Teaching SR without simultaneity

[robphy]

My interpretation of the Burke's point of view (which is essentially the radar method)
is that simultaneity is not primitive, but operationally defined by a radar measurement.

Assuming an inertial observer in Minkowski spacetime or a very small neighborhood in spacetime...

Given a distant event Q (not your worldline),
define the event on your worldline (call it Q') that you say is simultaneous with Q
by noting the clock reading $t_{send}$ when you must sent a light signal to reach Q
and noting the clock reading $t_{receive}$ when you receive the reflection (the echo).

Then you would assign Q to
have time coordinate $t_Q=\frac{1}{2}(t_{receive}+t_{send})$, (the "midway" time)
which is time $t_{Q'}$ of the clock reading of the local event Q' (the midpoint event of the send-receive segment).

Other observers would generally disagree that Q and Q' were simultaneous,
by making radar measurements from their worldlines.

So, simultaneity is a concept derived from a radar measurement.
Simultaneity has been de-emphasized, by demonstration.Another variation of this idea is to
determine the set of distant events that you say is simultaneous with a local event M.
Let M be the midpoint of a segment PF on your inertial worldline.
Locate the intersection of the past light cone of the future endpoint F
and the future light cone of the past endpoint P.
Those events are, according to you, simultaneous with M.
By varying the length of the segment with M as the midpoint,
you construct the hyperplane of simultaneity simultaneous with M according to you.

(In my light-clock-diamonds method, that's the the spacelike-diagonal of the clock-diamond.)

You can see the intersection of cones in this video

 

[Sagittarius A-Star]

It is. Of course, one doesn't completely get rid of it, because it is a useful concept afterall, but de-emphasizing its importance is exactly what Burke did in his book. In his own words:

... For us simultaneity will be only a defined notion. ...

I think, that is nothing new (and not specific to Burke):

§ 1. Definition of Simultaneity.
...
"time" common to A and B. This last time (i.e., common time) can now be defined, however, if we establish by definition that the "time" which light requires in traveling from A to B is equivalent to the "time" which light requires in traveling from B to A.

Source:
https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Bodies

 

[PAllen]

It seems to me that failing to mention the relativity of simultaneity doesn't mean that it is no longer a major pitfall for the unwary student.

Every student will at some point invoke the natural concept of simultaneous events unless they know that explicitly absolute simultaneity is gone.

It's like failing to mention there is a crocodile in the river. And if the students stay in the boat then no harm may come to them. But, it only takes the unwary student to unwittingly dip their toe in the water for disaster to strike.

The idea to get across is that simultaneity of Newtonian physics is replaced by something completely different - future, past, and "possibly now" or "elsewhen". Further, instead of relativity of something that has no physical relevance in relativity (simultaneity), we have invariance of the causal classification. Instead of absolute simultaneity, past and future (Newtonian), we have invariance of past, future, and "possibly now".

I do think you still need to introduce simulaneity as a useful convention only after 'debunking' it as something the universe cares about.

 

[Orodruin]

Instead of absolute simultaneity, past and future (Newtonian), we have invariance of past, future, and "possibly now".

This is a fundamental insight in my opinion. Together with the realisation that the spacetime no longer has a time order but only a partial time order.

 

[robphy]

This is a fundamental insight in my opinion. Together with the realisation that the spacetime no longer has a time order but only a partial time order.

You might be interested in the work of A.A. Robb (1914) A theory of time and space
who tried to axiomatize the structure of Minkowski spacetime using the "after" relation.
(Robb's (1911) Optical Geometry of Motion gave us the term "https://en.wikipedia.org/wiki/Rapidity".)

As a grad student, I studied aspects of causal sets in my dissertation.

 

[FactChecker]

The idea to get across is that simultaneity of Newtonian physics is replaced by something completely different - future, past, and "possibly now" or "elsewhen". Further, instead of relativity of something that has no physical relevance in relativity (simultaneity), we have invariance of the causal classification. Instead of absolute simultaneity, past and future (Newtonian), we have invariance of past, future, and "possibly now".

I do think you still need to introduce simulaneity as a useful convention only after 'debunking' it as something the universe cares about.

How is simultaneity different from length, distance, and elapsed time? Aren't they all frame-dependent?

 

[PAllen]

How is simultaneity different from length, distance, and elapsed time? Aren't they all frame-dependent?

I would understand elapsed time as proper time, so no. Length and distance are associated with specific bodies, and there is the notion proper length/distance associated with their mutual rest frame (and in this frame, you can measure without a simultaneity convention). It is certainly true that distance becomes ambiguous in GR, as does relative velocity of non-colocated bodies.

The issue with simultaneity is that while the notion is fundamental in Newtonian physics, what is fundamental in SR is its replacement by a whole region of events - a plane becomes a volume. This is the fundamental replacement of understanding that must occur.

I really have seen many misunderstandings in SR associated with beliefs that planes or lines of simultaneity have some significance beyond convention. Just as an aside, I will mention something I have researched - Einstein never once in all his writings on SR ever used the notion of lines or planes of simultaneity.

 

[robphy]

The issue with simultaneity is that while the notion is fundamental in Newtonian physics, what is fundamental in SR is its replacement by a whole region of events - a plane becomes a volume. This is the fundamental replacement of understanding that must occur.

In some situations, you may still want a [hyper]surface (not a volume),
e.g. an initial data surface for a differential equation.

 

[robphy]

Just as an aside, I will mention something I have researched - Einstein never once in all his writings on SR ever used the notion of lines or planes of simultaneity.

It seems to me that Einstein didn't think or reason in terms of spacetime diagrams.
I once asked someone from the Einstein papers project if there was some example of such reasoning.
That person didn't have an example.

 

[physicsworks]

I think, that is nothing new (and not specific to Burke):

I wonder, did you actually read the book? Although Burke's approach is somewhat relying on the radar method as @robphy pointed out and hence can be traced back to Bondi's k-calculus, who, in turn, popularized Milne's approach, it is a novel pedagogical approach and not just in teaching special relativity in particular. The book treats many other familiar concepts in a somewhat unique way. For instance, the metric tensor in general relativity is explained by analyzing the propagation of wave packets on the water surface (the metric is associated with the law of dispersion of the selected wave type). The rate of clocks is determined by oscillations of the water surface and light signals are represented by ripple waves having a different law of dispersion. Now, I would personally not adopt many of the novelties in the book to teach a course in relativity, but it is definitely thought provoking and by no means unoriginal.

 

[FactChecker]

I would understand elapsed time as proper time, so no. Length and distance are associated with specific bodies, and there is the notion proper length/distance associated with their mutual rest frame (and in this frame, you can measure without a simultaneity convention).

I think I see. In a rest frame, objects have size and length, regardless of whether it is measured by any person in any coordinate system. As far as I can imagine, the same can not be said of the concept of "simultaneity" at different locations.

 

[PAllen]

In some situations, you may still want a [hyper]surface (not a volume),
e.g. an initial data surface for a differential equation.

Agreed. Specifying initial conditions is one place where I do this all the time. In fact, a common refrain of mine, in relation to SR 'paradoxes' is that both frame's conclusions are correct (bomb explodes/ doesn't explode, rod fits through hole/ does not fit through hole) because each case represents a different problem set up - the initial conditions are different. Each different set up, then has a description in any other frame. And for this purpose, at least in SR, I certainly use the phrase "simultaneous in frame A".

I agree that it is necessary at some point to explain the utility of simultaneity conventions, and even the advantages of particular ones. Coupling simultaneity to an operational convention clarifies what is different about the initial conditions specified in different frames.

But this should all be distinguished from the fundamental change in going to SR - replacement of absoluteness of simultaneity with absoluteness of causal relations; and absence of any fundamental notion of simultaneity in relativity.

 

[Sagittarius A-Star]

I wonder, did you actually read the book?

No, I did not read the book, and I did not criticize the overall book. My comment was only regarding the one sentence from Burke, which I cited.

 

[Office_Shredder]

As someone who just worked through a very basic introduction to special relativity, maybe I have a uniquely uninformed perspective here, which might be useful.

Simultaneity of relativity for me clicked when I started thinking of it as relativity of clock time. Just the way that the distance between two points in a frame changes as your relative speed changes, the difference in clock time between those points changes as your speed changes. Clocks appear to run slow when your speed increases is a consequence of the change in clock times, and is not the fundamental fact you should just go resolve paradoxes (relativity of clock time is the right thing to focus on)

 

[vanhees71]

The important point is that, as Einstein discussed carefully in his seminal paper of 1905, to have a unique time at any point in an inertial frame you need a clock-synchronization convention, and the standard choice is to synchronize all clocks at rest in an inertial frame by using two-way light signals and assume that the signal from clock A to B needs as much time as back from B to A. This then implies that the clocks synchronized in this way wrt. another inertial frame $\Sigma'$, moving with constant velocity against the first inertial frame $\Sigma$, are not synchronized with the clocks synchronized in $\Sigma$. This leads to time-dilation.

Then, analyzing how the length of a rod at rest in $\Sigma$ is measured in the inertial frame $\Sigma'$, you get length contraction, and that's a consequence of the "relativity of simultaneity", i.e., the observer in $\Sigma'$ is marking the coordinates of the end points of the rod simultaneously in $\Sigma'$, but for the observer in $\Sigma$ these events are not simultaneous.

I think at this point (and in related more "funny" situations like the garage paradox) one has to discuss the "relativity of simultaneity", but it's indeed true that overemphasizing these apparent paradoxa is more confusing in the beginning than necessary, and a more straight-forward way is to use the geometry of Minkowski space to emphasize the consistency of the relativistic spacetime model rather than the apparent paradoxes, which only are felt to be paradox, because we are familiar to Newtonian spactime descriptions, because all velocities occurring in our everyday life are much smaller than the speed of light.

 

[Sagittarius A-Star]

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.

 

[Orodruin]

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.

 

[Ibix]

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.

 

[vanhees71]

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.

 

[Ibix]

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 traveling 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.

 

[FactChecker]

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.

 

[PAllen]

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!

 

[Sagittarius A-Star]

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.

 

[robphy]

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.

 

[Sagittarius A-Star]

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

 

[vanhees71]

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.

 

[Sagittarius A-Star]

I propose the sequence:

1. Define events and spacetime interval (timelike, lightlike, spacelike)
2. Stipulate that a one way-speed is isotropic in an inertial frame, although the universe cares only for 2-way speeds
3. Define (as preparation for the following), what a standard inertial coordinate system is
4. Describe the spacetime Geometry with the Minkowski metric, reverse triangle inequality
5. Derive the Lorentz transformation and velocity addition
6. Derive the Minkowski diagram from hyperbolic rotation, 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

 

[robphy]

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 Effect—Simultaneity of Spatially Separated Events—
Past and Future: Absolute and Relative—The 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 )

 

[vanhees71]

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.

 

[valenumr]

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.

 

[vanhees71]

And it's immediately resolved by drawing the corresponding Minkowski diagram (and I'm not too much in favor of Minkowski diagrams).

 

[Sagittarius A-Star]

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.

Source:
https://commons.wikimedia.org/wiki/File:Animated_Spacetime_Diagram_-_Length_Contraction.gif

 

[valenumr]

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

View attachment 295490

Source:
https://commons.wikimedia.org/wiki/File:Animated_Spacetime_Diagram_-_Length_Contraction.gif

Im still trying to fit my 12 foot ladder in my 10 foot garage

 

[Sagittarius A-Star]

Im still trying to fit my 12 foot ladder in my 10 foot garage

You must move it at least with
$v = c\sqrt{1-(\frac{10}{12})^2} \approx 0.55 c$.

 

[valenumr]

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?