Teaching Minkowski diagrams effectively

[pervect]

I've felt for some time that a primer on space-time diagrams and light clocks would be very helpful, but I was (and am) too lazy to do an exposition. Perhaps someday? But I can outline a sort of "homework exercise", a series of tasks that one can do to break down the larger task of "drawing a light clock" into a sequence of logical, smaller steps. There may be different or better ways of doing this, this is just my first shot at an outline.

1) Phase 1. Draw an empty space-time diagram. Identify the time axis "t", and the space axis, "x".

2) Phase 2. Draw an "event" at some specific time $t_0$ and position $x_0$ on the space-time you drew in part 1.

2) Phase 3. Draw a representation of a stationary object on the space-time diagram. A stationary object has the condition that it's coordinate, x, is constant. Or we can write $x = x_0$, where x_0 is some constant. Note that when you graph this on a space-time diagram, this curve is a line. This curve is called the "worldline"" of the object. Note that we can and are considering an "object" as an infinite set of "events". Each instant the "object" exists corresponds to one "event" on the space-time diagram.

3) Phase 4. Draw a representation of a boxes on a space-time diagram. Make a box with two walls, one at x=-1 and one at x=+1. Try making some boxes of different sizes, and at different positions.

4) Phase 5. Now that we can draw stationary objects on a space-time diagram, learn how to draw moving objects on a space-time diagram. We are interested in objects moving at some velocity "v". Hint: the equation of motion will be x = $x_0 + v\,t$.

4a) Draw light beams on a space-time diagram, which can be considered objects that move at some velocity c. At this point some discussion of scale is probably needed. The intent is to chose a scale such that light beams move at 45 degree angles. Sorry if this isn't really well explained.

Now comes the longer tasks, the end goal. All the other subtasks were designed to pave the way to be able to perform this more complex, longer task.

5) Draw the space-time diagram of a light clock in a stationary box, where the center of the box is at the origin of the space-time diagram. This is the space-time diagram of the box, plus a space-time diagram of two light beams (or perhaps 4 light beams). The light beams start in the center of the box, move in both spatial directions, hit the walls of the box and reflect. Considering the reflected light beams as different from the original non-reflected light beams makes 4 light beams, considering them as the same makes two light beams in this diagram. The diagram should show, that after they reflect, the light beams, which started at the center of the box, re-meet in the center of the box.

6) Identify, on the diagram, the length or size of the box. We want to identify the dimension of the box as the length between two events on some line that we draw on the space-time diagram. Repeat this process to identify the duration of one tick of the light clock, again as the length of some line on the space-time diagram.

7) Optional, but highly recommended. Read about Einstein's simultaneity convention, and apply it to the light clock to show that the two events of the light beams "hitting the walls of the box" are simultaneous.

8) Repeat the above process for a box that is moving, rather than is stationary. If you're still with us at this point, you might refer to some of the existing diagrams on this thread and others.

9) Optional, but recommended. Repeat step 7 for the moving box. Ideally, you'd notice that something unexpected is going on with simultaneity in the moving frame. It's different than it is in the stationary frame! Do some reading about the "relativity of simultaneity". This is probably the hardest step - it doesn't involve drawing any lines , but in interpreting the diagrams you've made.

 

[robphy]

7) Optional, but highly recommended. Read about Einstein's simultaneity convention, and apply it to the light clock to show that the two events of the light beams "hitting the walls of the box" are simultaneous.

8) Repeat the above process for a box that is moving, rather than is stationary. If you're still with us at this point, you might refer to some of the existing diagrams on this thread and others.

9) Optional, but recommended. Repeat step 7 for the moving box. Ideally, you'd notice that something unexpected is going on with simultaneity in the moving frame. It's different than it is in the stationary frame! Do some reading about the "relativity of simultaneity". This is probably the hardest step - it doesn't involve drawing any lines , but in interpreting the diagrams you've made.

It seems to me that "spacetime diagram" might be a turn-off to a beginner...
or to a teacher that prefers "moving boxcars in space" rather than "worldlines in spacetime".

One strategy that I'm been trying on and off is to emphasize that
spacetime diagrams are really "position-vs-time graphs"...
and sometimes I'll use "position-vs-time" in the beginning... and
hold off the use of "spacetime diagram" until later.


Among the various steps, step 8 is the really challenging part,
especially if one tries to motivate the result without directly appealing
to equations like the Lorentz Transformations.
So, I have been using the ideas given in my insight
https://www.physicsforums.com/insights/relativity-on-rotated-graph-paper-a-graphical-motivation/

 

[Nugatory]

Phase 4. Draw a representation of a boxes on a space-time diagram. Make a box with two walls, one at x=-1 and one at x=+1. Try making some boxes of different sizes, and at different positions.

Perhaps draw a representation of a rod first?

 

[Ibix]

One strategy that I'm been trying on and off is to emphasize that
spacetime diagrams are really "position-vs-time graphs"...
and sometimes I'll use "position-vs-time" in the beginning... and
hold off the use of "spacetime diagram" until later.

Yes - they're just position versus time graphs. But then we take them a bit more literally in relativity because we re-interpret them as maps of spacetime (or at least, a slice of it). And we can draw mapswith different directions drawn vertically on the page, just as you can rotate a map on your phone. That's all different choices of frame are (but Minkowski geometry isn't quite the same as Euclidean geometry and the rules are a bit different).

That seems to me like the sensible progression to motivate them.

 

[robphy]

One complaint about "spacetime diagrams" is that they involve a non-Euclidean geometry.

Part of my motivation to start off with using "position-vs-time graphs"
is to point out that
the PHY101 x-vs-t graph also involves a non-Euclidean geometry.

Euclidean trigonometry does not work on the x-vs-t plane.
The Pythagorean Theorem does not work on the x-vs-t plane.
In units where positions are measured in seconds,
and where (OA) and (OB) are the proper-times elapsed along those worldlines,
which are equal [(OA)=(OB)] on a x-vs-t graph
we have
$(OA)^2 +(AB)^2 \neq (OB)^2$.
In addition, in this geometry [Galilean geometry]
(AB) is Galilean-orthogonal to both (OA) and (OB):
$(AB) \perp_G (OA)$ and $(AB) \perp_G (OB)$.

So, using a "PHY 101 x-vs-t graph" already uses a non-Euclidean geometry.
Admittedly, we learn to read off the information without being aware of this non-Euclidean geometry.
But maybe being aware of it could help one understand spacetime-diagrams in special relativity.

 

[renormalize]

I'm am unfamiliar with this approach. It sounds intriguing. Can you help me understand?

In units where positions are measured in seconds,

In this Physics 101 context, how are $x$ positions in, say, meters converted to positions in seconds?

the proper-times elapsed along those worldlines

How is this "proper time" defined in a Physics 101 context? Is it distinct from the absolute time of Newton?

in this geometry [Galilean geometry]

Can you point me to a reference that explains non-Euclidean "Galilean geometry" in more detail? Thanks!

 

[Ibix]

In this Physics 101 context, how are $x$ positions in, say, meters converted to positions in seconds?

We usually call a "second of distance" a light second, $3\times 10^8\mathrm{m}$.

How is this "proper time" defined in a Physics 101 context?

It's the time measured by a clock following the specified path.

Is it distinct from the absolute time of Newton?

The possibility that two (ideal, correctly working) clocks measure different intervals is something one could check as a test of the Newtonian assertion that they never will. It's a bit of a loaded question given what we know now, but it's not unreasonable to ask.

 

[renormalize]

The possibility that two (ideal, correctly working) clocks measure different intervals is something one could check as a test of the Newtonian assertion that they never will. It's a bit of a loaded question given what we know now, but it's not unreasonable to ask.

That's all well and good.
What I'm questioning is the assertion by @robphy that "the PHY101 x-vs-t graph also involves a non-Euclidean geometry". How does the introduction of non-Euclidean geometry and a universal speed aid in understanding freshman-level classical particle motion in 1D? I'd like to read a literature reference that more fully explains, in this freshman-physics context, the statements about the Pythagorean Theorem "not working" in the $x-t$ plane and "Galilean orthogonality".

 

[robphy]

That's all well and good.
What I'm questioning is the assertion by @robphy that "the PHY101 x-vs-t graph also involves a non-Euclidean geometry". How does the introduction of non-Euclidean geometry and a universal speed aid in understanding freshman-level classical particle motion in 1D? I'd like to read a literature reference that more fully explains, in this freshman-physics context, the statements about the Pythagorean Theorem "not working" in the $x-t$ plane and "Galilean orthogonality".

If the goal is just to understand 1-D non-relativistic motion,
then a spacetime viewpoint is probably unnecessary.

If the goal is to understand spacetime kinematics,
then I think it is useful to find ways to build up
to the spacetime viewpoint in special relativity (and in general relativity).

One such way is to suggest that the PHY101-x-vs-t graph already has
some of the features of the spacetime diagram for special relativity.

• As mentioned, the x-vs-t graph already has a non-Euclidean geometry
  because of the way it is designed and used.
  Such a non-Euclidean geometry is not being "added"...
  rather, it is being "revealed".
• The expression of lengths in terms of a time does not imply the introduction of
  a physical universal speed... In the context of non-relativistic physics, one could use
  any speed [chosen by convention] to describe a length in terms of a travel-time using that speed.
  It is done in this context to set up the possibility of trying to do some kind of metrical geometry
  using common units on the x-vs-t graph.
• In "Galilean spacetime" (which could be googled) one could associate a universal speed [which is invariant under Galilean transformations].
  That speed is infinite, which could be visualized as the "opening up of the light-cone" as the invariant-speed in special relativity (called the speed of light, but maybe should be called the maximum-signal-speed) is increased to infinity in the "non-relativistic limit".
  
  
• For me and my approach, the origin of these ideas about and specific features of the Galilean spacetime can be traced back to the mathematician I.M. Yaglom (which is mentioned in an old thread https://www.physicsforums.com/threads/going-from-galilei-to-minkowski.930035/ ), which was discussed in a textbook by G.G. Emch mentioned in that thread [search for these names there].
  
  Yaglom's idea is based on "Galilean geometry" as one of the nine "Cayley-Klein geometries" in the plane. In the context of physics, one can look at Yaglom's "A Simple Non-Euclidean Geometry and Its Physical Basis" (1979).
  
  My approach for the teaching of relativity that builds on that idea can be found
  at AAPT Topical Workshop: Teaching General Relativity to Undergraduates (Syracuse, 2006)
  in a poster that I presented:
  "New Ideas for Teaching Relativity: Space-Time Trigonometry" (2006) https://www.aapt.org/doorway/Posters/SalgadoPoster/SalgadoPoster.htm
  
  You can see some of these ideas implemented in my Desmos visualization
  robphy's spacetime diagrammer for relativity v.8d-2020 (time upward)
  https://www.desmos.com/calculator/kv8szi3ic8

 

[renormalize]

@robphy Thank you for your explanation and the references.

 

[cianfa72]

But then we take them a bit more literally in relativity because we re-interpret them as maps of spacetime (or at least, a slice of it). And we can draw maps with different directions drawn vertically on the page, just as you can rotate a map on your phone. That's all different choices of frame are (but Minkowski geometry isn't quite the same as Euclidean geometry and the rules are a bit different).

Yes, we re-interpret them as maps of spacetime. However in these maps lines of constant coordinate "space" $x$ should be timelike while lines of constant coordinate time $t$ should be spacelike.

 

[pervect]

It seems to me that "spacetime diagram" might be a turn-off to a beginner...
or to a teacher that prefers "moving boxcars in space" rather than "worldlines in spacetime".

Possibly. My main thought is that the approach (which is of course your approach) in "Relativity on Rotated Graph paper" is a promising approach to those who want to learn a useful amount about Special Relativity but are intimidated by the math. But to use this approach does require the ability to draw and interpret space-time diagrams. And, I see this ability as probably lacking if it hasn't been learned and studied, and I expect there are a lot of PF readers who haven't learned and studied it.

While drawing space-time diagrams is a non-mathematical task, it will still require a certain amount of work and thought from the learner/student. How do we go about this? We can break the larger tasks into a sequence of smaller ones.

Much of the time in what I wrote was spent on the first goal, just being able to draw the space-time diagram that represents a light clock. I think there may need to be a few additional goals here that I didn't include - basically, to explain how a light clock is a combination of a clock, and a ruler (or rod). The "clock" part is in the name "light clock", so hopefully it doesn't need much more explanation. Still, it wouldn't hurt to stack multiple light clock diamonds to represent multiple ticks of a single clock. The only real issue here is basically how to do the stacking.

The "ruler" part seems lacking in exposition as I think about it more. Basically the idea is that the representation of a ruler on a space-time diagram is a group of parallel lines - a pair of parallel lines for a unit ruler, or unit rod. If we keep the idea of "stacks", then "stacks" of "clocks" allow us to measure larger units of time, while "stacks" of "rods" similarly allow us to measure distance.

It is somewhat of a leap to go from the stationary light clock to the moving light clock, I agree. The first step though is to be able to use light clocks to measure times and distances for a stationary observer. Then the hope is that the lessons learned will allow the reader/student to move on to the moving case, even if it is a bit of a jump, if the stationary case is understood well enough. Perhaps there is more that can be done to ease the transition.

One source of resistance will likely be the fact that people already know how to measure distances and times of stationary objects without light clocks, and may not see the point in learning how to do it with light clocks and space-time diagrams. I'm not sure there is an answer to this, though, except to explain that learning this approach will eventually allow one to work problems in relativity. For an actual class, one can add the motivation that one will be able to pass the class :). I still anticipate a majority of casual readers on PF who won't be motivated or interested enough to do the grunge work. But, we can perhaps help those that do have that motivational spark.

 

[Chicken Squirr-El]

One strategy that I'm been trying on and off is to emphasize that
spacetime diagrams are really "position-vs-time graphs"...
and sometimes I'll use "position-vs-time" in the beginning... and
hold off the use of "spacetime diagram" until later.

Has anyone ever considered ignoring the spacetime diagram convention and teaching everything in terms of time being on the x-axis since it's already familiar? Yes, textbooks will have them the usual way, but I wonder if swapping the axes ends up slowing down the learning process way more than one might think.

"Okay class, we're going to learn relativity, but first, let's switch it up a bit and have time go on the y-axis instead of the x-axis like everything else you've been taught so far." Their brains have already been trained to see it the other way. I still have to slightly force myself to not interpret them that way when I first look at one. Anyone else?

 

[Ibix]

"Okay class, we're going to learn relativity, but first, let's switch it up a bit and have time go on the y-axis instead of the x-axis like everything else you've been taught so far." Their brains have already been trained to see it the other way. I still have to slightly force myself to not interpret them that way when I first look at one. Anyone else?

I think it's a valid complaint, but your solution ends up with the x direction on the y axis which isn't great either. Unless you start teaching the Lorentz transforms in the y-t plane as well.

 

[robphy]

Has anyone ever considered ignoring the spacetime diagram convention and teaching everything in terms of time being on the x-axis since it's already familiar? Yes, textbooks will have them the usual way, but I wonder if swapping the axes ends up slowing down the learning process way more than one might think.

In my poster ""New Ideas for Teaching Relativity: Space-Time Trigonometry" (2006)
linked above in #9, the "t"-axis (for Galilean and Special Relativity) is along the horizontal,
as in the typical x-vs-t graphs.
I did that because the "Spacetime Trigonometry" approach is to show
the analogies among Euclidean geometry, Galilean geometry, and Minkowski spacetime geometry.

Furthermore, I use "y" along the vertical axis
so that I could use "y-vs-x" for Euclidean geometry and "y-vs-t" for the spacetime geometries.
Since "x" was along the horizontal for Euclidean Geometry,
I did not want it along the vertical for the spacetime-geometries.



In the "Relativity on Rotated Graph Paper" approach, the goal was to
decorate already existing spacetime diagrams [with t-upward]
with the ticks of light-clocks to reveal a computational method involving
light-cone coordinates and counting areas of causal diamonds.

In more recent introductions to my approach, I start with y-vs-t with t on the horizontal axis,
then eventually moving the t-axis to the vertical axis to match typical relativity diagrams.

 

[robphy]

Still, it wouldn't hurt to stack multiple light clock diamonds to represent multiple ticks of a single clock. The only real issue here is basically how to do the stacking.

The light-clock diamonds are traced out by the light-signals in a longitudinal light-clock
(which is different from the transverse light-clocks seen in the textbooks).
So, the "stacking" comes naturally from the progression of the ticking longitudinal light-clock.

The pre-cursor to counting longitudinal light-clock diamonds on a rotated graph paper
came from my visualization of a ticking "circular" light-clock.

"Visualizing proper-time in Special Relativity"
https://arxiv.org/abs/physics/0505134
Physics Teacher (Indian Physical Society), v46 (4), pp. 132-143 (October-December 2004)

( the videos use v=(4/5)c; the images are for v=(3/5)c )

(images are from my article
"Relativity on Rotated Graph Paper"
https://arxiv.org/abs/1111.7254
Am. J. Phys. 84, 344 (2016) [ https://doi.org/10.1119/1.4943251 ]

and from my insight
https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/ )

The "ruler" part seems lacking in exposition as I think about it more. Basically the idea is that the representation of a ruler on a space-time diagram is a group of parallel lines - a pair of parallel lines for a unit ruler, or unit rod. If we keep the idea of "stacks", then "stacks" of "clocks" allow us to measure larger units of time, while "stacks" of "rods" similarly allow us to measure distance.

I have recently referred to "ticks" (along timelike lines) and
"space ticks" (along spacelike lines), which I quickly shorten to "s-ticks" and to "sticks".
So that I can describe displacements in terms of "ticks" and "sticks", e.g,
3 sticks every 5 ticks describes the velocity of (3/5)c.

With "sticks" laid out end to end (like one would do with a meterstick")
we could measure a length in units of the "stick".


A key idea of the approach is that
the tick of a longitudinal light-clock provides the prototype cell
that can be used to tile spacetime (lay out a grid)...
and thus provide a set of coordinates
for the inertial observer carrying that longitudinal light-clock.

Furthermore, once the grid of diamonds is set up [hence the rotated graph paper],
it becomes easier to perform radar measurements
and visualize the results.