Is it time to "retire" time dilation and length contraction?

[m4r35n357]

Considering the enormous number of questions posed on this forum and other places, the concepts seem fundamentally flawed (because both are formally and practically unobservable). The calculations themselves (together with the Lorentz Transform) are highly error-prone and the results misleading (the "Mr Tomkinson" phenomenon) and unsatisfying (you can't directly see either except in very specific circumstances).

I realize this might come across as flamebait, but as an amateur learner I have not found either concept particularly useful to my studies. They seem most common in pop science "wow look how weird relativity is" presentations, and the fallout is seen in all of the forums that I visit (here, Physics Stack Exchange, reddit/AskScience etc)

As an alternative, I would propose approaches that I have seen in a few places, but which are not very common.

1) Formal derivation of the Lorentz Transform, leading swiftly to defnition of the space-time interval and space-time diagrams.

2) Introduction of four-vectors, including four-momentum and four-frequency

3) Extension of the 1+1 spacetime to 2+1 (which is necessary and sufficient for first-person calculations of what we can see/observe/measure)

4) Develop formulas for aberration/Doppler/headlight effects, so students will be able to calculate eg. the shift in apparent positions etc. of stars/galaxies

5) maybe introduce time dilation etc as a historical formality for the interested (or bored!) student

I'll leave it here for now, rest assure that I have other practical arguments in reserve if anyone deems this post worthy of comment!

 

[PAllen]

I think that is a plausible approach, consistent with de-emphasizing coordinate dependent features (which time dilation and length contraction are).

 

[PWiz]

1) Formal derivation of the Lorentz Transform, leading swiftly to defnition of the space-time interval and space-time diagrams.

2) Introduction of four-vectors, including four-momentum and four-frequency

I agree that this should be the approach for all students who wish to pursue GR into the future. It makes the algebraic to topological transition much easier later on.

 

[Nugatory]

I expect that no matter what approach is adopted, the essential challenge will be the same: The student has to let go of the notions of absolute time and absolute simultaneity first. Although I agree that length contraction and time dilation are widely misunderstood, I also find that these misunderstandings are almost always the result of the deeper misunderstanding of absolute time.

 

[m4r35n357]

I expect that no matter what approach is adopted, the essential challenge will be the same: The student has to let go of the notions of absolute time and absolute simultaneity first. Although I agree that length contraction and time dilation are widely misunderstood, I also find that these misunderstandings are almost always the result of the deeper misunderstanding of absolute time.

Agreed. In my case I did just that, but found that after letting go of absolute time and simultaneity I was left floundering, wondering what was left to trust ;) The thing the brought me back into the "reconstructive" phase was precisely the concept of proper time, which I felt had been hidden behind layers of deconstructive and tedious calculations (crushed spaceships and bloody trains!) in the majority of presentations.

Based on my own experience I am convinced that if a student has to let go of such fundamentals (and they must), they need to be given something else to hang on to!

I think that learning about proper time, together with calculating what we can actually see, were my salvation. Once learned, I could look back on all my failed attempts to understand SR and see that much of the teaching material itself had been the major obstruction in my case.

 

[m4r35n357]

I agree that this should be the approach for all students who wish to pursue GR into the future. It makes the algebraic to topological transition much easier later on.

Even for SR, in my opinion (regarding 1); If proof be needed, then surely the twin paradox is the golden example of where the space-time interval and diagrams are the right way to go.

Just reading people's explanations of what happens with simultaneity at the turnaround makes me cringe, and don't even get me started on explanations based on acceleration ;)

 

[phinds]

@m4r35n357, while I agree w/ your annoyance at all the time and energy that is spent on correcting unfortunate misconceptions (for which, I think, Nugatory has identified a deeper root cause), and I do not argue at all with your proposal, I would point out that is it not going to have any effect. Beginners, and even more to the point amateurs, are going to keep asking those same questions over and over whether you or I like it or not and I think part of the charter of PF is to help them clarify the issues by having those same damned long discussions every time a noobie asks the questions. For a lot of those folks the very phrase "Formal derivation of the Lorentz Transform" will make their eyes glaze over. I think you have to guide them to that, not hit them over the head with it up front even though doing so would be better for the more math-oriented.

 

[m4r35n357]

@m4r35n357, while I agree w/ your annoyance at all the time and energy that is spent on correcting unfortunate misconceptions (for which, I think, Nugatory has identified a deeper root cause), and I do not argue at all with your proposal, I would point out that is it not going to have any effect. Beginners, and even more to the point amateurs, are going to keep asking those same questions over and over whether you or I like it or not and I think part of the charter of PF is to help them clarify the issues by having those same damned long discussions every time a noobie asks the questions. For a lot of those folks the very phrase "Formal derivation of the Lorentz Transform" will make their eyes glaze over. I think you have to guide them to that, not hit them over the head with it up front even though doing so would be better for the more math-oriented.

My "annoyance, list of demands, whatever" was more aimed at courses, books, etc. that keep peddling the same old line than at what happens on this forum (the OP was tagged "teaching", but it's not really obvious). I only mentioned this forum (amongst others) in the sense that this is one of the places that the casualties turn up!

I'm not trying to tell anyone here what to do, just giving some idea of what I feel is likely to be effective. Although if I was starting by asking questions here I would want to learn about proper time first, purely because the concept is simpler (as are the calculations).

BTW I'm slightly amused that you seem to think that "Formal derivation of the Lorentz Transform" was aimed at "noobies" here ;)

 

[phinds]

My "annoyance, list of demands, whatever" was more aimed at courses, books, etc. that keep peddling the same old line than at what happens on this forum (the OP was tagged "teaching", but it's not really obvious). I only mentioned this forum (amongst others) in the sense that this is one of the places that the casualties turn up!

fair enough.

I'm not trying to tell anyone here what to do, just giving some idea of what I feel is likely to be effective. Although if I was starting by asking questions here I would want to learn about proper time first, purely because the concept is simpler (as are the calculations).

One of the challenges faced here is that we get SO many people who don't really want to study anything, they just want simple answers to, for example, what they heard Michio Kaku spouting on a TV show.

BTW I'm slightly amused that you seem to think that "Formal derivation of the Lorentz Transform" was aimed at "noobies" here ;)

I took your whole post as being addressed at noobies, since they are the ones that create the issue that I perceived you to be addressing. Certainly a noobie with a PhD in physics isn't likely to have the same level of difficulty as a much less knowledgeable noobie, but the latter are WAY more common here.

 

[m4r35n357]

I took you whole post as being addressed at noobies, since they are the ones that create the issue that I perceived you to be addressing. Certainly a noobie with a PhD in physics isn't likely to have the same level of difficulty as a much less knowledgeable noobie, but the latter are WAY more common here.

I trust we now all understand that my ire was aimed at the teaching materials and not the students . . .

 

[phinds]

I trust we now all understand that my ire was aimed at the teaching materials and not the students . . .

fair enough, but I don't think that was clear in the original post

 

[m4r35n357]

Hmm, I thought I was quite explicit that I was talking about my personal learning experience and how TD and LC (obvious abbreviations) hindered rather than helped.

Here is a more pointed attempt at the title of my original post (too long for a real title without the abbreviations):

"Should the concepts of TD and LC be regarded in same way as relativistic mass?"

Neither set of ideas is actually wrong, but they all serve to make the subject of SR harder to learn than it should be (in my experience of course).

 

[SlowThinker]

Considering the enormous number of questions posed on this forum and other places, the concepts seem fundamentally flawed
...
As an alternative, I would propose approaches that I have seen in a few places, but which are not very common.

1) Formal derivation of the Lorentz Transform, leading swiftly to defnition of the space-time interval and space-time diagrams.

From my experience (when I struggled to understand SR, or trying to explain it to others), the relativity of simultaneity (I call it "time's slope" ) is the first thing that needs to be explained and understood. Without it, the mutual length contraction and mutual time dilation don't make any sense, which is the main obstacle in accepting those 2 concepts.

Even noobies have heard about "time is relative", but if I say that my clock runs twice as fast as yours, and yours are running twice as fast as mine, they are left scratching their heads.

Sooner or later you'll need to introduce the Lorentz transformation, but starting with it isn't likely to be effective. IMHO.

 

[m4r35n357]

Without it, the mutual length contraction and mutual time dilation don't make any sense, which is the main obstacle in accepting those 2 concepts.

I've never computed either, and don't see why I would need to (nor have I ever computed relativistic mass).

My point is they don't really help much so why bother going to the extra effort (at least to start with)? All they tell you is some unobservable weirdness associated with two moving objects (in two frames of reference). Using the spacetime interval, a piece of paper and a ruler one can explain the twin paradox in words of one syllable (almost). That is a fairly complex example using three frames of reference, and the result is observable. Maybe it's a personal thing but I find the latter far more satisfying, as well as easier to learn.

 

[Smattering]

I think that is a plausible approach, consistent with de-emphasizing coordinate dependent features (which time dilation and length contraction are).

In the twin paradox, once the twins are re-united, there can be age differences that are not coordinate dependent.

So is it really correct to say that time dilation is a purely coordinate dependent feature?

 

[m4r35n357]

In the twin paradox, once the twins are re-united, there can be age differences that are not coordinate dependent.

So is it really correct to say that time dilation is a purely coordinate dependent feature?

The age difference, like the aging itself, is coordinate-independent (invariant). Time dilation does not enter into the discussion or the calculation.

 

[Dale]

We can't even seem to "retire" relativistic mass.

 

[Smattering]

The age difference, like the aging itself, is coordinate-independent (invariant). Time dilation does not enter into the discussion or the calculation.

And what term other than "time dilation" would you propose to describe the fact that one person has aged at a different rate than the other or some physical process has progressed at a different rate?

 

[m4r35n357]

And what term other than "time dilation" would you propose to describe the fact that one person has aged at a different rate than the other or some physical process has progressed at a different rate?

Difference in "length" of world lines.

 

[SlowThinker]

I've never computed either, and don't see why I would need to (nor have I ever computed relativistic mass).

Hmm so when I fly to Alpha Centauri at 0.9c, I only age 1.83 years.
How do you explain it without length contraction?
How can you even use Lorentz transformation without referring to length contraction?

 

[m4r35n357]

Hmm so when I fly to Alpha Centauri at 0.9c, I only age 1.83 years.
How do you explain it without length contraction?
How can you even use Lorentz transformation without referring to length contraction?

I calculate the "length" of the world line, t^2 - x^2 (and x = 0.9t). I certainly wouldn't go to the extra effort of calculating the Lorentz Transform, or bother to look up the formulas for time dilation or length contraction (I haven't memorized them either).

That is my point, in a nutshell.

 

[PeroK]

I've never computed either, and don't see why I would need to (nor have I ever computed relativistic mass).

My point is they don't really help much so why bother going to the extra effort (at least to start with)? All they tell you is some unobservable weirdness associated with two moving objects (in two frames of reference). Using the spacetime interval, a piece of paper and a ruler I can explain the twin paradox in words of one syllable (almost). That is a fairly complex example using three frames of reference, and the result is observable. Maybe it's a personal thing but I find the latter far more satisfying, as well as easier to learn.

I like to understand what I'm doing intuitively. So, the first step in SR for me was definitely to rebuild my intuition. Once that was done, I was happy to crunch things unthinkingly with Lorentz. If you start with Lorentz and don't mention the unmentionables, then you'll get fewer questions, but that's because you haven't confronted and challenged one's classical intuition.

If you start with Lorentz, sooner or later someone is going to notice that a high-speed particle appears to live longer than it should. I doubt that hitting the problem with Lorentz is going to be satisfactory. I would want to know why a particle appears to live longer and just plugging things into a Lorentz Transformation wouldn't be satisfactory. I fail to see how time dilation is unobservable: the lifetime of a high-speed particle; the atomic clocks in the Hafele-Keating experiment?

My other point is that time dilation is not weird: once you've rebuilt your intuition it's, well, blindingly obvious.

Hyperbolic spacetime with a non-Euclidean metric? Now that is weird!

 

[Smattering]

Difference in "length" of world lines.

I see. But I am pretty sure people will still be asking how a difference in the length of world lines can result in twins aging at different rates.

At what age do people get in contact with relativistic physics typically? I'd say not earlier than at the age of 16 maybe. So 16 years long they have been living in a Newtonian world, they have watched everything happen at non-relativistic speeds, and their brains have adopted to this world where absolute time is not a misconception at all, but a perfectly valid approximation.

And now you expect that there is some way to teach them relativistic physics without confusing them?

 

[m4r35n357]

Once that was done, I was happy to crunch things unthinkingly with Lorentz

That's my justification for step 1 in my OP. I wasn't happy to do that because it seems messy and over-complicated to me. Once derived, one can dispense with the LT and just use "pythagoras".

 

[m4r35n357]

I see. But I am pretty sure people will still be asking how a difference in the length of world lines can result in twins aging at different rates.

At what age do people get in contact with relativistic physics typically? I'd say not earlier than at the age of 16 maybe. So 16 years long they have been living in a Newtonian world, they have watched everything happen at non-relativistic speeds, and their brains have adopted to this world where absolute time is not a misconception at all, but a perfectly valid approximation.

And now you expect that there is some way to teach them relativistic physics without confusing them?

There is a minimum of pain in deriving the LT once, for all time (one 2x2 matrix inversion and about 5 lines of high-school algebra), then the spacetime interval is one or two lines of even easier algebra. They can then get results easily, and that is the major obstacle overcome. If they wish to contemplate the "internals" at more length (step 5) then they can go back and do it the "hard" way in the confidence that they can check their results easily.

Do not underestimate the practical difficulty of using the LT when all your preconceptions about time and simultaneity are laying in pieces on the floor!

BTW I was about 47 when I decided to learn this stuff ;)

 

[pervect]

Considering the enormous number of questions posed on this forum and other places, the concepts seem fundamentally flawed (because both are formally and practically unobservable). The calculations themselves (together with the Lorentz Transform) are highly error-prone and the results misleading (the "Mr Tomkinson" phenomenon) and unsatisfying (you can't directly see either except in very specific circumstances).

I realize this might come across as flamebait, but as an amateur learner I have not found either concept particularly useful to my studies. They seem most common in pop science "wow look how weird relativity is" presentations, and the fallout is seen in all of the forums that I visit (here, Physics Stack Exchange, reddit/AskScience etc)

As an alternative, I would propose approaches that I have seen in a few places, but which are not very common.

1) Formal derivation of the Lorentz Transform, leading swiftly to defnition of the space-time interval and space-time diagrams.

2) Introduction of four-vectors, including four-momentum and four-frequency

3) Extension of the 1+1 spacetime to 2+1 (which is necessary and sufficient for first-person calculations of what we can see/observe/measure)

4) Develop formulas for aberration/Doppler/headlight effects, so students will be able to calculate eg. the shift in apparent positions etc. of stars/galaxies

5) maybe introduce time dilation etc as a historical formality for the interested (or bored!) student

I'll leave it here for now, rest assure that I have other practical arguments in reserve if anyone deems this post worthy of comment!

I think this is rather similar to the approach in modern texts, such as Taylor & Wheeler's "Space-time physics", though Taylor& Wheeler attempt to motivate the Lorentz interval by considering first the fact that distance between two points on a map stays the same if you rotate the map. Then they draw an analogy to the way that the way the Lorentz interval stays the same if you change velocities. This is the "Parable of the Surveyor", I can provide quotes and/or weblinks to the first edition the first few chapters of which are available on Taylor's website.

For a lower-level introduction than the 4-vector approach (which is good for college level and as far as I know mostly standard nowadays at that level), I still prefer the K-calculus approach (such as Bondi's "Relativity and Common Sense". The K-calculus approach is by no means modern anymore).

However, in spite of the fact that standard college-level texts have been (afaik) teachingl relativity this way for some time now, we still get all these questions about time dilation, and I rather suspect we'll continue to do so. It seems that the average curious student does not start out with a college level textbook, but reads a popularization instead.

Typically, I've noticed that students attempt to interpret time dilation in terms of pre-existing notions of "absolute time", and that it's hard to get them to change their thinking. It's hard to get them to even understand what is meant by the phrase "absolute time" so that one can explain why it doesn't work. The notion is firmly implanted in their thoughts, and the concepts to talk about it do not seem to be there, one can repeat the phrase "time is relative and not absolute" as often as one likes, but if it's not understood, it does no good, the words are perhaps heard but they are not understood properly.

People have studied how to teach the relativity of simultaneity in the context of a college course, judging their success by how well students did on standardized tests. It's an interesting read, but I"m unsure how effective this approach is on forums like PF. Specifically, see Scherr's "The challenge of changing deeply held student beliefs about the relativity of simultaneity".

I may be over-focussed on the relativity of simultaneity issue. Sometimes the problem is even more basic. I may start another thread on this.

 

[m4r35n357]

I think this is rather similar to the approach in modern texts, such as Taylor & Wheeler's "Space-time physics", though Taylor& Wheeler attempt to motivate the Lorentz interval by considering first the fact that distance between two points on a map stays the same if you rotate the map. Then they draw an analogy to the way that the way the Lorentz interval stays the same if you change velocities. This is the "Parable of the Surveyor", I can provide quotes and/or weblinks to the first edition the first few chapters of which are available on Taylor's website.

For a lower-level introduction than the 4-vector approach (which is good for college level and as far as I know mostly standard nowadays at that level), I still prefer the K-calculus approach (such as Bondi's "Relativity and Common Sense". The K-calculus approach is by no means modern anymore).

However, in spite of the fact that standard college-level texts have been (afaik) teachingl relativity this way for some time now, we still get all these questions about time dilation, and I rather suspect we'll continue to do so. It seems that the average curious student does not start out with a college level textbook, but reads a popularization instead.

Typically, I've noticed that students attempt to interpret time dilation in terms of pre-existing notions of "absolute time", and that it's hard to get them to change their thinking. It's hard to get them to even understand what is meant by the phrase "absolute time" so that one can explain why it doesn't work. The notion is firmly implanted in their thoughts, and the concepts to talk about it do not seem to be there, one can repeat the phrase "time is relative and not absolute" as often as one likes, but if it's not understood, it does no good, the words are perhaps heard but they are not understood properly.

People have studied how to teach the relativity of simultaneity in the context of a college course, judging their success by how well students did on standardized tests. It's an interesting read, but I"m unsure how effective this approach is on forums like PF. Specifically, see Scherr's "The challenge of changing deeply held student beliefs about the relativity of simultaneity".

I may be over-focussed on the relativity of simultaneity issue. Sometimes the problem is even more basic. I may start another thread on this.

Yes, I distilled that list from lots of sources; there are lots of "good" texts like you describe, but they are usually well hidden amongst the chaff ;) I've heard a lot of good things about Bondi k-calculus, but like the other "good stuff" I didn't encounter it (freely online) until I had already suffered the pain, so it came too late for me!

I quote a lot from "Reflections on Relativity", but it's not really a textbook; more useful for consolidation of knowledge (although it contains the LT derivation I have mentioned above).

 

[PeterDonis]

what term other than "time dilation" would you propose to describe the fact that one person has aged at a different rate than the other or some physical process has progressed at a different rate?

"Differential aging".

 

[PeterDonis]

I am pretty sure people will still be asking how a difference in the length of world lines can result in twins aging at different rates.

And the answer is: because the length of the worldline is the "age".

Note, btw, that your description, "aging at different rates", itself obscures the key point. Everything "ages" along its own worldline at the same "rate"--one second per second. The difference in lengths of worldlines, which is the difference in ages, is due to spacetime geometry, not "different rates" of anything. In other words, the term "different rates" implies that there is some absolute standard according to which "rates" are measured without regard to the lengths of worldlines, and there isn't.

 

[robphy]

In my opinion, there is a lot of textbook inertia because intro texts always seem to follow the often told story of the ether, Michelson-Morley,... and Einstein's treatment of Special Relativity (which could be called a physicist's approach). Often, the notions of spacetime, light cones, and worldlines (all due to Minkowski) are rarely developed... It seems spacetime is too mathematical.

In my opinion, spacetime and its geometry is really the only way to understand relativity.

As pervect said, I think Bondi's approach is a better way to start. The radar method part is essential to give meaning to what we mean by time and space components. The k-calculus part is efficient (since one is secretly working in the eigenbasis) and is physical (Doppler factor)... But might be argued to be too advanced.

 

[Smattering]

"Differential aging".

O.k., then let's call it differential aging. Still, the concept will appear magical to most people who are not used to it.

And the answer is: because the length of the worldline is the "age".

Hm ... then I really need to look up how the length of a world line is defined, because I would have expected that the younger twin is the one with the longer world line and the older twin is the one with the shorter world line.

In other words, the term "different rates" implies that there is some absolute standard according to which "rates" are measured without regard to the lengths of worldlines, and there isn't.

I see what you mean, but I am not sure whether I can agree here. After all, there is no observer who sees them aging at the same average rate when averaging over the entire journey, is there? So although there is no absolute standard to which rates are measured, there is indeed an intersubjective agreement that the twins are aging at a different average rate between departure and reunification.

 

[Orodruin]

Hm ... then I really need to look up how the length of a world line is defined, because I would have expected that the younger twin is the one with the longer world line and the older twin is the one with the shorter world line.

The concept of "length" in space time does not follow the Pythagorean theorem you are used to but instead the Pythagorean theorem comes with a minus sign on the space component. This changes the behaviour of the geometry.

 

[PeterDonis]

let's call it differential aging. Still, the concept will appear magical to most people who are not used to it.

Sure, that's unavoidable. The only response is to show them the experimental evidence that says it's true, and then give them the theoretical model (SR and Minkowski spacetime) that correctly predicts the experimental results.

I really need to look up how the length of a world line is defined

It's defined as the integral of the line element $ds^2$ along the worldline. This isn't just true in spacetime; it's true in any Riemannian or pseudo-Riemannian manifold.

I would have expected that the younger twin is the one with the longer world line and the older twin is the one with the shorter world line.

I assume that's because you are looking at the worldlines as they are drawn on a spacetime diagram, and seeing that, in the Euclidean geometry of the diagram, the younger twin's worldline is longer. But the actual geometry of spacetime is not Euclidean; it's Minkowskian. So the spacetime diagram, which can only be drawn in a medium with Euclidean geometry, cannot possibly represent all worldlines in Minkowskian spacetime with their actual spacetime lengths. (This should be obvious if you consider the worldline of a light ray: it's a 45-degree line on a spacetime diagram, with some nonzero Euclidean length, but its actual Minkowskian length is zero.) That's why you need to compute the length of the worldline using the formula I gave above; you can't just eyeball it from the diagram.

there is no observer who sees them aging at the same average rate when averaging over the entire journey, is there?

No, of course not. But none of these observers can claim that their observations of the rates are the "right" ones, the ones we should use as an absolute standard. There is no absolute standard for observing "rates of aging". The only absolute is the actual differential aging that is observed when the twins meet up again, and that is not a rate, it's just two different ages.

although there is no absolute standard to which rates are measured, there is indeed an intersubjective agreement that the twins are aging at a different average rate between departure and reunification.

The "average rate" is not a direct observable; it's something that's calculated. The direct observable is the different ages of the twins when they meet up again. The "average rate" is really just a different way of representing this same observable.

Furthermore, "average rate" is not an absolute way of representing that observable, unlike the observable itself, the difference in ages. Why? Consider: what do you divide the different ages by in order to get the average rate? There is no answer; there is no absolute number that represents the "reference" amount of time elapsed between the two events (the twins separating and meeting up again). So there's no absolute number you can divide the different ages by to get different average rates. The different ages themselves are the only absolute numbers in the scenario.

 

[Vitro]

I assume that's because you are looking at the worldlines as they are drawn on a spacetime diagram, and seeing that, in the Euclidean geometry of the diagram, the younger twin's worldline is longer. But the actual geometry of spacetime is not Euclidean; it's Minkowskian. So the spacetime diagram, which can only be drawn in a medium with Euclidean geometry, cannot possibly represent all worldlines in Minkowskian spacetime with their actual spacetime lengths. (This should be obvious if you consider the worldline of a light ray: it's a 45-degree line on a spacetime diagram, with some nonzero Euclidean length, but its actual Minkowskian length is zero.) That's why you need to compute the length of the worldline using the formula I gave above; you can't just eyeball it from the diagram.

+1 on this, I was about to say the same thing. This is another common misunderstanding, confusing spacetime interval with the length of the line segment on a spacetime diagram.

 

[PWiz]

Hm ... then I really need to look up how the length of a world line is defined, because I would have expected that the younger twin is the one with the longer world line and the older twin is the one with the shorter world line.

It is the way you think - the accelerated twin has a "longer" worldline than the unaccelerated twin. Generalizing Newton's first law, we can say that the unaccelerated twin moves on a geodesic by taking the "shortest" route between two events (the event where the twins separate and the event where they meet again). But the "length" of this route is defined in a way different from your intuition: $ds^2 = dx^i dx^j - dt^2$ ($dx^i dx^j$ is the ordinary spatial distance between the two events). The minus sign in the equation is what is responsible for throwing you off; however, it is easy enough to see that if $ds^2$ is to be minimum (generalization of Newton's first law), the spatial distance must be minimum ( as expected) but $dt^2$ must be maximum. Any acceleration is going to result in $ds^2$ becoming greater than its minimum value, and $dt^2$ becoming less than its maximum value. This is why we say that the accelerated twin ages less. (Side note - the negative sign in the metric equation makes spacetime a pseudo Riemannian manifold, and it is the primary reason your intuition [which thinks in terms of Riemannian manifolds] fails.)