Teaching special relativity

  • #1

Main Question or Discussion Point

Hundred years of special relativity have generated papers with titles like
"Derivation of the Lorentz transformations from the Maxwell equations"
"From m=mcc to the Lorentz transformations via the law of addition of relativistic velocities"
in order to quote only to of them.
Do you think that they have pedagogical power or they are only algebra game?
Are the relativistic postulates the most powerfull approach?
 

Answers and Replies

  • #2
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bernhard.rothenstein said:
Hundred years of special relativity have generated papers with titles like
"Derivation of the Lorentz transformations from the Maxwell equations"
"From m=mcc to the Lorentz transformations via the law of addition of relativistic velocities"
in order to quote only to of them.
Do you think that they have pedagogical power or they are only algebra game?
Are the relativistic postulates the most powerfull approach?
Most of these papers have very limited value, some of them (the ones published in Galilean Electrodynamics or Apeiron, both fringe journals) are downright wrong. In many instances, the authors use the fact that the result is allready known and they manipulate the data in such a way to prove their points. One author in particular, N.Hamdan has made a career in publishing such papers in both aforementioned journals.
 
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  • #3
Hurkyl
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If you want my opinion, the most important thing to understanding SR is understanding Minowski geometry. That means space-time diagrams.

Once I realized that space-time diagrams are a means to do geometry, SR instantly became clear to me. Of course, it took a while before I had realized this -- I had originally thought of them as being merely a plot of time vs space. It probably didn't help that I had to learn this on my own. (But then again, maybe it did)
 
  • #4
robphy
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In my opinion, some of these papers have pedagogical value.
Some assorted thoughts on this...

In my humble opinion, I am not convinced that Special Relativity is best learned by following Einstein's original presentation. In accord with Hurkyl's comment, I feel that the spacetime diagram and aspects of its Minkowskian geometry must be used. (I wonder why such tools are not in many introductory textbooks. My guess: early on, Einstein didn't use them and didn't like them.)

As I suggested in the https://www.physicsforums.com/showpost.php?p=924467&postcount=6", one could use the many symmetries of Minkowski spacetime to formulate Special Relativity with an alternate set of postulates. (Similarly, Euclid's postulates are not the only way... or arguably even the best way... to formulate Euclidean geoemtry.)

An alternate non-historical formulation (i.e., a carefully crafted fairy tale) may be helpful in developing for the student the important ideas in SR.... and possibly avoid the same stumbling blocks encountered by the original formulations and presentations. For example, from an earlier https://www.physicsforums.com/showthread.php?t=96931", I gave references to a paper that develops a Galilean-invariant electromagnetic theory and suggests that one may teach that first to motivate an appreciation of a "relativity principle". Then, further experimental results [taken out of historical order] could motivate a modified theory [i.e., Maxwell] and a modified relativity principle [Einstein's]. History could have turned out that way.

With the many features of Minkowski (possibly blurred and obscured by its many symmetries), one may want to know "Which structure is most primitive? or most fundamental?" in the grand scheme of things. Then, by weakening or removing the other structures, one can generalize to other situations. This was the game played to obtain GR... and it is being used to guide various approaches toward a quantum theory of gravity.

To me, the causal structure is the most fundamental. There have been various attempts to use the causal structure to formulate SR (i.e., construct or characterize Minkowski spacetime) or even more general spacetimes. (related refs: Robb, Alexandrov, Zeeman, Ehlers-Pirani-Schild, etc...)

My $0.05, a penny for each thought.

[EDIT: This old post is relevant to this thread:
https://www.physicsforums.com/showpost.php?p=694535&postcount=8 ]
 
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  • #5
in my humble oppinion the best way of teaching special relativity is to derive the Lorentz-Einstein transformations from the two postulates and without using supplementary assumptions. I consider very powerfull pedagogically the approaches of Peres and Kard. Each of them derives the fundamental equations of special relativity but fail to show that each of the particular results (time dilation, addition law of relativistic velocities could lead directly to the transformation equations.
 
  • #6
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Frankly, considering that SR is usually taught during the 4th or 5th semester to a Physics student, I think that the approach is practical. Few Junior level Physics students are going to be prepared to tackle non-Euclidean geometry using standard Mathematical (that is to say metric space) techniques.

-Dan
 
  • #7
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topsquark said:
Frankly, considering that SR is usually taught during the 4th or 5th semester to a Physics student, I think that the approach is practical. Few Junior level Physics students are going to be prepared to tackle non-Euclidean geometry using standard Mathematical (that is to say metric space) techniques.

-Dan
Berhard was speaking of special relativity. The only difference the student will encounter is a non-euclidean metric at best. Even that the student doesn't need to know.

Pete
 
  • #8
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Just my 2 cents;
I actually find the Minowski space-time geometry an unnecessary and often inappropriate addition to SR. Einstein didn’t like it in SR either, but then he was early to decide that he couldn’t get SR to produce a complete solution and that is where it was useful in considering an extra dimension for the GR solution.

SR is classical and Minowski tends to imply that it is not, which can be a problem sometimes. I’ve never seen a true SR problem that required a Space-time solution that couldn’t be resolved as well with a good reference frame comparison as in trains of different fixed speeds Einstein’s classic method.

I think Minowski should be saved for GR after a good understanding of SR.
 
  • #9
Hurkyl
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I’ve never seen a true SR problem that required a Space-time solution that couldn’t be resolved as well with a good reference frame comparison as in trains of different fixed speeds Einstein’s classic method.

...

I think Minowski should be saved for GR after a good understanding of SR.
Of course, it's also true that there isn't a Euclidean geometry problem that cannot be solved using coordinates.

But just because problems can be solved with coordinates does not mean that coordinates are a good way to understand the theory. In fact, it's widely believed that relying on coordinates as a crutch is a severe obstacle to understanding geometric concepts.

I see nothing about SR that suggests it is an exception. In fact, I can testify from personal experience that SR instantly made sense once I discovered Minowski geometry.
 
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  • #10
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topsquark said:
Frankly, considering that SR is usually taught during the 4th or 5th semester to a Physics student
lessee, i was an engineering student, and we got it in our 3rd semester of physics. we had classical mechanics, then classical E&M, then a semester of "modern physics" that introduced SR and QM and the H atom. oh, but we started the first physics in our second semester, so it was the 4th semester.

for me, the postulates that no inertial frame is qualitative different (or "better") than any other inertial frame of reference and that we can't tell the difference between a "stationary" vacuum and a vacuum "moving" past our faces at a high velocity, that there is no difference and that Maxwell's Equations should work the same for any and all inertial frames so then the speed of E&M must be measured to be the same in all inertial frames, even if it is the same beam of light viewed by two observers moving relative to each other. from that, we got time dilation, then length contraction, then Lorentz transformation, then mass dilation, then reletivistic kinetic energy and from that E=mc2.

it seems pedagogically straight forward to me.
 
  • #11
robphy
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RandallB said:
Just my 2 cents;
I actually find the Minowski space-time geometry an unnecessary and often inappropriate addition to SR. Einstein didn’t like it in SR either, but then he was early to decide that he couldn’t get SR to produce a complete solution and that is where it was useful in considering an extra dimension for the GR solution.
Unfortunately, this seems to be an attitude reflected in many introductory physics textbooks. Many seem to prefer a "functional approach" (i.e. formula-based algebra and analysis), in the style of the first SR papers by Einstein. Not a spacetime diagram is to be seen anywhere. By contrast, there are many "distance vs time" graphs in the first few Galilean/Newtonian kinematics chapters. It's not fully appreciated that those are [Galilean] spacetime diagrams... yes, with a time dimension attached. Of course, these graphs are interpreted as merely plots of functions with little awareness of the underlying [Galilean spacetime] geometry. It may be probably just a matter of time for the textbooks to catch up... but I'm not holding my breath.

In my opinion, if there is an underlying geometrical structure, it's best to bring some attention to it. (For example, I think vector calculus and drawings of vector fields (as opposed to merely a system of scalar PDEs) help us understand and interpret Maxwell's Equations.) It provides some structure as to what can make mathematical [and hopefully physical] sense.

Of course, there are many aspects to Minkowskian geometry (as there are in Euclidean geometry). Those aspects need not be introduced all at once. One could have spacetime diagrams (akin to "distance vs time" graphs and to constructions on the Euclidean plane) at one level. At another, one can introduce vectors, and later, the metric,... groups, tensors, lie algebras, ...as needed. Certainly, students of Euclidean geometry get things gradually... so can students of special relativity.

RandallB said:
I think Minowski should be saved for GR after a good understanding of SR.
In my opinion, that good understanding comes from appreciating at least the spacetime diagrams of Minkowski. "A spacetime diagram is worth a thousand words".
 
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  • #12
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robphy: Could you embellish upon your statement above? "To me, the causal structure is the most fundamental. There have been various attempts to use the causal structure to formulate SR (i.e., construct or characterize Minkowski spacetime)"

Thanks

Yogi
 
  • #13
robphy
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causal structure

yogi said:
robphy: Could you embellish upon your statement above? "To me, the causal structure is the most fundamental. There have been various attempts to use the causal structure to formulate SR (i.e., construct or characterize Minkowski spacetime)"

Thanks

Yogi
First, take a look at the attachment on this old post: https://www.physicsforums.com/showpost.php?p=694535&postcount=8, which diagrams some of the approaches to deriving the Lorentz Transformation. Causality is in the upper left corner.

Earlier in this thread, I mentioned that the numerous symmetries enjoyed by Minkowski spacetime allow many starting points. For various reasons one may prefer a certain core set of ideas from which the rest follow.

It can be argued that the "causal structure" is among the most fundamental ideas in relativity. The causal structure is the [point] set of events together with its causal ordering (a list of relations, indicating for each event P, a list of events inside P's future "light cone"... that is, the list of events that can be influenced by P). Mathematically, this ordering relation is called a http://www.google.com/search?q=partial+order", which is a very simple structure... more primitive [and more arguably physical and constructive from experimental measurements] than (say) a metric tensor with Lorentzian signature.

Soon after Einstein and Minkowski, Alfred Robb (some interesting facts I just googled about http://answers.google.com/answers/threadview?id=573896" using a betweenness relation and a set of postulates. [This evening, I happened to be reading Robb's book.]

Other results along these lines have been achieved by Alexandrov (see http://link.aip.org/link/?JMAPAQ/25/113/1" [Broken].

Part of the appeal of causal structure is the idea that the causal structure (i.e. the light cone structure) determines the metric at each event in a general spacetime up to a conformal ("scale") factor (see Hawking and Ellis, p 60-61). Said another way... since in 4 spacetime dimensions, the metric has 10 independent components per event, knowledge of the light cone yields 9/10 of what the metric gives. (Due to the symmetries of Minkowski, its causal structure determines everything except a single [constant] conformal factor.)

So, the question becomes... how little [and with what more "natural" and physically meaningful structures] does one need to recover the spacetime? Some work along these lines (continuing from Alexandrov, Zeeman, etc...) was done by http://link.aip.org/link/%3Fjmp/17/174" [Broken] .

Some more references on http://www.phy.olemiss.edu/~luca/Topics/c/causal_st.html" [Broken] (in addition to the references included within the above papers and links).
 
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  • #14
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Ref: Minowski space-time geometry - unnecessary to SR
robphy said:
Unfortunately, this seems to be an attitude reflected in many introductory physics textbooks. Not a spacetime diagram is to be seen anywhere.

In my opinion, that good understanding comes from appreciating at least the spacetime diagrams of Minkowski. "A spacetime diagram is worth a thousand words".
Where you want to declare that a bad thing, I consider it a good thing to help reach a complete understanding of SR. Not only for new students, but many who have been around for some time are still unable to resolve some fundamentally simple SR issues, (like GPS twins).

I agree a spacetime diagram can be very useful, but not for SR.
It’s a lead to GR and should help focus on how SR and GR are two different things.

To often many seem to forget the point that SR is Classical, all the way to the point of using only 3D not 4D. Bringing it in on SR only gives the impression that some “underlying” geometry that helps build some “causal structure” are an integral part of SR. It is not, that comes with GR. I really think to understand that it helps to recognize the difference between SR & GR early on.

I’m convinced that even many advanced students don’t fully appreciate the distinction that there are THREE major theories SR, GR & QM not two SR/GR vs QM.
 
  • #15
selfAdjoint
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This whole thread is just demonstrating that people have different internal ways of understanding. That's why there are algebraists and geometers in math, and very very few mathematicians have ever shone in both fields.

There is no problem of this in research; the world is wide there seems to be no end of research problems for both kinds of minds. But a problem arises in teaching: the teacher cannot do his or her best job except by leading from their own strength. But there is no guarantee that the student shares these strengths. How can this dilemma be resolved? That is to me a more interesting topic than ever-repeated dogmas that one way or the other is "better".
 
  • #16
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But the algebraists understand that geometry is important, and the geometers understand that algebra is important. In fact, it's sometimes said that algebraic geometry was invented precisely so that algebraists could do geometry, and so that geometers could do algebra!!!


But RandallB's issue isn't algebra vs geometry: his is purely geometric. He seems to assert that SR is all about foliating space-time into RxR³, and most certainly not about using Minowski geometry.

RandallB: how do you defend your assertion? In classical mechanics, it's easy: time is universal, which gives us a canonical way to split space-time into RxR³.

But that's not so in Special Relativity -- foliating space-time is entirely unphysical: it depends on which reference frame you're using.


I think you're confusing mathematical techniques with physical theories. Special relativity is a physical theory: it makes some assertions about the universe.

Using an RxR³ foliation of space-time is a mathematical technique you might use to study Special Relativity. So is synthetic Minowski geometry. So is full-blown differential geometry.

But whatever mathematical technique you use to study it, it's still SR.

The reason Minowski geometry is appealing is because it is just as elementary as Euclidean geometry, but it doesn't require any unphysical choices, such as a foliation into space and time.

Rejecting Minowski geometry for studying space-time in favor of RxR³ foliations is analogous rejecting Euclidean space for studying 3-dimensional space, and saying we should instead foliate space into Euclidean planes!
 
  • #17
robphy
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selfAdjoint said:
There is no problem of this in research; the world is wide there seems to be no end of research problems for both kinds of minds. But a problem arises in teaching: the teacher cannot do his or her best job except by leading from their own strength. But there is no guarantee that the student shares these strengths. How can this dilemma be resolved? That is to me a more interesting topic than ever-repeated dogmas that one way or the other is "better".
One possible solution: Let each approach be developed as far as possible [i.e. write it up, give talks, educate, and demonstrate] and make them available and accessible (i.e. digestable) for students to pick and choose from. Ideally, the educator should be aware of the various approaches and be tuned in enough to guide each student to what may work best for that student.

All I am advocating is that the geometrical viewpoint be given a fair chance in introductory physics. If we are happy with the way SR is taught, learned, and understood, then I would probably not mess with it.... But if we are not or if we want to see if a better job can be done, then let's try something new with them... (After all, are the introductory textbooks of 25 or 50 years ago like those of today? The Physics Education community is hard at work trying to make improvements in numerous topics.)

But why the geometric viewpoint? It's part of the intuition of the modern relativist. It helped me understand the subject. It might help my students do the same.
 
  • #18
i have heard the following story:
the misionary who was eaten by the natives did his job well?
no! because he has not propagated his faith well or he has imposed it by force!
I think the situation can be extended to the teacher-learner relationship!
 
  • #19
teaching

selfAdjoint said:
This whole thread is just demonstrating that people have different internal ways of understanding. That's why there are algebraists and geometers in math, and very very few mathematicians have ever shone in both fields.

There is no problem of this in research; the world is wide there seems to be no end of research problems for both kinds of minds. But a problem arises in teaching: the teacher cannot do his or her best job except by leading from their own strength. But there is no guarantee that the student shares these strengths. How can this dilemma be resolved? That is to me a more interesting topic than ever-repeated dogmas that one way or the other is "better".
I have heard the following problem:
Did the missionary who was eaten by the natives his job?
No, because he did not well his job or he tried to impose by force his dogmas.
The situation can be extended to the teacher learner relationship.
 
  • #20
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Hurkyl said:
He seems to assert that SR is all about foliating space-time into RxR³, and most certainly not about using Minowski geometry.

RandallB: how do you defend your assertion?
What “assertion” are you imagining I made?
I thought you understood me that “Space-time” did not belong in SR but in building GR. In SR, distance and time are observed and experienced relative to motion - that’s it. Working it all out may not be easy, but that’s it. SR has no ‘space-time’ to be foliating or using a mathematical technique on.

An abstraction like space-time may look friendlier, but ultimately a misleading analogy, because as you said “foliating space-time is entirely unphysical”. That is part of why GR ultimately leads to seeing an indeterminate background. (QM is indeterminate as well but I think that can be attributed to HUP.)

SR (without space-time or foliating it) is a determinate background theory, as you’d expect of a classical theory. Sure, it’s not a complete solution – that’s why there are two more theories. Dealing with background differences shouldn’t be included in early classes.
But, linking SR & GR with a common use of space-time in the long run can make it harder to see those differences clearly later on.

I’m only suggesting that the early recognition that SR and GR are different here is useful. Not that creative mathematical techniques on space-time shouldn’t be used, just use them in relation to understanding GR, both the theory and its history.
 
  • #21
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RandallB said:
I thought you understood me that “Space-time” did not belong in SR but in building GR.
I can't see how anyonw could come to a conclusion like this. Spacetime is merely the union of space and time regardless if we're talking SR or GR. In fact Minkowski, the inventor of spacetime for relativity, spoke of it in terms of SR.

Pete
 
  • #22
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pmb_phy said:
In fact Minkowski, the inventor of spacetime for relativity, spoke of it in terms of SR.
His error, and Einstein for one saw that. The teacher did not catch up with the student here, even if it's popular to think so. Einstein understood he was leaving the classical and SR as he developed GR, I'm not convinced that Minkowski even realy understood simultaneity like Einstein did, but then he didn't get to work with relativity for very long.
 
  • #23
Hurkyl
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I’m only suggesting that the early recognition that SR and GR are different here is useful. Not that creative mathematical techniques on space-time shouldn’t be used, just use them in relation to understanding GR, both the theory and its history.
You're doing more than suggest that "the early recognition that SR and GR are different here is useful" -- you advocate denying students the opportunity to learn a powerful, yet elementary, tool that is widely used in the modern treatment of special relativity.

And from your last couple posts, I don't think the reason has anything to do with learning SR -- you merely wish to artifically inflate the difference between SR and GR.


An abstraction like space-time may look friendlier, but ultimately a misleading analogy, because as you said “foliating space-time is entirely unphysical”.
Space-time is physical. The foliation is unphysical (in SR).

Of course, in prerelativistic mechanics, the decomposition of space-time into space and time is physical -- that's what makes SR different from prerelativistic mechanics and the various Lorentz ether theories.

The whole point of using Minkowski geometry to do special relativity is that all of its fundamental elements have physical interpretations in SR. E.G.: Points are events. Timelike lines are precisely the trajectories of inertial point-like particles. Two timelike line segments are congruent if and only if a clock travelling the first from start to finish reads the same time as a clock travelling he second from start to finish. Et cetera.


I'm not sure what you mean by (in)determinate background -- I can't figure out a possible interpretation that is consistent with your usage. (SR is determinate, while GR and QM are not)
 
  • #24
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Hurkyl said:
I'm not sure what you mean by (in)determinate background -- I can't figure out a possible interpretation that is consistent with your usage. (SR is determinate, while GR and QM are not)
I don’t understand what your saying here:
If you saying (SR is determinate, while GR and QM are not)
that what I said too, thought I was clear.
OR if you understand that I said
(SR is determinate, while GR and QM are not) but don’t understand what I mean.

I see “determinate” as “background dependent” and
“indeterminate” as “background independent”
SEE Smolin’s:

http://arxiv.org/PS_cache/hep-th/pdf/0507/0507235.pdf [Broken]

or google (background dependent perimeter smolin) for more.

As to the rest I see that originating back at the space-time idea.
If others do not or disagree that OK.
 
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  • #25
Hurkyl
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I see “determinate” as “background dependent” and
“indeterminate” as “background independent”
SEE Smolin’s:
That was my first thought you meant, but the quantum stuff used to model the universe is background dependent. Part of the whole hullabaloo in theoretical physics is that it's hard to come up with a useful quantum theory in a background independent way.

But yes, I would that that SR is background dependent: it requires that space-time be Minowski; it specifies both the topology and the metric!
 

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