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

[oldman]

Special relativity tells us that twins age differently if after birth they part, follow different paths through spacetime and then reunite. The twin who follows the longest worldline ages least. At least this is how I imagine the twin paradox, which I thought I understood.

Here's a silly question. Why doesn't a difference in say, height, develop between the twins as they travel? What aspect of the time dimension makes it so different from the space dimensions, even in relativity where dimensions are treated democratically? Is it just the mysterious fact that "time flies" while space dimensions don't?

 

[member 141513]

Ya, after 20th century,thanks to great scientists,we know that
time and space are just concepts of human beings .
Before we thought that they were natural laws,especially time.
since now we describe physical laws using (ct,x,y,z) coordinates.
time,before it is a natural law, can now be "used" to describe the outside world natural laws.
growth of human is a natural law. so BY the well-known kinematic equation:
NATURAL LAW(I)=TIME(T)-SPACE(S)

I is SS
T is (ct )(ct)
S is XX+YY+ZZ

if I is the natural law,we see that when "speed"(X/ct) increases ,that means XX increases or T decreases or some of their combination , we should not change I.

therefore our "sense" of space should be decreased and "sense" of time should be increased.
but the natural law(growth of the twins) should be just like as usual and is still there~.

PROVIDED that they have delicious food and toys to play and air-conditioning too. just as our daily life~
in fact it is not too bizarre when we use the time concept after 20th century developed

but if we get stuck in the middle ,we may think "WHY NATURAL LAW changes?", always in mind and mind

 

[Jorrie]

Why doesn't a difference in say, height, develop between the twins as they travel? What aspect of the time dimension makes it so different from the space dimensions, even in relativity where dimensions are treated democratically?

My take is that a difference in 'height' (in the direction of travel, i.e., twins are traveling head-first) does develop as far as observations from one frame to the other is concerned, just like a difference in clock rates are observed. Then, just as observed differences in clock rates vanish when the two twins are brought to the same frame again, observed height differences vanish.

However, elapsed time is an integration of clock rates and will differ if the twins took different spacetime paths.

 

[Saw]

Special relativity tells us that twins age differently if after birth they part, follow different paths through spacetime and then reunite. The twin who follows the longest worldline ages least. At least this is how I imagine the twin paradox, which I thought I understood.

Here's a silly question. Why doesn't a difference in say, height, develop between the twins as they travel? What aspect of the time dimension makes it so different from the space dimensions, even in relativity where dimensions are treated democratically? Is it just the mysterious fact that "time flies" while space dimensions don't?

"Height" (length in the direction perpendicular to relative motion) is not affected by SR effects. During the whole trip both twins agree that they are the same height.

"Width" (length in direction parallel to relative motion) would be a different thing. During the trip, in A frame the width of B is shorter and vice versa. At the end of the trip, when they reunite, it is still so. This means, certainly, a difference. As to time, one twin is younger than the other, whilst as to length both twins keep holding that the other is thinner. This, in my opinion, is due to the way one measures: ages or clock readings can be compared in one single spacetime point, while length measurement requires two distant synchronised clocks. To measure the length of a moving object you need either (i) two observers with synched clocks who see the back and the front of the object at the same time (and then you infer that the distance between them is the length of the object) or (ii) one single observer who measures the time it takes for the two ends of the object to pass by (and then length = vt, but in order to measure v beforehand you also needed two synched clocks). So, since simultaneity is relative in SR = synch operations give different results in different frames, the twins are doomed to get different measurements as to their respective widths, whether they are together or not

 

[matheinste]

Special relativity tells us that twins age differently if after birth they part, follow different paths through spacetime and then reunite. The twin who follows the longest worldline ages least. At least this is how I imagine the twin paradox, which I thought I understood.

Here's a silly question. Why doesn't a difference in say, height, develop between the twins as they travel? What aspect of the time dimension makes it so different from the space dimensions, even in relativity where dimensions are treated democratically? Is it just the mysterious fact that "time flies" while space dimensions don't?

It does on the face of it seem a bit odd. I think the answer is roughly along these lines.

When the twins reunite and compare the proper time as shown on their clocks, they are reading the number of ticks, which effectively represents the result of the clock's integration of time during their respective journeys. If each carried some sort of device which measured their distance traveled during their journeys, say one click per meter, these results, representing an integration of their distances journeyed, would also be different. Their clock rates and measuring stick lengths are the same as each other when they reunite and so any length measurements made after reuniting are the same, as are any clock measurements made after reuniting.

Perhaps it is easier to think of the proper time as a count of the number seconds laid along their time intervals between parting and reuniting. In the same way the distance traveled can be thought of as a count of the number of meter sticks laid end to end along the path length. Both cumulative effects.

I think there is confusion here between the height of a twin and the cumulative distance travelled. One is the same after reuniting, the other is not.

I am not entirely happy with this so I too would like to see a definitive answer.

Matheinste.

 

[atyy]

The ideal twins are points to illustrate proper time and the clock hypothesis.

If they are not points, then you have to specify their rigidity, details of acceleration, etc which as we all know, are messy. In a baroque scenario where both twins accelerate, can you have one end of one twin age more than the other, and the other end age less?

 

[matheinste]

The ideal twins are points to illustrate proper time and the clock hypothesis.

If they are not points, then you have to specify their rigidity, details of acceleration, etc which as we all know, are messy. In a baroque scenario where both twins accelerate, can you have one end of one twin age more than the other, and the other end age less?

I thought it wouldn't be simple.

Matheinste.

 

[atyy]

I thought it wouldn't be simple.

Matheinste.

Let's make each twin a bell spaceship string. One accelerates, and one doesn't. The one that snaps will be shorter?

 

[matheinste]

Let's make each twin a bell spaceship string. One accelerates, and one doesn't. The one that snaps will be shorter?

I must have dozed off for a moment, For some reason I had a dream about a hornet's nest.

Matheinste.

 

[atyy]

I must have dozed off for a moment, For some reason I had a dream about a hornet's nest.

Matheinste.

 

[yossell]

What aspect of the time dimension makes it so different from the space dimensions, even in relativity where dimensions are treated democratically? Is it just the mysterious fact that "time flies" while space dimensions don't?

I'm not sure what 'time and space are treated democratically' amounts to here. But somehow, because of it, you're expecting a symmetry?

This may be a contentious answer, so read with due caution!

I don't think distinctions between time and space do completely disappear in SR. There are time-like and space-like lines in SR - and these are distinguished by having a positive or negative Minkowski separation. Spatial and temporal coordinates make different contributions to the invariant 4-d Minkowski metric: in x^2 + y^2 + z^2 - c^2 t^2 (or, as some textbooks prefer, the negative of this) - the time coordinate has a different sign from the spatial ones.

If the separation between two events is + (or (-)) in one frame it will be + (or -) in all. Clocks are understood as moving along time-like lines whereas an extended body has a separation which is space-like in any frame. No amount of frame changing, or accelerating will take a clock on a time-like curve to one on a space-like curve.

This may not be enough to answer your feeling that, somehow, the distinction between time and space needs to be *explained*. However, such demands always for further explanation need to be treated with care - for every theory, explanations must eventually give out and we can only say this is how it is. And hopefully the above may be enough to see that there isn't a straightforward argument from Relativity's treatment of space and time, to the thought that there should be shape dilation analogous to time dilation.

 

[Al68]

What aspect of the time dimension makes it so different from the space dimensions, even in relativity where dimensions are treated democratically? Is it just the mysterious fact that "time flies" while space dimensions don't?

The "space" dimension in the direction of travel is shorter for the traveling twin, in the exact same proportion as the elapsed time.

Time elapsed is shorter, distance traveled is shorter. Sounds like they're more same than different to me.

 

[Dmitry67]

Age is 'accumulated' proper time.
This is why the difference in age does not disappear when twins meet again.

 

[Al68]

Age is 'accumulated' proper time.
This is why the difference in age does not disappear when twins meet again.

Exactly. If each twin had a clock and an "odometer", they would show both less time and distance for the traveling twin.

 

[Dale]

IMO, this has more to do with the difference between "age" and "height" than the difference between time and space. When you ask for someone's height you are asking for their total spatial extent from head to foot. When you ask for age you are not asking for their total temporal extent in the birth to death direction, but just the extent between birth and "now". A comparable measure for space would be the vertical distance from "here" (e.g. my kitchen table top) to the top of a person's head. If one twin were standing bent over at the waist that measure would be different, not because the bent twin was "really" shorter but simply because the measurement itself captures a different piece of information. For time, the measurement comparable to height would be lifespan.

 

[Borg]

"Height" (length in the direction perpendicular to relative motion) is not affected by SR effects. During the whole trip both twins agree that they are the same height.

So, if either person lays down (parallel to the direction of relative motion), they appear shorter to the other?

 

[matheinste]

IMO, this has more to do with the difference between "age" and "height" than the difference between time and space. When you ask for someone's height you are asking for their total spatial extent from head to foot. When you ask for age you are not asking for their total temporal extent in the birth to death direction, but just the extent between birth and "now". A comparable measure for space would be the vertical distance from "here" (e.g. my kitchen table top) to the top of a person's head. If one twin were standing bent over at the waist that measure would be different, not because the bent twin was "really" shorter but simply because the measurement itself captures a different piece of information. For time, the measurement comparable to height would be lifespan.

Along those lines, in someones own rest frame, while living, their age is increasing, while their meter stick remains the same length. However at the end of their life, neither is inceasing.

So in the case of the twins, let us assume that their non separated lifespans would have been equal. Then, after separating and reuniting they would have the same lifespan as measured by their own clocks at their respective deaths. But their actual deaths would not be simultaneous in their reunited rest frame inasmuch that the stay at home would die first.

So, same proper times at respective deaths and same meter stick lengths.

Does that seem reasonable.

Matheinste.

 

[cfrogue]

The ideal twins are points to illustrate proper time and the clock hypothesis.

If they are not points, then you have to specify their rigidity, details of acceleration, etc which as we all know, are messy. In a baroque scenario where both twins accelerate, can you have one end of one twin age more than the other, and the other end age less?

The answer would seem to be affirmative.

http://articles.adsabs.harvard.edu//full/1992ASPC...30...1L/0000001.000.html

 

[oldman]

My take is that a difference in 'height' (in the direction of travel, i.e., twins are traveling head-first) does develop as far as observations from one frame to the other is concerned, just like a difference in clock rates are observed. Then, just as observed differences in clock rates vanish when the two twins are brought to the same frame again, observed height differences vanish.

However, elapsed time is an integration of clock rates and will differ if the twins took different spacetime paths.

Thanks, Jorrie --- and the rest of you, for taking the trouble to consider my silly question. I agree with the way you and matheinste think, Jorrie. There seems to be a fundamental difference in the way distance and time intervals are measured. Distance measurements involve the concept of simulteneity -- no integration needed here (but the time dimension is invoked) --- wheras measured time involves duration, needing integration over flowing time. The concept of simultaneity and duration are two observer-assigned attributes of the time dimension; namely a significant origin and a scale. The space dimensions, on the other hand, seem to lack significant origins analagous to time's observer assigned 'now'.

But, like matheinste:

I am not entirely happy with this so I too would like to see a definitive answer.

 

[atyy]

The proper time, which is the time involved in the twin paradox, isn't the time that is a dimension. Time as a dimension is coordinate time. 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. Proper time is a geometric invariant, and you can rotate your spacetime coordinate axes any way you want, even use non-inertial coordinates, and the proper time will always be the same.

 

[atyy]

Furthermore, that the proper time is related to real time requires an additional hypothesis above the principle of relativity, and the constancy of the speed of light.

 

[oldman]

...I don't think distinctions between time and space do completely disappear in SR. ... No amount of frame changing, or accelerating will take a clock on a time-like curve to one on a space-like curve...

...there isn't a straightforward argument from Relativity's treatment of space and time, to the thought that there should be shape dilation analogous to time dilation.

Yes, I agree with you. The distinctions between space and time don't disappear in relativity, although Minkowski's statement: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" may give some that impression. Time flows and can be accumulated, space doesn't and can't. Why we experience time as a flow seems to me still mysterious, despite the growing acceptance of ideas about block universes.

 

[Al68]

Why we experience time as a flow seems to me still mysterious...

Well, if you ever figure that out, tell me so I can write a paper on it and win the Nobel prize.

 

[Dale]

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.

 

[oldman]

...Furthermore, that the proper time is related to real time requires an additional hypothesis above the principle of relativity, and the constancy of the speed of light.

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?

I still need educating, please.

 

[oldman]

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.

 

[Rasalhague]

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.

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

 

[Rasalhague]

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.

 

[Dale]

You might have a look at The End of Time by Julian Barbour.

I was very dissapointed in that book.

 

[yossell]

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]

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?

I still need educating, please.

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.

 

[Rasalhague]

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.

 

[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?

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.

 

[member 141513]

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.