Relativity Lorentz factor in textbooks on Special Relativity

[Trysse]

I recently had a look at some textbooks to see how they make use of the Lorentz factor.

I have seen that many textbooks introduce γ purely as a mathematical shorthand. On the other hand, some books that delve a little deeper into γ use graphs to show how γ and v are related.

I have found different graphs and I am curious, which graph you think best captures how $\gamma$ respectively relativistic effects behaves? Which do you think is best in terms of clarity?

Do you think that the Lorentz factor is essential and/or helpful in teaching special relativity?

The graphs are from Rindler, Resnick, Strohm and Rahaman. The last is a bit of an outlier as the graphs don't refer to $\gamma$ instead it is meant to display the quantitative effect of length contraction and time dilation.

Rindler:

Resnick

Strohm:

Rahaman:

 

[PeroK]

Rindler says all that needs to be said, IMO.

 

[Demystifier]

Rindler says all that needs to be said, IMO.

They all do.

 

[weirdoguy]

Do you think that the Lorentz factor is essential and/or helpful in teaching special relativity?

Well, writing it out explicitly is tiring, for everyone. So I guess it is essential. But at the end of the day it's still just notation.

 

[robphy]

Rindler says all that needs to be said, IMO.

They all do.

For $\gamma$ as a function of $v/c$, maybe that's all.
But $\gamma$ is so much more than that function of $v/c$.

Is this graph all that needs to be shown to help students understand Euclidean geometry?

 

[Trysse]

Thank you for your answers so far! I appreciate the feedback.

Since nobody has so far pointed this out, I have a question: Has anyone noticed that the scales in these graphs are distorted? If yes, what did you think about it? If no, what do you think about this now? Do you think it’s a problem?

What I mean by distorted is this: Both axes $\beta = v/c$ on the horizontal and $\gamma=1/\sqrt{1-v^2/c^2}$ on the vertical—are unitless quantities. So, both should use the same scale to faithfully represent the relation between $\beta$ and $\gamma$. However, in most textbook graphs, β is stretched (or equivalently, γ is flattened), which distorts the visual representation.

Because of this distortion, I don't like any of the diagrams. I only like the first part of Rahaman’s because it is undistorted and correctly shows the quarter-circle. However, in combination with his graph of $\gamma$, this is an especially bad example because his γ-graph looks as if it were just $\gamma^{-1}$ rotated by $90°$, which gives the completely wrong impression.

What do you think?

 

[weirdoguy]

I think it's not a problem.

So, both should use the same scale to faithfully represent the relation between β and γ.

Using different scales on both axes is as normal as it could get and it happens in literally every part of physics on every level of education.

 

[robphy]

Since nobody has so far pointed this out, I have a question: Has anyone noticed that the scales in these graphs are distorted? If yes, what did you think about it? If no, what do you think about this now? Do you think it’s a problem?

As @weirdoguy says, it's not a problem since you are just looking at a functional dependence,
which (as I was hinting above) is a very narrow (myoptic) view of special relativity,
where relativistic quantities are often regarded as "factors" or "corrections".

(in desmos, "t" is used for parametric functions)

(This is essentially my earlier graph of cosine as a function of slope (rather than angle).)


(This is essentially your graph of the gamma-factor as a function of velocity.)


In my opinion, one needs to see the geometry and trigonometry of spacetime,
just like one needs the circular trig functions to understand Euclidean geometry.
It is here where scaling should not be distorted (which is why one often uses "geometric units").

 

[Trysse]

I am not sure the distortion of the graphs poses no problem.

Consider the following question: What is the didactic intention of putting the graphs into the textbooks? I assume it is to build intuition in students. So the question is, what intuition do students get from a distorted graph?

If I don't think of $c$ as $1$ without units, the choice of scales seems to be arbitrary. On the vertical axis, there are no units and the values grow without bounds. On the horizontal axis, the units are $time/distance$ and go up to around $3.000.000.000 m/s$. My choice of the scales might be influenced by two goals: To make the graph fit nicely on a page and to support the narrative, that relativistic effects can be neglected at low relative speeds but somehow explode at speeds close to $c$.
However, with $\beta=v/c$, there are no units on both axes. The scales should not be arbitrary.

Imagine you have a regular cartesian coordinate system but the axes are not scaled equally. Would you say it doesn't matter if you wish to represent quadratic growth in such a system? Or would you rather prefer a system with equally scaled axes?

Is it necessary to exaggerate the narrative? Wouldn't it be helpful to show that relativistic effects also apply at low speeds (even if the effects can be neglected for most practical purposes)?

 

[weirdoguy]

So the question is, what intuition do students get from a distorted graph?

My 17 years of teaching physics at high-school and undergraduate level tells - correct intuition. Distorted graphs are used in all branches of science on a daily basis. And my advice would be to not overthink things that don't need to be overthought. There are tons of other things in relativity that you can put this energy in

 

[Trysse]

There might be other fields, but right now I am interested in this one.

So what is the intuition you want convey to your students? Or what do you think is the intuition they build?

What do you think students remeber 10 years after having learned about relativistic effects if Special Relativity has not become their major field of interest but they have not forgotten everything?

This question is not meant to dismiss your response. It is asked out of genuine interest.

 

[robphy]

There might be other fields, but right now I am interested in this one.

So what is the intuition you want convey to your students? Or what do you think is the intuition they build?

What do you think students remeber 10 years after having learned about relativistic effects if Special Relativity has not become their major field of interest but they have not forgotten everything?

This question is not meant to dismiss your response. It is asked out of genuine interest.

I'm not sure if the question is directed just to @weirdoguy ...
but I'll chime in with an answer....
which is what I wrote in post #8.

Appreciating and understanding the
spacetime geometry of the "position-vs-time" graph of special relativity
is the primary intuition I want to convey... because
the spacetime geometry encodes both
the numerical results and the physical interpretation of special relativity.
A plot of $\frac{1}{\sqrt{1-(v/c)^2}}$-vs-$(v/c)$ is one of the numerical results
but it does not encode the physical interpretation.

 

[weirdoguy]

but right now I am interested in this one.

And I, as a teacher, and also my students, are interested in everything, since physics is a very 'holistic' science. In Poland SR is tought in the very end of 4-year high school, and by then everyone has seen tons of graphs with different scalings. Frankly, it's the first time I see someone has a problem with this, so I don't necesarrily see at the moment how to respond other than that.

Also, intuition is not only built on graphs. It is built by solving tons of different problems. Graphs can help, but they are not the very basis.

 

[vela]

So what is the intuition you want convey to your students? Or what do you think is the intuition they build?

I'd like to know what intuition you want to convey to students with these plots. You're asserting that the "distorted" axes obscures some sort of intuition, but you never exactly say what this intuition is.

 

[Trysse]

@weirdoguy I think I got your point. However, I take it more as a personal position than an argument in this discussion.
Although the problem of distorted graphs is mostly discussed in the context of statistical data and not in the context of mathematical functions, I believe the same principle applies. The following quote supports your statement that students can handle distorted graphs (i.e. extract data from a graph), but it also points out, that students may not be aware of how a distorted graph can be misleading.

ALTHOUGH MOST MIDDLE SCHOOL STUDENTS HAVE learned how to read and extract information from various types of graphs, it might not be obvious to them how particular graphs can be misleading or inaccurate.
Regarding your comment:

And I, as a teacher, and also my students, are interested in everything, since physics is a very 'holistic' science. In Poland SR is tought in the very end of 4-year high school, and by then everyone has seen tons of graphs with different scalings. Frankly, it's the first time I see someone has a problem with this, so I don't necesarrily see at the moment how to respond other than that.

Also, intuition is not only built on graphs. It is built by solving tons of different problems. Graphs can help, but they are not the very basis.

This thread is not about physics in general, not even about special relativity in its entirety. It is about a very specific aspect of teaching special relativity: The question, of whether the relation between relative speed and relativistic effects is well represented by the graphs $\beta /\gamma$ as they can be found in many textbooks. So, you can put all your energy into all other areas of physics and special relativity which you find more rewarding. However, in this thread, this is the only thing that matters.
I agree with you, that intuition is not only built on graphs. However, if graphs are present, to some extent they influence intuition. So I would accept any arguments that graphs are not necessary and quite a number of textbooks come without graphs of $\gamma$. But if graphs are used, I think they should be well-designed.

That this is the first time, someone has brought this point to your attention should rather be an argument to consider this question at least once. You can of course just carry on as usual (and put your effort into arguing why this need not be considered).
For me personally, the distorted graphs pose no problem. I am neither teaching SR nor writing textbooks about it. To me, this is just a curiosity I like to think about. So I am in fact far less invested in this topic than you are as a teacher.

@robphy The question was aimed at everybody. So Thank you for your response.

As @weirdoguy says, it's not a problem since you are just looking at a functional dependence,
which (as I was hinting above) is a very narrow (myoptic) view of special relativity,
where relativistic quantities are often regarded as "factors" or "corrections".

But I see the graphs not so much as a representation of some functional (i.e. mathematical) dependence. I see them as the representation of some underlying fundamental relation between relative speed and the magnitude of relativistic effects. Of course, this relationship is expressed also in its mathematical formulation (i.e. the function). However, this relation encoded in the formulation of the function is not accessible to all (students) in the same way. This is why I think the undistorted representation matters.

I also don't see $\gamma$ as a correction factor. It does not correct, there is nothing wrong to correct. But this might be nit-picking on words.

@vela I hope I have answered your question already. In fact I don't want to convey any intuition, I am rather wondering what intuition distorted graphs instil in others (and reflect what intuition they had originally instilled in me). And of course I think what would be a better intuition.

I think the intuition that is usually conveyed with the above graphs, is that there is a distinction between low (non-relativistic) speeds and high (relativistic) speeds. However, I think it would be much better to convey an intuition, that relativistc effects continuously increase, that even if they are not detectable, they also apply to us. So not: "Look, here everything is fine and over there everything is behaving weird."

To give an obvious example: The below diagram shows graphs for $y=x$, $y=x^2$ and $1=x^2+y^2$. I think nobody would argue, that the graphical representation of these equations is helpful to give onlookers a good idea about what these equations mean.

Und just to show what we are talking about, the following picture shows a distorted and undistorted graph of $\gamma(\beta)$. The left graph tells the usual story: Nothing happens at low speed before $\gamma$ (suddenly) explodes (e.g. Rindler writes: Note its slow initial increase and its asymptote at v = c) . In the right graph, the increase appears more continuous and less abrupt. You may argue, that the left graph is better to read. In this case, I say that reading a value from a graph that can only be constructed by calculating some values and interpolating the graph is a futile exercise.

 

[Meir Achuz]

It's funny that some people prefer graphs, and others, like me, prefer equations.
There are many functions that look like those graphs. The important thing is the square root.

 

[PeroK]

It's funny that some people prefer graphs, and others, like me, prefer equations.
There are many functions that look like those graphs. The important thing is the square root.

I like that Rindler explicitly shows the vertical asymptote. Otherwise, we could be looking at the graph of an exponential.

 

[WWGD]

It's funny that some people prefer graphs, and others, like me, prefer equations.
There are many functions that look like those graphs. The important thing is the square root.

And that's music to Meir ;).

 

[BVirtual]

My take is physics students should be explicitly taught how to read graphs to not only hone their intuitive skills, but to be able to recognize when a graph has 'distorted' axis or axes, that might effect their intuition.

Not just in math class, with no physics. Explicit "physics" lessons in making a 'reality' physics curve/graph and then reading that curve in order to derive a "reality intuition." In many cases, it is not just the min, max, axis intercept points, but also elbows where the curvature becomes different, passing through 45 degrees, becoming much steeper, is an area of interest in physics, but not in geometry classes. That approaching light speed and going above 5% of the speed of light does not mean you need to change your equation to add relativistic complexity. When at 95% of c that is true. How does one read these Lorentz factor curves to see that type of intuition?

For example, such a *physics* lesson might be to compare a geometrical graph with the same curve on a logarithm graph, with paragraphs that point out the distortion, point out the differences in how one's 'physics" intuition is changed by each type of graph, and how to overcome this issue.

Why do I feel this way? Well, I had to learn this 'difference' by long exposure to it over many years in school. Just like I had to learn how to visualize how a graph would change when one of its many terms had a variable change. I wish someone in K12 had taught me a lesson of drawing graphs for multi term equations where one variable changed. Of course, such equations were not common in K12. I had to learn it my self education. Now, I am quite good at it. I would have liked this skill upon my college sophomore year as it would have made learning physics from a visualization aid easier.

Early explicit lessons in reading a "physics" graph, and making a "physics" graph, with multiple terms is what I think is needed. Not just single variable graphs, as those are just the first few pages of such a lesson. Most important is where are the "important" parts of the curve? Min and max, crossing the axes, yes were all taught, and the method of determining these important parts were taught. But that was pure math, with no physics and the reality intuition that must be learned from not just those parts, but also where the 'elbow' is, when does one have to take into account approaching light speed for contraction or for time dilation.

This entire thread is a good example of what would be included in such explicit lesson chapter of about 30 to 50 pages, with ever increasing complexity. I think such a lesson would increase a young adult's curiosity and desire to learn more physics. When does a car start skidding? And why? And how to correct it? With graphs of the equations of the friction, etc. Why does a pitched baseball curve? And how to throw it, and a speed ball? How to spin a ping pong ball so when it bounces on the other side, it does not go straight, but its spin takes it to one side. These things would have interested me.