Teaching SR without simultaneity

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Summary:
De-emphasizing simultaneity in SR curriculum. Thoughts? Experiences?
This thread, some earlier ones, and my signature got me thinking a bit about whether it would be a good idea to de-emphasize simultaneity when teaching SR since it is a frame dependent concept. I did not do much thinking myself yet but would like to hear people’s opinions and thoughts on how possible it would be to do and if anyone has tried it. I believe Lorentz transformations would still be of high relevance, but was thinking of removing focus from the meaning of events being simultaneous, time dilation, and length contraction and instead focus on spacelike separation between events, differential ageing and clocks measuring worldlines, and the geometry of spacetime. The other concepts would probably still need to be covered, but in a less prominent fashion.

Any thoughts? Did anyone try something similar?
 
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Dale
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I have not tried it before, but I also thought about it. The relativity of simultaneity is the most difficult concept for students to learn, and the source of most so-called paradoxes. So on one hand it would be easier to teach SR without it. But on the other hand the students would probably retain the concept of absolute simultaneity and would not know how to resolve the standard paradoxes.
 
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Orodruin
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But on the other hand the students would probably retain the concept of absolute simultaneity and would not know how to resolve the standard paradoxes.
Sure, this is why I think it cannot be skipped completely, but rather would like to de-emphasize it and relegate it to perhaps a single lecture after covering the other concepts.
 
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Dale
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Sure, this is why I think it cannot be skipped completely, but rather would like to de-emphasize it and relegate it to perhaps a single lecture after covering the other concepts.
That sounds like an interesting compromise. If you teach it in a predominantly spacetime/covariant approach then it may not be such a stretch to convince people that simultaneity is relative since it is just a coordinate line.
 
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IMO, it does much more than explaining paradoxes. It is the fundamental problem with frame-dependent measurements of distance and elapsed time. For beginners, it seems to be the most often overlooked aspect of relativity and, if anything, needs to be emphasized more.
 
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IMO, it does much more than explaining paradoxes.
If you do not start by introducing and discussing the simultaneity idea there are not many paradoxes to explain.
It is the fundamental problem with frame-dependent measurements of distance and elapsed time.
As above, those frame dependent measurements would not be there to behin with because they are, well, frame dependent. The discourse would be more focused on the geometry of spacetime and its geometrical structure.

For beginners, it seems to be the most often overlooked aspect of relativity and, if anything, needs to be emphasized more.
We draw different conclusions then. I would say it needs to be less emphasized precisely because it is prone to misunderstandings rather than being in any way fundamental.
 
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vanhees71
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Summary:: De-emphasizing simultaneity in SR curriculum. Thoughts? Experiences?

This thread, some earlier ones, and my signature got me thinking a bit about whether it would be a good idea to de-emphasize simultaneity when teaching SR since it is a frame dependent concept. I did not do much thinking myself yet but would like to hear people’s opinions and thoughts on how possible it would be to do and if anyone has tried it. I believe Lorentz transformations would still be of high relevance, but was thinking of removing focus from the meaning of events being simultaneous, time dilation, and length contraction and instead focus on spacelike separation between events, differential ageing and clocks measuring worldlines, and the geometry of spacetime. The other concepts would probably still need to be covered, but in a less prominent fashion.

Any thoughts? Did anyone try something similar?
I think the problem is that to understand relativity you need to discuss the transformations between inertial reference frames, and then all these "kinematical effects", which are in principle not very interesting from a physical point of view, automatically become an issue. I thought a lot about to avoid all these socalled paradoxes as much as you can, but then came to the conclusion that this would imply to just introduce the Minkowski spacetime manifold as a postulate, but I found this not very attractive, i.e., you'd loose all the thinking in terms of symmetries, which has been explicitly introduced into physics by Einstein in his famous paper of 1905.

That's not so much the case with the standard way we teach and learn about Newtonian mechanics. There's no problem to completely ignore Galilei transformations without loosing much content, although of course to miss Noether's theorem is a missed opportunity to convey one of the most aesthetical aspects of physics.
 
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vanhees71
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That sounds like an interesting compromise. If you teach it in a predominantly spacetime/covariant approach then it may not be such a stretch to convince people that simultaneity is relative since it is just a coordinate line.
That's what I also thought, but then I had no idea, how to convincingly introduce Minkowski spacetime. So I start with the special principle of relativity (existence of inertial frames as in Newton's Lex I) and the independence of the speed of light from the velocity of the light source, as in Einstein's paper.

There are two approaches from these postulates: One is to just use the gedanken experiment with a short flash of light from a point-like light source leading to the invariance of the equation ##(c t)^2-\vec{x}^2=0## in all IFs, i.e., in another IF it looks the same ##(c t')^2-\vec{x}^{\prime 2}=0##. From this you get to the Lorentz transformations for general spacetime four-vectors, ##(t,\vec{x})## very easily, and almost immediately you only work in a manifestly covariant way. Then the apparent paradoxes are no paradoxes anymore to begin with, because everything is covariant (or even invariant, if you work in a basis-independent formalism). This is, however also somewhat abstract for a first encounter with SR.

The other approach is the one by Einstein in his original paper of 1905, i.e., using light signals to synchronize clocks and then derive also the measurement of spatial distances. Then of course you emphasize more the (1+3)-dimensional formalism and from the very beginning the relativity of simultaneity.
 
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robphy
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In my opinion, the effectiveness or ineffectiveness of the
highlighted use of various concepts (like "simultaneity" or "time-dilation")
depends on the primary mode of reasoning,
which can be crudely classified as
  • verbal, with necessarily technical-definitions for previously-everyday words
    in very-carefully crafted sentences
  • formulaic and symbolic (what's primed and unprimed?)
  • spatial-geometric (frame-dependent, using moving boxcars with clocks)
  • spacetime-geometric (frame-independent with a spacetime viewpoint,
    possibly relying on some analogs of one's Euclidean-intuition)
Ideally, one would like to easily translate among these modes.

In some modes, various concepts are easier to recognize.

For beginners, I think the Bondi k-calculus (which is lightly algebra-based and is operational [using radar-methods]) is the best approach. The spacetime view is developed (secretly) with the Doppler effect [that's why it's called "k"-calculus] (and, even more secretly, the eigenbasis of the Lorentz transformation). Traditional formulas (Lorentz transformations and velocity-composition) come later.

Bondi k-calculus
https://en.wikipedia.org/wiki/Bondi_k-calculus
https://archive.org/details/relativitycommon0000bond
and, yes,
there's an Insight for that: https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/ :wink:
 
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  • #10
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If you do not start by introducing and discussing the simultaneity idea there are not many paradoxes to explain.
I think that people will still come up with the same confusing scenarios, but will be less prepared to see the solution in any convincing way.
As above, those frame dependent measurements would not be there to behin with because they are, well, frame dependent. The discourse would be more focused on the geometry of spacetime and its geometrical structure.
That may be true as long as a short class lasts, but eventually, people will try to understand SR in familiar reference frames and will be just as confused and less prepared.
 
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I need to moderate my statements. I think that it is essential to point out how a person's familiar concepts of distance and elapsed time depend on simultaneity. They just need to be convinced that their long-held reference-frame concepts have problems in SR. That should allow them to more easily accept the geometry of SR. IMO, it is difficult to completely accept the new geometry until one is convinced that the old one will not work. Beyond that, I do not think that the "relativity of simultaneity" needs to be emphasized.
 
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Ibix
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I always say this, but Minkowski diagrams were revolutionary for me. Being able to see the transforms acting closely analogous to Euclidean rotations, and to see that the time axis and simultaneity plane are as arbitrary as the x and y axis, yet that there's a reason to pick my own rest frame for my own work, was what made the whole thing click. Then you can state and resolve things like the twin paradox as analogous to travel on a Euclidean plane, and see the change of simultaneity just like the change of definition of "normal to my path" when I turn a corner.

I think you probably do need something like radar coordinates (aka k-calculus) to derive the Lorentz transforms. Drawing light cones on a Minkowski plane is straightforward, and leads where you want to go. Actually, I think part of the problem is how conceptually trivial all this is - all of the SR "weirdness" is just a minor variation on rotation matrices.

I would definitely want to link this to electromagnetism in terms of the history of it, but I'd be almost inclined to present Maxwell -> Einstein -> Minkowski as a short historical introduction, then draw a Minkowski diagram, derive the boosted axes using radar, derive the transforms from geometry, and then maybe look at resolving some "paradoxes" (because they will come up, like it or not).
 
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Not entirely relevant, but David Tong wrote this comment in his electromagnetism lectures.
Remember all those wonderful things that you first learned about in special relativity: time dilation and length contraction and twins and spaceships so on. You’ll never have to worry about those again. From now on, you can guarantee that you’re working with a theory consistent with relativity by ensuring two simple things
  • That you only deal with tensors.
  • That the indices match up on both sides of the equation
It’s sad, but true. It’s all part of growing up and not having fun anymore.
 
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  • #14
vanhees71
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As I teach the high-school-teacher students, it's easy to discuss such didactical issues with them, and after giving the intro to SR for the 3rd time now, the way to construct Minkowski diagrams seems indeed to help them a lot. May description, after having derived the Lorentz transformations via Einstein's "light-clock method" (which one can use also in high-school lessons and which is also nicely described in a few good high-school textbooks, though it's hard to find the; the best one I know of is the newest edition of "Metzler Physik" (of course in German)) and having established the "Minkowski pseudo-metric" is as follows:

Draw the ##x^1##-axis to the right and the ##x^0=ct##-axis upwards (i.e., as a usual Cartesian coordinate system in the plane, but then forget about anything related to Euclidean geometry of the plane, and that's the most difficult point, at least for me). The "tic marks" in these axes are of course arbitrary, defining units of lengths and "times ##\times c##". Then draw the light cone and the world line of the origin of another inertial frame, which is the ##x^{\prime 0}=c t'## axis of the new frame. According to Einstein's 2nd postulate ("constancy of ##c##") the ##x^{\prime 1}## axis is then the ##x^{\prime 0}## axis such that the light cone is the bisecting line. To get the correct scales for the tic marks on the new axes draw the hyperbolae ##(x^0)^2-(x^1)^2=\pm n^2## with ##n \in \mathbb{N}##. The intersections with the constructed ##x^{\prime 0}## (hyperbolae with ##+n^2##) and ##x'## (hyperbolae with ##-n^2##) then give the correct tick marks for these new axes. This clearly demonstrates again the importance to forget about "Euclidean thinking" when looking at a Minkowski diagram.
 
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This is a good (non-mathematical) HS-level 'intro to the landscape' book:

1641667683883.png
 
  • #16
robphy
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... To get the correct scales for the tic marks on the new axes draw the hyperbolae ##(x^0)^2-(x^1)^2=\pm n^2## with ##n \in \mathbb{N}##. The intersections with the constructed ##x^{\prime 0}## (hyperbolae with ##+n^2##) and ##x'## (hyperbolae with ##-n^2##) then give the correct tick marks for these new axes. This clearly demonstrates again the importance to forget about "Euclidean thinking" when looking at a Minkowski diagram.
While what you say is true,
it might be more useful to say
slightly generalize your "Euclidean thinking".
The Minkowskian hyperbolas suggest that the notion of "circle" is based on a similar looking quadratic form.

The direction orthogonal to a timelike-vector
is defined by the tangent line where the vector meets the quadratic form.
(This is associated with the notion of "simultaneity".)
  • This was true for Euclidean geometry with its circle.
  • This is true for Minkowskian geometry with its [generalized] or [pseudo-]circle (the hyperbola).
  • This is true for Galilean geometry with its [generalized] or [pseudo-]circle (the "t=1" hyperplane, which is associated with the so-called "[degnerate] temporal metric of the Galilean geometry" ).
"Angles" can be defined using the areas of sectors, which could define arc-lengths along spacelike curves and define associated trig-functions. etc.. etc.. etc...

So, it might be useful to not-completely-forget Euclidean thinking....
...but, instead, think about how we start from a circle and get to (say) the rotation matrices to develop, by analogy, the Minkowskian and Galilean spacetime geometry and their associated constructions and formulas.
 
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Orodruin
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I am currently on my phone so unfortunately have not had the time to sit down to really go through the replies. So just let me make a couple of comments and remarks.

This is, however also somewhat abstract for a first encounter with SR.
The target level I had in mind was students in 4th year taking a specialized SR course. They have already been somewhat exposed to relativity in the typical modern physics course in year 2. Of course, the idea and its applicability to other levels is interesting as well.

While what you say is true,
it might be more useful to say
slightly generalize your "Euclidean thinking".
The Minkowskian hyperbolas suggest that the notion of "circle" is based on a similar looking quadratic form.

The direction orthogonal to a timelike-vector
is defined by the tangent line where the vector meets the quadratic form.
(This is associated with the notion of "simultaneity".)
  • This was true for Euclidean geometry with its circle.
  • This is true for Minkowskian geometry with its [generalized] or [pseudo-]circle (the hyperbola).
  • This is true for Galilean geometry with its [generalized] or [pseudo-]circle (the "t=1" hyperplane, which is associated with the so-called "[degnerate] temporal metric of the Galilean geometry" ).
"Angles" can be defined using the areas of sectors, which could define arc-lengths along spacelike curves and define associated trig-functions. etc.. etc.. etc...

So, it might be useful to not-completely-forget Euclidean thinking....
...but, instead, think about how we start from a circle and get to (say) the rotation matrices to develop, by analogy, the Minkowskian and Galilean spacetime geometry and their associated constructions and formulas.
This is very close to what I had in mind for discussing the geometry of the theory. Of course, one also needs to arrive to Minkowski space, which I usually do through considering the invariance of ##dx = \pm dt## for a light signal and through there arguing for the invariance of the line element ##ds^2##. From there geometry takes over.
 
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robphy
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This is very close to what I had in mind for discussing the geometry of the theory. Of course, one also needs to arrive to Minkowski space, which I usually do through considering the invariance of ##dx = \pm dt## for a light signal and through there arguing for the invariance of the line element ##ds^2##. From there geometry takes over.
Here's an old poster where some of these ideas are being developed
http://www.aapt.org/doorway/Posters/SalgadoPoster/SalgadoPoster.htm
from the poster section of https://www.aapt.org/doorway/TGRU/
(PDF version: https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-GRposter.pdf )
and slides from a talk
http://visualrelativity.com/papers/Salgado-CayleyKlein-slides-UWLmath20141031.pdf

I use something I call the "Trilogy of the Surveyors" (inspired by Taylor & Wheeler's "Parable of the Surveyors") to motivate the "circle"... the light cone in (1+1) form the asymptotes of the Minkowski circle.
 
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  • #19
PAllen
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I think the most important thing to get across is that simultaneity is not physics at all, but convention, and this is the radical departure from Newtonian physics. Instead of simultaneity, you have that events can be causally related or not, and nothing else matters. Relativity of simultaneity, as often taught, seems to lead to misleading notions there is physical meaning to planes of simultaneity, which must be unlearned for GR.

However, it is worth teaching that if you adopt any reasonable convention for synchronizing clocks, and apply it to two pairs of clocks in relative motion (each pair at mutual rest), then each pair will not be synchronized according to the other pair.
 
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  • #20
robphy
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I think the most important thing to get across is that simultaneity is not physics at all, but convention, and this is the radical departure from Newtonian physics. Instead of simultaneity, you have that events can be causally related or not, and nothing else matters. Relativity of simultaneity, as often taught seems to lead to misleading notions there is physical meaning to planes of simultaneity, which must be unlearned for GR.

However, it is worth teaching that if you adopt any reasonable convention for synchronizing clocks, and apply it to two pairs of clocks in relative motion (each pair at mutual rest), then each pair will not be synchronized according to the other pair.
Pedagogically...
(akin to my reluctance to banish "Euclidean thinking" so soon),
I'm not sure if I'd go that far with demoting simultaneity like that [so soon]
to those just trying to get a handle on relativity....
unless you are prepared to explain many things in terms of causal relations alone [since nothing else matters].

While distant simultaneity may be a convention,
the related notion of orthogonality [of 4-vectors] [in the tangent space] at an event is not a convention.

I think it is more important to first develop spacetime intuition
while connecting (via the Correspondence principle) to what we think we know with our Galilean intuition.
Work on gradually reshaping their Galilean intuition into special-relativistic intuition,
then possibly to general-relativistic intuition and beyond (quantum-gravity intuition?).

In my opinion, it's too drastic to discard various concepts so soon.
It's better to peel away layers of Galilean thinking and
use the remaining structure to lead them to special-relativity and byeond.

My $0.02.
 
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  • #21
vanhees71
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While what you say is true,
it might be more useful to say
slightly generalize your "Euclidean thinking".
The Minkowskian hyperbolas suggest that the notion of "circle" is based on a similar looking quadratic form.
It's not suggested by but following from the introduction of the indefinite fundamental form. As I said, the spacetime geometry is derived before the Minkowski diagram is introduced. This is much better to understand than if you try it the other way around, because then this subtlety with how to define the "tic marks" on the new axes is more cumbersome to derive, and it becomes less clear.
The direction orthogonal to a timelike-vector
is defined by the tangent line where the vector meets the quadratic form.
(This is associated with the notion of "simultaneity".)
  • This was true for Euclidean geometry with its circle.
  • This is true for Minkowskian geometry with its [generalized] or [pseudo-]circle (the hyperbola).
  • This is true for Galilean geometry with its [generalized] or [pseudo-]circle (the "t=1" hyperplane, which is associated with the so-called "[degnerate] temporal metric of the Galilean geometry" ).
"Angles" can be defined using the areas of sectors, which could define arc-lengths along spacelike curves and define associated trig-functions. etc.. etc.. etc...
There are of course analogies. You still have Euclidean geometry for the standard time slices for any inertial observer, and there you can use the Euclidean 3D geometry, which is of course important to operationally define the observables known from Newtonian physics. Also the rapidity is a very natural parameter to describe the hyperbolae, ##x^0= n \cosh \eta##, ##x^1=n \sinh \eta##. They are also the natural parameters for the Lorentz boosts. This is in very close analogy to defining circles in the Euclidean plane with an angle and rotations around an axis with a rotation angle.
So, it might be useful to not-completely-forget Euclidean thinking....
...but, instead, think about how we start from a circle and get to (say) the rotation matrices to develop, by analogy, the Minkowskian and Galilean spacetime geometry and their associated constructions and formulas.
You must forget Eulcidean thinking to correctly read a Minkowski diagram though! That's the greatest challenge the Minkowski diagram introduces.

I was surprised when in discussions with my students it came out that for the Minkowski diagrams are very helpful, even given the difficulty with the non-Euclidicity necessary for their interpretation. Before I avoided Minkowski diagrams at all costs and only introduced them very briefly with the excuse that they occur in many high-school textbooks, and sometimes in a very weird form, where they even work with Euclidean angles instead of the more appropriate notion of rapidity.

Somewhat artificially you can also use Loedel diagrams to stay within Euclidean geometry, but these only work if you deal with only two inertial reference frames. For me they are utmost confusing and never convinced me enough to try them out in teaching.

Of course, also your "rotated-graph-paper approach" is a viable alternative. The idea behind this is of course just to use light-cone coordinates and the fact that hyperbolae are described as ##l_1 l_2=\text{const}## with them. I've not tried yet this approach in the classroom either, because of lack of time, and this approach is not used in the high-school context at all.
 
  • #22
Ibix
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Relativity of simultaneity, as often taught, seems to lead to misleading notions there is physical meaning to planes of simultaneity, which must be unlearned for GR.
Certainly happened to me. I had real trouble initially with there not being a meaningful "time when something crosses the event horizon according to a distant observer" because I kind of believed that SR frame changes had to be physically meaningful because they had physical effects like length contraction (I am aware that's not accurate nor necessarily totally consistent, but it was a viewpoint I did hold). Where did all that meaningfulness go in the transition to GR? Once I got the 4d view of SR and the analogy to Euclidean space and could see what a frame change was then it was much easier to accept GR's near-total abandonment of simultaneity as a concept.
 
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  • #23
Dale
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I was surprised when in discussions with my students it came out that for the Minkowski diagrams are very helpful, even given the difficulty with the non-Euclidicity necessary for their interpretation.
Personally, it was when I finally discovered Minkowski diagrams and four-vectors that SR "clicked" in my mind.
 
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For me it was the opposite. I learnt about SR in highschool, where heavy use was made of Minkowski diagrams, and I never knew, how this issue with the unit tic marks on the axes really works. For me it made click, when I first learnt about the algebraic foundations in terms of the "Minkowski product" and got the idea with the hyperbolae ;-).
 
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robphy
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Personally, it was when I finally discovered Minkowski diagrams and four-vectors that SR "clicked" in my mind.
For me, I also needed
the dot-product (for efficient coordinate-free linear-algebra calculations)
and radar-methods (for operational definitions to define space and time components)
and how they are related.
 
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