This week we began readings from chapter 36 (24MB) of Arnold Arons’ 1965 calculus-based textbook Development of Concepts of Physics (rare, but occasionally found on the used market…I have two copies and I hope to get Dover to reissue the book in paperback). This is the chapter on special relativity, and in my opinion forms the best foundation for relativity for any introductory physics course regardless of whether it’s calculus-based or algebra-based. Using mainly words and delightfully simple prose, Arons begins with Newton’s conceptualizations of space and time and then moves through a detailed discussion of clock syncoronization.
The first bit of physics is the concept of a frame of reference. This week I discovered that this Wikipedia entry now uses the term observational frame of reference to mean what I have always previously defined as a frame of reference. The first sentence of the article now conflates the concepts of reference frame and coordinate system, and I think this is a significant conceptual error. One can refer to a reference frame without referring to a coordinate system, and the converse is probably true (I need to think more about that…).
Next we addressed how to name and draw reference frames. We largely follow historical convention here: O or S is the name of a frame, primes indicate a frame moving relative to another frame, axes in standard configuration favoring the righthand direction. It occurred to me that in every textbook I can think of, we’re used to seeing two frames of reference in introductory relativity, a stationary S (or O) frame and a moving S’ (or O’) frame, which neglects one important frame, namely that of the reader! Yes, I understand that the reader is assumed to be in either of the two given frames, but that certainly is not how the diagrams are always, or even usually, drawn. Therefore, I also introduce the S” (or O”) frame and call it your frame. So in all, we have the S frame, the S’ frame, and your frame.
This image shows the naming conventions. Many, if not most, textbooks erroneously dictate that the S’ (or O’) label must apply to “the” moving frame. Aside from the use of “the” there is something else troubling about this edict, namely that if adhered to it requires changing that frames label during the reasoning for the current situation. Remember that labels are just that, arbitrary labels, but the important thing is to be consistent within the scope of a particular situation (I am trying to avoid the word “problem” becuase I think it’s, well, problematic.). Once a set of labels has been chosen and assigned, keep it intact for the duration of that situation and everything should be fine.
I have one huge concern with my drawings, and it is that by drawing a visual representation of a frame in standard configuration in this conventional way, I have committed the error of conflating the concepts of reference frame and coordinate system. I need a way out of this so I don’t feel guilty about it! I’m open to suggestions! Must we draw coordinate axes? Given that there is always a relative velocity involved, perhaps some coordinate-free geometry could be somehow employed…I don’t know, but this issue bothers me. As I write this, another option occurs to me. Perhaps we could just draw the physical entities involved and label them with their velocities and let the drawing as a whole represent the frame of reference, with no need to to draw or otherwise visualize coordinate axes. What do you think? This really bothers me.
While I’m on the issue of drawing, here’s an example we formulated in class.
The situation is “an object moves to the left relative to you.” Note that I drew the S” frame so as to suggest that it includes or encompasses the other frames. This, I think, is more in keeping with how the S and S’ frames are shown in textbooks from the reader’s perspective.
Note that I’m setting the stage for proper vector notation here too by using velocity magnitude, and I’m careful avoid the word “speed”. Familiar though it is, “speed” sticks out like a sore thumb because it’s the only term I know of that is a special name for a particular vector’s magnitude. We don’t do it with momentum, acceleration, force, etc. Pedantic? Okay pedantic.
I’m thinking a good assessment of understanding at this point might be to state the situation and have students draw the appropriate diagrams for all relevant observers.
We also use manipulatives to help visualize these ideas. Students are given homemade (by me) kits with reference frames and other assorted objects. These can be used to act out the scenarios describes in the Arons reading.
I used a short assessment in class that asked students to act out a scenario for which an observer in the right hand frame reports the ball bearing is moving to the left. Of course there are many such scenarios that will work, and sure enough different students came up with different, and equally valid, scenarios. For example, the ball bearing might be stationary in the left hand frame while the right hand frame moves to the right. The right hand frame may be stationary while the left hand frame moves to the left carrying the ball bearing with it. Try asking students to demonstrate different scenarios making the ball bearing move to the right, move to the left, and not move at all relative to them (the S” frame).
No discussion of reference frames is complete with out showing the famous 1960 Frames of Reference video featuring Patterson Hume and Donald Ivey. There are several links to it on YouTube; I tried to find one with no ads becuase I don’t like shoving ads in students’ faces. Apologies in advance if the one linked below displays ads or if the link rots. Notify me and I’ll fix it.
We wrapped up class time this week with a discussion of the breakdown of simultaneity as a result of the invariance of the magnitude of light’s velocity. This was basically a group demonstration using a “skateboard” and an apparatus consisting of a small stick (about 1 light-nanosecond long) and two plastic square mirrors. We imagined what would happen if a light at the stick’s center suddenly turned on, sending a pulses toward either end of the stick. Which mirror would catch its light first? Well, of course the answer depends on whether or not one is in the frame for which the stick is at rest. Students are to read the details in Aarons for next week.