Teaching SR without simultaneity

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  • #51
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Summary:: De-emphasizing simultaneity in SR curriculum. Thoughts? Experiences?

... but was thinking of removing focus from the meaning of events being simultaneous, time dilation, and length contraction ...
Time dilation and length contraction can also be calculated without emphasizing simultaneity, by using a moving, L-shape light clock.
 
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  • #52
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Time dilation and length contraction can also be calculated without emphasizing simultaneity, by using a moving, L-shape light clock.
Calculated, yes. However, it is quite relevant to understanding length contraction.

Edit: In some sense I would say it would even make things worse as it is computing length contraction without making the differences in simultaneity explicit.
 
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  • #53
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I've been thinking a bit about my own experience of SR as an undergrad, and have a couple more observations.

SR was taught very differently from physics before and after that. Almost all non-relativistic physics boils down to either conserving energy and momentum, or applying forces. Relativity was taught to me almost entirely as "here are the Lorentz transforms and here are some scenarios we'll attempt to understand" - the scenarios being mostly the usual paradoxes. We had done Galilean frame changes for solving particle collision problems (transform to zero momentum frame, solve, transform back), but it wasn't really something I connected with the Lorentz transforms. We were just talking about frame changes, not doing the same physics as we did in Newtonian mechanics. Or at least, that's what I remember - it's been a couple of decades...

I think it would be interesting to stress what's the same: maybe just state that the four momentum exists and do a few collision problems. The process is the same although the algebra's nastier. You can do that without even thinking about simultaneity, because all we're doing is transforming four vectors at a point. After that, get into the implications of the Lorentz transforms - that they imply a 4d structure, and show that a line of constant ##x'## or ##t'## is slanted in ##x,t## coordinates and look at how frame coordinates relate to one another. You could even then draw a spacetime diagram of a couple of collisions between particles and discuss the lack of meaning of simultaneity.

I guess the thing I'm getting at was the disconnect between relativistic physics ("here are some transforms") and pre-relativistic physics ("here are some forces") and trying to bridge the gap. I know you don't want to go too deep into forces in relativity because it gets messy quickly, but conservation laws still work and showing up front that they still work, albeit modified, before getting into the weirdness would (IMO) help to link relativity to pre-existing student knowledge.
 
  • #54
vanhees71
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That's an important point. Overemphasizing the paradoxes overemphasizes funny puzzles which don't play much of a role for physics. Another problem is that you can do very little point-particle mechanics, practically only single-particle motion in external fields (charged particles in external electromagnetic fields as a real-world example), and even this is only an approximation given that we neglect the notorious radiation-reaction problem.
 
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  • #55
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Another problem is that you can do very little point-particle mechanics, practically only single-particle motion in external fields (charged particles in external electromagnetic fields as a real-world example), and even this is only an approximation given that we neglect the notorious radiation-reaction problem.
Sure, but you can do the same "two billiard balls of mass m and M travelling with velocity v and V collide without spinning" problems you do in first year undergrad. You probably don't want to spend too long on it because, as you note, realistic problems get messy very quickly. But showing students that they can solve simple problems with only fairly minor modifications to tools they already have would, I think, be useful. You could then demonstrate that the problems rapidly get intractable, so here's Lagrange/Hamilton for more realistic situations.
 
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  • #56
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When I read an established "bible" on a subject, it often has a motivation, details, and a thorough discussion that other, more modern books do not have. (Feller in probability, Knuth in CS, etc.) I love those books. I think that such a book on SR would include similar motivation and a discussion of how the relativity of simultaneity allows the speed of light to be measured as constant. It does more than explain the paradoxes. It would make a lot of things seem dirt-simple.
 
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When I read an established "bible" on a subject, it often has a motivation, details, and a thorough discussion that other, more modern books do not have. (Feller in probability, Knuth in CS, etc.) I love those books. I think that such a book on SR would include similar motivation and a discussion of how the relativity of simultaneity allows the speed of light to be measured as constant. It does more than explain the paradoxes. It would make a lot of things seem dirt-simple.
Two of my favorites as well!
 
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  • #58
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Edit: In some sense I would say it would even make things worse as it is computing length contraction without making the differences in simultaneity explicit.

Yes. However, in this scenario you can derive the relativity of simultaneity afterwards from the calculated length contraction.

The horizontal and vertical moving light pulses leave the left/bottom edge of the L-shape clock at the same time (tick-event).
  • In the rest frame of the clock, these light pulses arrive at the right mirror and the top mirror at the "same time":
    ##\frac{L_0}{c} - \frac{L_0}{c} = 0##
  • In the reference frame, in which the clock is moving, these light pulses arrive at the right mirror and the top mirror at different times:
    ##\require{color}\frac{L_0/\color{red}\gamma}{\color{black}c-v} - \gamma\frac{L_0}{c} = \gamma\frac{vL_0}{c^2} (\frac{c^2-v^2}{v(c-v)} - \frac{c}{v} * \color{blue}\frac{c-v}{c-v}\color{black})= \gamma\frac{vL_0}{c^2}##.
Of course I understand, that you want to avoid this.
 
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  • #59
robphy
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Yes. However, in this scenario you can derive the relativity of simultaneity afterwards from the calculated length contraction.
It seems to me that one needs a definition of simultaneity (or at least orthogonality)
before one has a definition of length (in a plane of simultaneity)...
or any other spatial quantity... like, e.g., an electric field.
 
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  • #60
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It seems to me that one needs a definition of simultaneity (or at least orthogonality)
before one has a definition of length (in a plane of simultaneity)...
or any other spatial quantity... like, e.g., an electric field.

Yes. I should have mentioned, that the L-shape light clock scenario is described with reference to a standard inertial coordinate system. The time coordinate of it is defined by a grid of Einstein-synchronized clocks at "rest".

The same is also required and should be made explicit for the Minkowski metric (and its invariance), which one would use to describe the spacetime geometry:

##(\Delta s)^2 = c^2(\Delta t)^2 - (\Delta x)^2-(\Delta y)^2-(\Delta z)^2= c^2(\Delta t')^2 - (\Delta x')^2-(\Delta y')^2-(\Delta z')^2##

I like the idea to de-emphasize something, the universe does not care about. But I think that implementing this will become difficult.

Edit: It is also possible to fulfill (almost) the OP's headline "Teaching SR without simultaneity", but then the Lorentz transformation needs be generalized to include Reichenbach's ##\epsilon##:
https://en.wikipedia.org/wiki/One-w...ansformations_with_anisotropic_one-way_speeds
 
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  • #61
vanhees71
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It seems to me that one needs a definition of simultaneity (or at least orthogonality)
before one has a definition of length (in a plane of simultaneity)...
or any other spatial quantity... like, e.g., an electric field.
Yes, and that's why you need a clock-synchronization convention, and the standard one is that defined by Einstein via light signals and via local measurements with one clock via light signals sent back and forth via this one reference clock and all the other clocks, all at rest wrt. the reference clock (and thus also wrt. each other). The kinematical effects (relativity of simultaneity, time dilation, length contraction) are then implications of this synchronization convention.
 
  • #62
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I propose the sequence:
  1. Stipulate that a one way-speed is isotropic in an inertial frame, although the universe cares only for 2-way speeds
  2. Define (as preparation for the following), what a standard inertial coordinate system is
  3. Describe the spacetime Geometry with the Minkowski metric, reverse triangle inequality
  4. L-shape light clock: time dilation, length contraction, relativity of simultaneity
  5. Derive the Lorentz transformation and velocity addition
  6. Show the Minkowski diagram, light cone
  7. Describe 4-momentum and 4-current / 4-potential to shows the unification of momentum and energy, magnetism and electricity.
  8. Four-frequency, light aberration and Doppler effect
  9. Uniformly accelerated reference frame, pseudo-gravity, gravitational time-dilation
  10. Rotating reference frame, Sagnac effect
 
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  • #63
robphy
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I propose the sequence:
  1. Stipulate that a one way-speed is isotropic in an inertial frame, although the universe cares only for 2-way speeds
  2. Define (as preparation for the following), what a standard inertial coordinate system is
  3. Describe the spacetime Geometry with the Minkowski metric
  4. L-shape light clock: time dilation, length contraction, relativity of simultaneity
  5. Derive the Lorentz transformation and velocity addition
  6. Show the Minkowski diagram
  7. Describe 4-momentum and 4-current / 4-potential to shows the unification of momentum and energy, magnetism and electricity.
  8. Four-frequency, light aberration and Doppler effect
Bondi would essentially start with the Doppler effect, the invariance of the speed of light, and the principle of relativity on a Minkowski diagram... then (oh, by the way) the rest of the standard textbook stuff follows.

https://archive.org/details/relativitycommon0000bond
(bolding mine)
I. ...The Concept of Force—The Evaluation of Acceleration
II. Momentum
III. Rotation
IV. Light - Faraday and the Polarization of Light—Maxwell and the Electromagnetic Theory of Light—Using Radar to Measure Distance—The Units of Distance—The Velocity of Light
V. Propagation of Sound Waves
The Doppler Shift—The Sonic Boom
VI. The Uniqueness of Light , A Hypothetical Ether—The Absurdity of the Ether Concept—Measuring Velocity—The Michelson-Morley Experiment
VII. On Common Sense - The Experience of Everyday Life —Time: A Private Matter—The
"Route-Dependence' of Time
VIII. The Nature of Time - The Peculiarities of High Speeds—
The Relationships of Inertial and Moving Observers...
The Value of k: A Fundamental Ratio

IX. Velocity -Einstein's Long Trains—
Determining Relative Velocities by the Radar Method—
The Relationship between k and v—Velocity Composition-
Proper Speed—The Unique Character of Light
X. Coordinates and the Lorentz Transformation ...
XI. Faster Than Light? Cause and EffectSimultaneity of Spatially Separated Events—
Past and Future: Absolute and RelativeThe Light Cone
XII. Acceleration - Acceleration and Clocks—The Twin "Paradox"-How Far Can We Travel
in Space?
XIII. Putting on Mass The Stretching of Time—Increasing Mass—Accelerating Protons—Einstein's Equation—Theory and Observation

(umm...well... we can let some terms in XIII slide )
 
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  • #64
vanhees71
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I just taught the introductory parts. I start with Einstein's two postulates motivated by the historical problem that Maxwell electrodynamics is not Galilei invariant but that on the other hand there's no empirical evidence for any preferred frame or an aether. Then I derive the Lorentz transformation using light clocks and clock-synchronization a la Einstein (physics track), which leads to the kinematical effects as conclusions from the necessity to operationally synchronize clocks and how to measure space-time intervals. Finally you get the Lorentz transformation. Then I discuss Minkowski spacetime and the property of the Lorentz transformation to transform Minkowski-orthonormal bases (tetrads) into each other (in analogy to rotations in Euclidean vector space). This is then used to explain how Minkowski diagrams are constructed (geometrical approach).

Then some elementary point-particle mechanics follows, using the heuristics to generalize Newton's ##\vec{F}=\dot{p}## which is an approximation valid in the momentaneous rest frame of the particle, leading to the introduction of proper time and four-dimensionally co-variant equations of motion, ##\mathrm{d}_{\tau} p^{\mu}=K^{\mu}## with the constraints ##p_{\mu} p^{\mu}=m^2 c^2## and the implication from this that ##p_{\mu} K^{\mu}=0##. As an example I derive ##K^{\mu}=q/c F^{\mu \nu} \mathrm{d}_{\tau} x_{\nu}##, with ##F^{\mu \nu}(x)## antisymmetric and its usual mapping to ##(\vec{E},\vec{B})##. This can then be used to get the Poincare-covariant formulation of classical electrodynamics, showing that Maxwell theory indeed is a relativistic classical field theory.
 
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I think the ladder "paradox" is perhaps one of the best illustrations of length contraction and simultaneity. Once one wraps their head around the resolution of the dilemma, it should open up some broader thinking.
 
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  • #66
vanhees71
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And it's immediately resolved by drawing the corresponding Minkowski diagram (and I'm not too much in favor of Minkowski diagrams).
 
  • #67
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And it's immediately resolved by drawing the corresponding Minkowski diagram (and I'm not too much in favor of Minkowski diagrams).

Maybe an animation helps the intuition, although it is redundant to the time axis.

Animated_Spacetime_Diagram_-_Length_Contraction.gif


Source:
https://commons.wikimedia.org/wiki/File:Animated_Spacetime_Diagram_-_Length_Contraction.gif
 
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  • #69
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Im still trying to fit my 12 foot ladder in my 10 foot garage 😂
You must move it at least with
##v = c\sqrt{1-(\frac{10}{12})^2} \approx 0.55 c ##.
 
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You must move it at least with
##v = c\sqrt{1-(\frac{10}{12})^2} \approx 0.55 c ##.
But seriously, can it ever fit with both doors closed simultaneously?
 
  • #71
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But seriously, can it ever fit with both doors closed simultaneously?
Yes and no.
  • In the rest frame of the garage, yes.
  • In the rest frame of the ladder, no.
 
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Yes and no.
  • In the rest frame of the garage, yes.
  • In the rest frame of the ladder, no.
Wait, what? I was being facetious. But now I'm confused. The ladder can't be contained by the garage, at least that's what I thought.
 
  • #73
Ibix
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Wait, what? I was being facetious. But now I'm confused. The ladder can't be contained by the garage, at least that's what I thought.
If it's moving fast enough and is length contracted enough then it will fit as viewed in the garage frame. Briefly. Then it'll slam into the end of the garage at a large fraction of lightspeed and leave a sizeable crater.
 
  • #74
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If it's moving fast enough and is length contracted enough then it will fit as viewed in the garage frame. Briefly. Then it'll slam into the end of the garage at a large fraction of lightspeed and leave a sizeable crater.
Ugh... It's late here. I'm still conceptually struggling with the idea that length contraction is "real" TBH. The non-simultaneity of the garage doors feels more natural to me. But I guess the fact that in one frame the ladder is entirely in the garage and observes both doors to be closed.... It hurts my brain. I guess that's why I brought it up as an example.
 
  • #75
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Wait, what? I was being facetious. But now I'm confused. The ladder can't be contained by the garage, at least that's what I thought.

Maybe it helps, if you look at the animated Minkowski diagram in above posting #67.

Imagine, the garage is at rest in the unprimed frame (with black coordinate-axes) and has the rest-length equal to ##\overline{OC}## in this frame. In this frame, the events ##O## and ##C## happen simultaneously.
 

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