper time with the coordinate time of a Lorentz frame (the kind of spacetime coordinate system used in Special Relativity). We call this coordinate system the object's "rest frame" because, in this frame, the object is at rest (not moving). But we don't have to use this frame; we could use a frame in which the object is not at rest but moving at some nonzero constant velocity. The object's proper time would not be parallel to the time axis of a frame where the object is moving, so its proper time would differ from coordinate time in that frame.

If the object accelerates during the interval in question (as in the twins "paradox" when the traveling twin turns around), then there is no single Lorentz frame whose coordinate time is equal to the proper time along the object's entire trajectory through spacetime.

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Dale
Mentor
You might have a look at The End of Time by Julian Barbour.
I was very dissapointed in that book.

Coordinate time is a dimension because one observer's coordinate time can be another observer's coordinate space - just as in Euclidean space where coordinate x and coordinate y axes are dimensions, one observer's coordinate x axis can be another observer's coordinate y axis.
I thought this was not the case. Yes, in the classical situation, A's x axis can be B's y axis and vice versa - and each have perfectly good coordinate systems. But in SR, it's not true that A's t axis can be B's x axis. No matter how much you boost or rotate or translate an SR reference frame, you can't map a temporal axis to a spatial axis.

It is true that the temporal difference between events differs in different frames, as does the spatial separation - perhaps that is all you meant by the quote?

DrGreg
Gold Member
Coordinate time is a dimension because one observer's coordinate time can be another observer's coordinate space - just as in Euclidean space where coordinate x and coordinate y axes are dimensions, one observer's coordinate x axis can be another observer's coordinate y axis.
I thought this was not the case. Yes, in the classical situation, A's x axis can be B's y axis and vice versa - and each have perfectly good coordinate systems. But in SR, it's not true that A's t axis can be B's x axis. No matter how much you boost or rotate or translate an SR reference frame, you can't map a temporal axis to a spatial axis.
Yes, you are right; atyy went a bit too far.

It is true that a dimension that is pure time to one observer can be part-time-part-space to another observer (but never a pure space). And a dimension that is pure space to one observer can be part-space-part-time to another observer (but never pure time).

By "part-time-part-space" I mean in the same way that the direction "north north west" could be described as "part-north-part-west".

atyy
I thought this was not the case. Yes, in the classical situation, A's x axis can be B's y axis and vice versa - and each have perfectly good coordinate systems. But in SR, it's not true that A's t axis can be B's x axis. No matter how much you boost or rotate or translate an SR reference frame, you can't map a temporal axis to a spatial axis.

It is true that the temporal difference between events differs in different frames, as does the spatial separation - perhaps that is all you meant by the quote?
An epsilon bit more than that:

Yes, you are right; atyy went a bit too far.

It is true that a dimension that is pure time to one observer can be part-time-part-space to another observer (but never a pure space). And a dimension that is pure space to one observer can be part-space-part-time to another observer (but never pure time).

By "part-time-part-space" I mean in the same way that the direction "north north west" could be described as "part-north-part-west".
Yes, thanks.

atyy
It's interesting that you identify the time dimension with "coordinate time" (which I take to be local time measured with a constellation of remote clocks synchronized with light signals, as set up by say, an inertial observer) and not with proper time (the time measured by a clock fixed to a {perhaps moving} particle). And yet proper time is marked by events separated by intervals, and can constitute a coordinate axis along which duration can be gauged in an abstract way.

Does this not make proper time a dimension, too? or do dimensions in relativity comprise only
intervals that can seem time-like or space-like to different observers? I'm puzzled by this, but perhaps it is just a semantic distinction made by professionals.

It's also interesting that you distinguish "proper" and "coordinate" times from "real" time. I still
have the feeling that if it flows, it's time --- and if not, it's not. What other hypothesis do were you referring to?

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.

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 amount of ageing in the example is determined by the proper time (the Minkowski arc length of either twin's worldline), though, isn't it? The twin for whom less proper time elapses between the events of them parting and meeting up again is the twin who ages less.

atyy
The amount of ageing in the example is determined by the proper time (the Minkowski arc length of either twin's worldline), though, isn't it? The twin for whom less proper time elapses between the events of them parting and meeting up again is the twin who ages less.
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.

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.

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 any one 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.

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