B Newtonian vs Relativistic Mechanics

[DrStupid]

Unmodified Newtonian force can easily accelerate to v>c, so you have to make modifications to prevent that. You can hide the modifications inside F or p, but either way you have to modify Newton's laws.

No, that doesn't require a modification of Newton's laws of motion. Acceleration above c is already prevented by Lorentz transformation.

 

[David Lewis]

You can talk about the speed of light in two senses - either as "the invariant speed (at which light happens to travel)" or as "the speed at which light travels...

You've nicely highlighted a crucial distinction here, which may apply to other branches of physics.
In the first sense, c is a property of space. If the universe contained no light, c would still be the same.

 

[Ibix]

Yes. I like to think about this stuff as in effective field theories. In the standard model we also probably (:P ) neglect physics beyond a certain energy scale, say particles much heavier than a certain parameter E. Renormalization is then just sending E to infinity and decoupling physics beyond E from physics up to E, i.e. we pretend these heavier particles we don't know yet can not contribute to our path integral. I regard the c --> oo limit of special relativistic field theories likewise; high energy ('special relativistic') processes like particle creation, propagation of EM-waves and time dilation are decoupled and thrown away.

But I admit one has to be careful in distinguishing between c as some sort of parameter which can be contracted (in the underlying Lie algebra or eqn's of motion) and the measured value in experiments. So the phrase "c is infinite in Galilean/Newtonian relativity is mend (by me) as a limiting procedure on the parameter c, certainly not as a historical claim!

That's an interestingly different perspective from mine. I guess we really need to define what we mean by "Newtonian physics". Do we mean the historical situation where we genuinely believe Newtonian mechanics to be an accurate description of the world, and things like Maxwell's equations (and hence the speed of light) are puzzle pieces that don't quite fit? Or do we mean what we get if we start with a modern understanding and make formal approximations in the appropriate limits?

In the latter case, attempts to measure the speed of light are presumably (?) out-of-bounds because we've formally stated that $v<<c$ and we therefore can't play around near $c$. In the former case attempting to measure $c$ will eventually force us into a re-examination of the accuracy of Newtonian mechanics.

Anyway - I suspect this is a bit much for the OP, if he's still reading.

 

[Dale]

No, that doesn't require a modification of Newton's laws of motion. Acceleration above c is already prevented by Lorentz transformation.

The Lorentz transformation is not compatible with Newton's laws, that is precisely the point.

 

[Dale]

No. But we also measure gravitational waves, while in Newtonian gravity these don't exist. The difference between the measured value of c and the limit c --> oo are post-Newtonian effects. So neglecting these effects is similar to treating c as inifinite.

I'd say it is not confusing when one reason from the underlying symmetry groups.

It is confusing because you are using the phrase "the speed of light" to refer to "the invariant speed" rather than to "the speed of electromagnetic waves".

The terminology is standard, but it is inherently confusing, and you are making it more confusing by not acknowledging it but rather pretending like a novice should be aware of things like underlying symmetry groups. The poor OP asked for responses at a "B" level and not only are you not targeting your responses to such a level you are actively making it more difficult for those who are.

 

[PeterDonis]

Newtonian force (F=dp/dt) doesn't need to be modified to be compatible with relativity.

Yes, it does; as Dale pointed out in a previous post, it gets changed to $F = dP / d\tau$, where $F$ and $P$ are now 4-vectors and $\tau$ is proper time, not coordinate time.

 

[vanhees71]

The confusion comes from looking at the historical development rather than the structure of physical theories. Einstein's breakthrough in 1905 was not the mathematics of the Lorentz transformation, which is due to Voigt in the late 1880ies and FitzGerald, Poincare, and Lorentz in the early 1900s, but his discovery that the "speed of light" is a universal natural constant defining the mathematical structure of the description of space and time. This was finally understood completely in 1908 by Minkowski who formulated the space-time symmetry as a pseudo-Euclidean affine manifold.

From this perspective, it's more natural to introduce SRT by analyzing the symmetries of a space-time manifold, where there exist (global) inertial reference frames, such that for any inertial observer space and time are homogeneous and space is Euclidean. This analysis leads, up to equivalence, to two possible mathematical structures, namely Galilei-Newtonian spacetime with the (inhomogeneous) Galilei group as the symmetry group or Einstein-Minkowski spacetime with the Poincare group as the symmetry group. In the latter case a universal "limiting speed" is introduced, which governs all physical laws since it's part of the underlying spacetime structure.

Whether or not the speed of light (i.e., the phase velocity of electromagnetic waves in vacuo) is equal to this limiting speed then is an empirical question. In a modern way one can formulate it as the question, whether the electromagnetic field is really exactly massless. Today the upper limit of the photon mass is $m_{\gamma}<10^{-18} \mathrm{eV}/c^2$. In the Standard Model of elementary particle physics we take the photon as exactly massless.

 

[Ilja]

In fact, Poincare was quite close in the spirit to the Minkowski interpretation even years before 1905 (but, sorry, don't ask me to provide quotes, this is out of memory of my youth). So one can find quotes of Poincare such what it is hard to find a difference to the spacetime interpretation. The guy who believed that only one of the many mathematically equivalent frames is the true rest frame of the ether was Lorentz, so it is quite correct to name this interpretation the Lorentz ether.

On the other hand, one can also make a case that Poincare has not transcended the limits of classical spacetime philosophy. I would summarize this as that he was thinking in the direction of spacetime interpretation but had enough philosophical background not to take these ideas too seriously. It is, essentially, one thing to recognize that I cannot decide, by observation, which of two theories (say about which frame is the true rest frame) are true, and the idea that it follows that there is no such truth, that above theories are equally valid.

 

[Grimble]

Thank you Gentlemen! Yes I am still reading.
I am limiting myself to Special Relativity as there seems little point in studying and trying to understand GR until I understand SR.

I like Occam's approach and like things to fit without assumptions, or guesses.

Surely the difference between Newton and Relativity is not about whether Newton thought the speed of light was infinite - as has been pointed out, he didn't, but rather, that it was not a general limit to the speed of everything.

Is the rotation of a moving Frame of Reference not the rotation of the time axis due to the movement of the traveling frame?

I have come to think the invariance of the speed of light to be easily explained:
light travels between two events at c; the speed of light in gravity free space.
Those two events exist in every Frame of Reference; only their coordinates differ.
Every observer is at rest in their own local Frame of Reference, therefore the locations of those two events are fixed for each and every Frame...
Therefore the locations of the two events are fixed in every frame - locations of events cannot move...
Therefore light will travel between the same two locations in the same time - proper time (as that is the time measured locally at the origin of any inertial FoR.
Therefore light, must always travel at the same speed - relative to Spacetime - in every FoR; which is how the speed of light is measured - it would be pointless to try and measure the speed of light relative to anybody in space, wouldn't it?

That just seems to me to be the simple straightforward way to view it...

Also I understood that at least much of Newtonian mechanics (apart from the speed of light) was true as a limiting case that it only differs as speed increases?

 

[PeroK]

I have come to think the invariance of the speed of light to be easily explained:
light travels between two events at c; the speed of light in gravity free space.
Those two events exist in every Frame of Reference; only their coordinates differ.
Every observer is at rest in their own local Frame of Reference, therefore the locations of those two events are fixed for each and every Frame...
Therefore the locations of the two events are fixed in every frame - locations of events cannot move...
Therefore light will travel between the same two locations in the same time - proper time (as that is the time measured locally at the origin of any inertial FoR.
Therefore light, must always travel at the same speed - relative to Spacetime - in every FoR; which is how the speed of light is measured - it would be pointless to try and measure the speed of light relative to anybody in space, wouldn't it?

That just seems to me to be the simple straightforward way to view it...

So many words, so little maths!

What you have written may or may not makes sense. I suspect you've used words to hide the lack of mathematical consistency in your argument. You're obviously clever, but you need to hone your methods and thought processes.

 

[Dale]

Surely the difference between Newton and Relativity is not about whether Newton thought the speed of light was infinite - as has been pointed out, he didn't, but rather, that it was not a general limit to the speed of everything.

Yes, that is what we were describing above with the discussion of the invariant speed.

proper time (as that is the time measured locally at the origin of any inertial FoR.

Proper time is defined along a particular worldline and is the time on a clock moving on that worldline. Reference frames and origins aren't part of it.

 

[Grimble]

Proper time is defined along a particular worldline and is the time on a clock moving on that worldline. Reference frames and origins aren't part of it.

Yes, I understand that.
But can you explain to me how the time axis for a Frame of Reference is not the World-line of the substantial point (to quote Minkowski) that is at the origin of the Frame of Reference?
A Frame of Reference is no more than a map of Spacetime based upon that origin. It is, according to that map, that Frame of Reference, at rest and the time on a clock residing at the origin of the Frame of Reference is one moving along that worldline, which must consist of a single location in that substantial point's (and that imaginary Clock's) Frame of Reference the only coordinate changing through the life of that point or clock being the time coordinate - in that Frame of Reference.

Given that reasoning, how can the origin of a Frame of Reference not be the Worldline of the origin of the Frame of Reference and how could that clock keep any time other than Proper Time? - (It is after all "doing exactly what it says on the Tin" )

Or is there some specific Frame of Reference that a worldline has to be plotted on? But then there is no preferred Frame of Reference; is there? So that would only leave the concept of some sort of divine view of spacetime...

Let me assure you, Dale, for I have great respect for you, I am not trying to rewrite relativity, or how it is understood, but only pulling bits together and reasoning (to try and understand better) how they must work. (Isn't that what scientists are supposed to do?)

 

[Ibix]

I think you are trying to understand this backwards, @Grimble. You aren't wrong, but I think you are heading in a confusing direction.

All clocks show proper time. "Proper" in this context is coming from the same root as "property", something which is the clock's own. In the twin paradox, for example, there are two clocks. One shows ten years elapsed, the other shows twenty. That means that along one path the proper time was ten years and along the other it was twenty years.

You are correct that the proper time shown by a clock at rest in some reference frame is the same as the coordinate time in that frame. But saying a clock shows proper time is a tautology.

You compared a frame of reference to a map. This is right - and coordinate time and the spatial coordinates are like grid references on that map. Proper time is the distance along some arbitrary (possibly non-straight) line on the map (strictly I should say a space-like line). If that line is a straight line representing an object at rest in some frame then the proper time along the worldline is the same as the coordinate time at that location, true. But that's because we defined coordinate time as the proper time of a set of clocks at relative rest.

Hope that makes sense.

 

[PeterDonis]

can you explain to me how the time axis for a Frame of Reference is not the World-line of the substantial point (to quote Minkowski) that is at the origin of the Frame of Reference?

It can be, but it doesn't have to be, any more than the Earth's equator or prime meridian has to have actual markers all along it.

 

[Grimble]

It can be, but it doesn't have to be, any more than the Earth's equator or prime meridian has to have actual markers all along it.

I am sorry but I don't see how it can not be the worldline of the origin of the frame of Reference - In that map of spacetime it is at rest and therefore only the time will change - isn't the worldline just the path that the subject takes? So i a Frame of Reference how can the origin be any other than the worldline and the clock be any other than proper time?

 

[Dale]

But can you explain to me how the time axis for a Frame of Reference is not the World-line of the substantial point (to quote Minkowski) that is at the origin of the Frame of Reference?

A frame of reference is a mathematical object, not a physical object. In GR it is formalized as something called a tetrad, in SR it is less formally used to refer to a coordinate system. Either way it is a mathematical object, not a physical object. A reference frame is part of the analysis, not part of the experiment.

You can use a frame of reference regardless of whether or not there is any substance located at the origin. Many times it is convenient to use reference frames where no object is at rest (e.g. the center of momentum frame) so no object is located at the origin other than "in passing".

From a theoretical standpoint proper time is the spacetime interval along a given timelike worldline, which is experimentally measured by a clock traveling on that given worldline. Neither the theoretical nor the experimental meaning of proper time mentions a frame of reference nor even "inertial".

 

[the_emi_guy]

I am sorry but I don't see how it can not be the worldline of the origin of the frame of Reference - In that map of spacetime it is at rest and therefore only the time will change - isn't the worldline just the path that the subject takes? So i a Frame of Reference how can the origin be any other than the worldline and the clock be any other than proper time?

Grimble,
The fundamental difference between Newton and special relativity is quite easy for someone at your level to grasp. Some of the previous posts may be delving into the consequences of relativity which can get very complicated indeed. (Newtonian mechanics itself can get very complicated too...principal of least action, calculus of variations etc etc.).

The postulate of SR is that the speed of light is constant in all frames of reference. Let's explore in a very simple way what this means.

The only equation we need is d = (v)(t) (distance = velocity x time) and some high school geometry.

Consider that you are inside a train that is moving along at 1m/s. You are standing up holding a ball 1 meter above the floor. Next you move the ball toward the floor at a constant velocity of 1m/s. One second later the ball has reached the floor.

What do you observe? The ball moved vertically 1 meter and it took 1 second to do so. d = (v)(t) --> 1meter = (1m/s)(1s).

What does someone outside the train observe? During the one second that the ball was in motion, it moved 1 meter vertically and 1 meter horizontally due to the train's motion, total distance moved is 1.414 meters (Pythagoras). It also moved faster than 1m/s since it has both vertical and horizontal motion, velocity= 1.414m/s (by vector addition of horizontal and vertical velocities). d = (v)(t) --> 1.414m = (1.414m/s)(1s).

So these two observers disagree about how far the ball has moved, and how fast it moved. They agree on is how long it took the ball to get to the floor (1s).

The same would apply if the moving object were a wave. Back inside the train you pick up a 1m long vertical pipe and launch a sound wave which travels within the pipe, using the air in the pipe as its medium. Within the train, you will see a vertically moving sound wave moving 1 meter. The observer outside will see the soundwave moving faster since the medium (air in the pipe) has an additional horizontal component to its velocity.

Finally, consider a similar experiment with a flashlight held 1 meter above the floor of the train. You switch on a flashlight and observe the light traveling toward the floor. After 3.3nanoseconds the light will have moved down 1 meter to the floor. What does the outside observer see? The outside observer sees the lightwave travel a distance greater than 1 meter (the lightwave has a horizontal component to its motion due the motion of the train) and, according to Newton, the lightwave must have traveled faster in order to cover this longer distance during the same 1 second interval, just as was the case with the ball and soundwave.

However, according to Einstein, both observes must see the light traveling at the same velocity.

So, if the outside observer sees the lightwave travel a longer distance, but traveling at the same exact velocity, then according to d=(v)(t) the outside observer sees the lightwave take longer than 1s to get to the floor.

Thus the the two observers disagree about the how far the lightwave moved, and the amount of time it took to get to the floor, but agree on how fast it was traveling.

This is the "root cause" of all of the strange, counter-intuitive stuff that happens in SR: Two observers are moving relative to each disagreeing about the amount of time that elapses between events because they are forced to agree about the speed of the lightwave.

You can actually derive the Lorentz transformation directly using nothing more that the above thought experiment and the Pythagorean theorem
- try it, I think it would be a good exercise for you.

Cheers

 

[Grimble]

You can actually derive the Lorentz transformation directly using nothing more that the above thought experiment and the Pythagorean theorem
- try it, I think it would be a good exercise for you.

Thank you. I have done that and it does work very simply.
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Mechanics%20compared.png?attachauth=ANoY7cp6yoIyeGFRapn5Qs4B7duBpjmEDFM5fzI_DLZ-ERtGRvW9pDP-om9pcekWU2pMUTBqFBS3BFVouGwnsy3D3f81IZMR_1DRalDIy449NhNH3ihzUJuRU4fzd18RpBmoqtqKMfGEf0JsFgvrsck_dec7Hh75CWJPOY0Hep5hcTbPsHQdRsmcRC521qCuFgVDxXkT_YasVP2V6djmjaTXoSaQq2U0FcLhRND1T1coizy59L12HvZMAN2i-S1fOhIi4LyjlI8jAqsZqQkRm8MTI7z_2y9YuA%3D%3D&attredirects=0

In Newtonian Mechanics the path of the light in Clock B measured from Clock A lies at angle α and travels 1.166ls in 1 second.
In Relativistic Mechanics - the second diagram - the light in Clock A, measured from Clock B, as we know travels 1ls in 1 second; while also traveling 0.6ls horizontally from Clock B. Thus arriving at point (-0.6,0.8) after 1 second, giving us a different path angle β.

The important difference I see here is that in Newtonian Mechanics it is tan α = vt/ct but in Relativistic Mechanics it is sin β = vt/ct.

That is the effect of the invariance of c.

And as you say the second diagram gives us the time measured for the light to reach the mirror in a resting clock = 1 second, the time in a clock moving at 0.6c = 1.25 seconds so ct = √ ( (ct)'² - (vt')² )= ct'√ ( 1-v/c² )
where c = 1,
v = 0.6
this gives t = t'√ ( 1-v/c² ) or t'/γ

t = 0.8 t' or t' = 1.25 t which we see in diagram 2 and the Lorentz Factor for 0.6c is 1.25

 

[the_emi_guy]

Thank you. I have done that and it does work very simply.

Nice job, I'm impressed!
Nice diagrams too!

Next to consider is that even though the observer on the platform measured 1.25 seconds (his "coordinate time"), he is not oblivious to how long the the observer in the train measured. Using the time he measured (hypotenuse, 1.25sec) and the velocity of the train, he can work Pythagorean backwards and compute the time elapsed on train observer:
1.25² - 0.75² = (train observers time)² = 1²

We call this the "proper time" \tau
Δ\tau² = Δt² - (Δx/c)²

Proper time is a useful concept because it is "invariant", all observers will agree on the "proper time" between events irrespective of their relative motion.

Armed with this insight, I recommend the following YouTube video:



Cheers

 

[Ilja]

All clocks show proper time. "Proper" in this context is coming from the same root as "property", something which is the clock's own.

Yes, the original German term is "Eigenzeit". The word "eigen" translates also as "private", "own", "separate", "distinctive". I think every of these translations would have been more appropriate than "proper", which has also the association of true, correct, genuine, appropriate, adequate. Associations which are misleading, and not present in the German "eigen", which also has "peculiar" as a possible translation. http://www.dict.cc/?s=eigen

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

 

[Grimble]

A frame of reference is a mathematical object, not a physical object. In GR it is formalized as something called a tetrad, in SR it is less formally used to refer to a coordinate system. Either way it is a mathematical object, not a physical object. A reference frame is part of the analysis, not part of the experiment.

You can use a frame of reference regardless of whether or not there is any substance located at the origin. Many times it is convenient to use reference frames where no object is at rest (e.g. the center of momentum frame) so no object is located at the origin other than "in passing".

From a theoretical standpoint proper time is the spacetime interval along a given timelike worldline, which is experimentally measured by a clock traveling on that given worldline. Neither the theoretical nor the experimental meaning of proper time mentions a frame of reference nor even "inertial".

Yes, I understand the Frame of Reference is not a physical thing. It is essentially the map of Spacetime from the perspective of a real or virtual observer at the origin of the frame.

If one takes that map of Spacetime, for a virtual observer at the origin (or null point if you prefer) of the map, then that observer's worldline - plotted on the map depicted by that frame of reference - would be a straight vertical line as that observer is permanently at rest in that frame, because the null point is his position.

Surely if,

proper time is the spacetime interval along a given timelike worldline, which is experimentally measured by a clock traveling on that given worldline

then isn't that the definition of the time axis of the map of spacetime that is a frame of reference, whatever motion that virtual, real or imaginary object at the origin has relative to any other object, particle, substantial point or body in Spacetime? For the Time axis of a Frame of Reference IS the path of a virtual clock at rest at the origin of that Frame of Reference.

Please, I am not trying to redefine anything but seeing a relationship, something that fulfils the definition - I will try to understand if that is wrong, but there must be someway in which it doesn't work...

The most confusing thing for me - and, I can only guess, for others is using a term like Proper Time because it implies a fixed and rigid scale. That Proper Time implies a specific rate that is the same for every observer; that time passes at the same rate for any observer on a clock that is at rest relative to them. (And maybe only for those which are non accelerating, with zero gravity)

As I say I am not trying to redefine anything I am just commenting on what the term Proper Time seems to mean vide:

Yes, the original German term is "Eigenzeit". The word "eigen" translates also as "private", "own", "separate", "distinctive". I think every of these translations would have been more appropriate than "proper", which has also the association of true, correct, genuine, appropriate, adequate. Associations which are misleading, and not present in the German "eigen", which also has "peculiar" as a possible translation. http://www.dict.cc/?s=eigen

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

 

[Dale]

Please, I am not trying to redefine anything but seeing a relationship, something that fulfils the definition - I will try to understand if that is wrong, but there must be someway in which it doesn't work...

Ah, OK. I got the mistaken impression that you were trying to redefine things. It seemed that you were trying to define proper time in terms of an inertial frame, instead of the other way around. Sorry I misunderstood.

The time measured by *any* clock is the proper time along its worldline. So a clock measures it's proper time regardless of whether or not it is at rest at the origin.

There is indeed a relationship between coordinate time (in an inertial frame) and proper time. That is $d\tau^2=dt^2-dx^2-dy^2-dz^2$. For a clock at rest anywhere in the frame (not just the origin) we have $0=dx=dy=dz$ so $d\tau=dt$.

 

[PeroK]

Yes, the original German term is "Eigenzeit". The word "eigen" translates also as "private", "own", "separate", "distinctive". I think every of these translations would have been more appropriate than "proper", which has also the association of true, correct, genuine, appropriate, adequate. Associations which are misleading, and not present in the German "eigen", which also has "peculiar" as a possible translation. http://www.dict.cc/?s=eigen

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

As an English speaker, "proper time" sounds right to me. "Private time", for example, would be time a particle spends alone, without any interference from observers! And "peculiar time" or "distinctive time" don't sound right at all. Possibly, "intrinsic" or "natural" time would be alternatives. But, really, if someone is blaming the term "proper" for a lack of understanding of SR, they are really clutching at straws! (I'm not sure how you'd say that in German!)

"Too many words, not enough maths" is the problem here, IMHO.

 

[Ilja]

But, really, if someone is blaming the term "proper" for a lack of understanding of SR, they are really clutching at straws! (I'm not sure how you'd say that in German!)

My point is not about blaming something, but about improving the understanding some common misunderstandings. If you don't understand what someone is misunderstanding, you cannot help him to reach a better understanding.

I prefer to name proper time "clock time". This avoids the misunderstanding inherent in proper time as well.

 

[stevendaryl]

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

I'm not sure if that gets at the reason people are confused by the twin paradox. To me, the misconception is that (apparently) each twin can view the other one as aging faster, which seems like a logical contradiction. I don't think that the phrase "proper time" is relevant in explaining why people find it confusing---the first introduction to the twin paradox usually doesn't even mention proper time.

 

[robphy]

I like Taylor & Wheeler's "wristwatch time".

 

[vanhees71]

Well, I also never understood why somebody has a problem with the twin paradox. At least after you have understood the necessity for Minkowski spacetime instead of Galilean spacetime (in SRT), it should be very clear that the difference in proper times of particles following different world lines is a logical consequence. It's not more mysterious than the fact that traveling between two places along different routes means to travel a different distance. There's a "proper" distance defined as the length of the shortest geodesic connecting the two points. The same you have here for "distances" in the sense of the Minkowski pseudo-metric.

 

[pervect]

Well, I also never understood why somebody has a problem with the twin paradox. At least after you have understood the necessity for Minkowski spacetime instead of Galilean spacetime (in SRT), it should be very clear that the difference in proper times of particles following different world lines is a logical consequence. It's not more mysterious than the fact that traveling between two places along different routes means to travel a different distance. There's a "proper" distance defined as the length of the shortest geodesic connecting the two points. The same you have here for "distances" in the sense of the Minkowski pseudo-metric.

My belief is that the main issue people have with understanding the twin paradox is that they don't understand that simultaneity is relative, and that they also don't understand what one mean if/when one says that simultaneity is relative, nor do they understand what one means if/when one says that "time is not absolute".

This may not be the only issue. Sheer, Shaefer and Vokos talk about the issue of understanding the relativity of simultaneity in their paper "The challenge of changing deeply-held student beliefs about the relativity of simultaneity", https://arxiv.org/abs/physics/0207081. While the title of the paper focuses on the issue that I've mentioned, reading the paper shows other gaps in student understanding that make it difficult for them to reach a proper understanding. So one may need to fill in those OTHER gaps in student understanding , before one can effectively address the issue of understanding the relativity of simultaneity, after which point one is finally ready to talk about the twin paradox.

Some other issues the authors have identified are very simple ones related to students not understanding aspects of non-relativistic physics, rather basic issues such as how reference frames work, how to properly account for propagation delays of signals that move at a finite speed, and even the idea that the order of two events that happen to a pointlike observer does not depend on the reference frame one chooses.

Given the lack of understanding of these points about non-relativistic physics, students have a difficult time dealing with the twin paradox. They mainly rely on their intuition, which does not work at all for relativity, and they have difficulty following the formal steps needed to work the problem in order to change their incorrect intuitions. I'm not aware of any good solution to the problem - I think Scherr's paper gives some good advice based on practice and observation in the context of a classroom, but the methods that work in a classroom do not necessarily work in a forum such as PF. A rather general remark by the autors about student errors is "However, in many cases, conceptual difficulties seemed to prevent students from answering correctly". It may be the case that untangling these various conceptual errors may simply require more effort and thought and study than most casual readers are able to give. It also suggests that the solution may have to be done one student at a time, i.e. individual attention to the student is required.

 

[robphy]

Given the lack of understanding of these points about non-relativistic physics, students have a difficult time dealing with the twin paradox. They mainly rely on their intuition, which does not work at all for relativity, and they have difficulty following the formal steps needed to work the problem in order to change their incorrect intuitions. I'm not aware of any good solution to the problem - I think Scherr's paper gives some good advice based on practice and observation in the context of a classroom, but the methods that work in a classroom do not necessarily work in a forum such as PF. A rather general remark by the autors about student errors is "However, in many cases, conceptual difficulties seemed to prevent students from answering correctly". It may be the case that untangling these various conceptual errors may simply require more effort and thought and study than most casual readers are able to give. It also suggests that the solution may have to be done one student at a time, i.e. individual attention to the student is required.

One solution is to provide the student with the right tool: the spacetime diagram.
In my opinion, "a spacetime diagram is worth a thousand words".
To me, an obvious follow-up study is to see
if students reason better with "worldlines on spacetime diagrams" (presuming they have been appropriately developed)
rather than "diagrams of space and spatial trajectories in one frame of reference" (which are featured prominently in their study).
Most introductory textbooks seem to avoid the spacetime diagram... or else merely mention it in passing
(maybe because it seems that Einstein did not reason with them...
in fact, he initially hated the idea of spacetime as something the superfluous that the mathematicians dreamt up...
until, of course, he realized that he needed it for general relativity).

 

[vanhees71]

Thanks, pervect. This comes very timely, because right now the semester started, and I teach a lecture called "Mathematical additions to theoretical physics 2", which was introduced some years ago to help the students with the mathematics needed for the theory lecture, which start at our university already in the very first semester, which is challenging for both teachers and students, because the German high-school education in mathematics is a desaster. What's called math at high school is not what you understand under that name as a physicist (let alone a mathematician). Now in Theoretical Physics 2 the professor starts with special relativity. So I'll give in my math class an introduction to Minkowski space. So perhaps this physics-didactic papers give me some ideas how to make the business easier to understand ;-)).

@robphy: Yes, that's what I thought. So yesterday I started right away with Minkowski space and draw the usual space-time "plane". I think here the greatest difficulty for the students is that the axes of (at least) one of the inertial observers is not "orthogonal in the Eucildean sense" and to forget about the Euclidean structure of the plane you usually associate with the plane you draw the space-time diagram on. You have to substitute it in your thinking by the "Minkowski geometry", which is pseudo-Euclidean rather than Euclidean. I hope this becomes clear by drawing the time- and spacelike hyperbolae defining the "unit mesh" on this Minkowski plane. On the other hand, the Lorentz transformation in this plane (a Lorentz boost of course), becomes very intuitive from the 2nd Einstein postulate (constancy of the speed of light for all inertial observers): You construct the world line of Bob in Alice's referene frame, which must be in the forward light cone of the origin (velocity of Bob relative to Alice less than the speed of light). Then the Bob's spatial vector must point such that the world line of the light front is the bisecting between Bob's time-like and the space-like angle. Then you just need to normalize these two vectors in the sense of the Minkowski pseudo-metric.