Teaching Relativity in a College Physics course

[Herman Trivilino]

I teach a bit of special relativity to non-majors enrolled in the typical two-semester introductory college physics sequence. These are my goals.

1. Have the students develop an understanding of the basics such as length contraction, time dilation, relativity of simultaneity, energy, and momentum.

2. Spend only one week of class time.

You may immediately conclude that both goals cannot be acheived. I'm willing to budge a bit on either one, but if possible I'd like to go as far as I can towards acheiving both.

I have only two meetings during this week, so I start with a reading assignment and an online homework assignment, both of which are to be completed before the first meeting. The reading assignment is half of the chapter on relativity found in the typical College Physics textbook. (The other half of the chapter is the reading assignment for the week's second meeting). The online homework assignment consists of numerical calculations involving the relationships between $v$, $\beta$, and $\gamma$.
$$\beta=\frac{v}{c}$$
$$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$
During the first meeting I go over the stuff I discussed above in Goal 1. In the past I've not been happy with the outcome and I'm thinking that a bit of spacetime geometry may help. The reason I think that is because the geometry is another representation, and multiple representations lead, in my belief, to a deeper understanding. Or they at least offer an alternative way to understand. Anyone who's studied relativity knows how valuable spacetime diagrams are, and the textbooks I've used do not take advantage of that.

So, with that in mind I've written up what I plan to be the lesson for that second class meeting. It's the attached PDF. If you take the time to read it and have feedback I'd appreciate hearing from you. Thanks.

 

[Andrew Mason]

You do not appear to explain or even mention the fundamental principles/postulates of SR. You also do not deal much with inertial frames of reference. I am not sure why these are omitted. It is impossible to understand anything about SR if you don't explain that the speed of light is the same in all inertial frames of reference and that all inertial frames are equivalent i.e. the laws of physics are the same into all inertial observers. I don't know how you expect anyone to understand the proper time, time dilation and length contraction without first explaining the Lorentz transformation.

If I were teaching this I would start by explaining the results of Michelson Morley experiment and compare the approach of Lorentz and Fitzgerald to that of Einstein. Then I would explain how Einstein's approach - abandoning the concept of the ether - explains the Michelson Morley results. Then I would introduce space-time diagrams and the Lorentz transformations. This allows one to calculate time dilation and length contractions. If you can do that in one week, good luck!

AM

 

[Herman Trivilino]

If I were teaching this I would start by explaining the results of Michelson Morley experiment and compare the approach of Lorentz and Fitzgerald to that of Einstein. Then I would explain how Einstein's approach - abandoning the concept of the ether - explains the Michelson Morley results. Then I would introduce space-time diagrams and the Lorentz transformations. This allows one to calculate time dilation and length contractions. If you can do that in one week, good luck!

Much of that is addressed in both the reading assignment and the first lecture. That's the reason you don't see it in the attachment.

 

[nrqed]

I teach a bit of special relativity to non-majors enrolled in the typical two-semester introductory college physics sequence. These are my goals.

1. Have the students develop an understanding of the basics such as length contraction, time dilation, relativity of simultaneity, energy, and momentum.

2. Spend only one week of class time.

You may immediately conclude that both goals cannot be acheived. I'm willing to budge a bit on either one, but if possible I'd like to go as far as I can towards acheiving both.

I have only two meetings during this week, so I start with a reading assignment and an online homework assignment, both of which are to be completed before the first meeting. The reading assignment is half of the chapter on relativity found in the typical College Physics textbook. (The other half of the chapter is the reading assignment for the week's second meeting). The online homework assignment consists of numerical calculations involving the relationships between $v$, $\beta$, and $\gamma$.
$$\beta=\frac{v}{c}$$
$$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$
During the first meeting I go over the stuff I discussed above in Goal 1. In the past I've not been happy with the outcome and I'm thinking that a bit of spacetime geometry may help. The reason I think that is because the geometry is another representation, and multiple representations lead, in my belief, to a deeper understanding. Or they at least offer an alternative way to understand. Anyone who's studied relativity knows how valuable spacetime diagrams are, and the textbooks I've used do not take advantage of that.

So, with that in mind I've written up what I plan to be the lesson for that second class meeting. It's the attached PDF. If you take the time to read it and have feedback I'd appreciate hearing from you. Thanks.

Good job. But I do not understand the nee to introduce units with c=1 in that context. I don't see the benefit and I see a lot of possible confusion. Those units are just useful for research scientists who prefer not to have to remember where to put factors of c but it is overkill and just confusing for first time students. I would get rid of that and include all factors of c.

Also, in the muon example I think it is absolutely necessary to emphasize the fact that $\Delta x'=0$ (with primed coordinates attached to the muon). My students are always puzzled at first, even though it is obvious to us.

But like the other poster said, I think it is important to show that the invariance of the spacetime interval follows from imposing that the speed of light is the same in all frames.

 

[Andrew Mason]

It might help if you provided us with your plan for the first lecture. It is very odd that you don't relate anything in the second lecture to the first. Frankly, I found your approach very unclear and confusing. Your explanation of speed as minutes/minute and time in minutes is very odd and confusing. I would also suggest using space-time diagrams to show the way the axes (of distance and time in other inertial frames) are transformed using the Lorentz transformations.

AM

 

[vela]

I suggest you look at chapter 1 of Spacetime Physics by Taylor and Wheeler. I just covered relativity in my class trying a spacetime geometry approach, and there's a lot of blanks to fill in for the students. The students also have their textbook, which uses the more typical approach that doesn't really mention geometry, so they got to see the theory from two perspectives.

I don't have a problem with $c=1$ for the geometrical approach. Typically, if the Lorentz transformations are mentioned at all in the usual textbooks, they are written as
\begin{align*}
t' &= \gamma(t - \frac{vx}{c^2}) \\
x' &= \gamma(x - vt)
\end{align*} which makes it look like $t$ and $x$ are on different footing. In contrast, when written as
\begin{align*}
t' &= \gamma(t - \beta x) \\
x' &= \gamma(x - \beta t)
\end{align*} where $t$ is measured in units of length, the symmetry between $x$ and $t$ is readily apparent. It's not terribly difficult for students to understand that when you say t = 1 m, you're talking about the time that it takes light to travel 1 meter. It's really no different than when someone says "Disneylan is an hour away." People know that an hour isn't literally a distance, but they understand it's the distance that corresponds to traveling for that amount of time.

 

[nrqed]

I suggest you look at chapter 1 of Spacetime Physics by Taylor and Wheeler. I just covered relativity in my class trying a spacetime geometry approach, and there's a lot of blanks to fill in for the students. The students also have their textbook, which uses the more typical approach that doesn't really mention geometry, so they got to see the theory from two perspectives.

I don't have a problem with $c=1$ for the geometrical approach. Typically, if the Lorentz transformations are mentioned at all in the usual textbooks, they are written as
\begin{align*}
t' &= \gamma(t - \frac{vx}{c^2}) \\
x' &= \gamma(x - vt)
\end{align*} which makes it look like $t$ and $x$ are on different footing. In contrast, when written as
\begin{align*}
t' &= \gamma(t - \beta x) \\
x' &= \gamma(x - \beta t)
\end{align*} where $t$ is measured in units of length, the symmetry between $x$ and $t$ is readily apparent.

I still do not see the advantage, at the introductory level. There are enough subtleties to grasp without adding an extra layer which does not serve much purpose, it seems to me (and time and space really are on different footing, as the difference of sign in the metric makes clear).

In any case, even setting c=1 in the equations does not dispense totally from needing to use c. If the muon's lifetime is give as 2.2 microseconds, the students will then have to covert that to a distance using c before plugging the result in the equations. If I had only one work or so to teach relativity, I would not want to spend some time discussing this layer of complication that does not bring in much insight, I think (and I know that from *our* point of view it is easy and useful, but from *their* point of view, I don't see the advantages.) To me, it is more pedagogically sound to try to introduce as few new concepts a at a time, especially when they are extremely counterintuitive. I think it is challenging enough to explain relativity or simultaneity, time dilation, etc without adding what seems an unnecessary layer at that level.

Just my opinion, of course.

 

[Herman Trivilino]

It might help if you provided us with your plan for the first lecture. It is very odd that you don't relate anything in the second lecture to the first.

The first post gives an overview of what's covered not only in the first lecture, but more importantly what the students are exposed to in the way of the reading assignment and the homework assignment. You can look, for example, at the OpenStax College Physics book by downloading the PDF for free.

One of the things covered there is time dilation, and that is referred to in the second lecture. Another is the relation between $\beta$ and $\gamma$.

The second lecture focuses on proper time and time dilation.

 

[Herman Trivilino]

Good job. But I do not understand the nee to introduce units with c=1 in that context. I don't see the benefit and I see a lot of possible confusion. Those units are just useful for research scientists who prefer not to have to remember where to put factors of c but it is overkill and just confusing for first time students. I would get rid of that and include all factors of c.

Thanks. The reason for using units where c = 1 is so that the geometry is clear. Otherwise it's obscured. I believe that's the reason Vela mentioned Taylor and Wheeler's book. It's not clear from the opening paragraph in my attachment, but I plan to spend time in class expanding on the notion that the invariance of the length of line segment in 2-d Euclidean space would be obscured if each dimension were measured in different units.

Also, in the muon example I think it is absolutely necessary to emphasize the fact that $\Delta x'=0$ (with primed coordinates attached to the muon). My students are always puzzled at first, even though it is obvious to us.

Yes, I spent some time thinking about that. I decided I'd see if I could get by without ever introducing x' and t'. It may not work, but if I want to meet Goal 2 (spend one week) I have to sacrifice something. I won't introduce invariance of the interval except insofar as to use proper time as the length of timelike intervals. Spacelike intervals are another thing that I sacrifice.

But like the other poster said, I think it is important to show that the invariance of the spacetime interval follows from imposing that the speed of light is the same in all frames.

I will definitely stress in both lectures that everything is a consequence of the two postulates. That is also something stressed in the reading assignments in the textbook, too.

My philosophy of teaching is to not necessarily be the first presenter of everything. Some things are presented to the student in reading assignments and homework assignments. I ask myself what is the best way to have a topic presented, and make my choice of methods based on that.

 

[Herman Trivilino]

I suggest you look at chapter 1 of Spacetime Physics by Taylor and Wheeler. I just covered relativity in my class trying a spacetime geometry approach, and there's a lot of blanks to fill in for the students.

That's what I'm wanting to try, too. If I can get students to understand time dilation and the twin paradox I'll be happy. Well, happier, anyway.

I don't know if you're familiar with Paul Hewitt's old cartoon video called Time Dilation. It's really well done, and makes use of the Doppler effect to demonstrate the twin paradox. It matches the treatment in his textbook. The numbers in my lesson match the ones Hewitt uses, so the students will see both treatments (Doppler and spacetime geometry).

 

[Andrew Mason]

The first post gives an overview of what's covered not only in the first lecture, but more importantly what the students are exposed to in the way of the reading assignment and the homework assignment. You can look, for example, at the OpenStax College Physics book by downloading the PDF for free.

One of the things covered there is time dilation, and that is referred to in the second lecture. Another is the relation between $\beta$ and $\gamma$.

The second lecture focuses on proper time and time dilation.

My concern about this approach is that in a one-week introduction to SR you are going to leave students confused unless you relate the space-time geometry to the underlying physics. The essential element is the constancy of the speed of light for all inertial observers.

Consider the Einstein light train thought experiment where a light signal from the origin occurs when the tail of the moving train coincides with the origin in the rest frame at time t=0 for observers A and A' in the respective frames. There is a mirror at the nose of the train and mirror at x = L in the rest frame, where L is the proper length of the train. A measures the time the signal reflected from his stationary mirror at x=L reached him as $t_0 = 2L/c$. The A' observer measures the same time for the signal to reach him: $t_{train} = 2L/c$.

But to the stationary observer, it took longer than L/c to reach the mirror and less than L/c to reflect back to A'. This is because A sees the light traveling farther than L in order to reach the mirror at the nose of the train and shorter than L to return. Working out the actual time difference is not trivial. It involves derivation of the Lorentz transformation. Einstein took several pages to show this in his 1905 paper.

The reason $(\Delta t_0)^2 = (\Delta t)^2 + (\Delta x)^2$ (as you choose to write it with units of x being units of time rather than distance) is not easy to see. Yet you make it look so simple that the students are going to think they are stupid if they don't see it. And that is not a good thing from a pedagogical point of view.

But the reason there must be a time difference is not difficult to see. And that is probably what the students are going to remember about special relativity.

AM

 

[nrqed]

Thanks. The reason for using units where c = 1 is so that the geometry is clear. Otherwise it's obscured. I believe that's the reason Vela mentioned Taylor and Wheeler's book. It's not clear from the opening paragraph in my attachment, but I plan to spend time in class expanding on the notion that the invariance of the length of line segment in 2-d Euclidean space would be obscured if each dimension were measured in different units. [/tex]

But all terms have the same units, of course, even if we keep c not equal to 1. Then there is $(c \Delta t)^2$ in the square of the spacetime interval which clearly has the same units as $(\Delta x)^2$. I still don't see the advantage of having c=1. With that convention, two things happen: A) when students see $(\Delta x)^2 + (\Delta t)^2$, they have to remember that this weird system of units is used in order to see that the two terms have the same dimension and B) every time they have a time given in a problem, they cannot leave it in seconds in the equations, they first have to convert it to meters before being able to use the equations. For people who will become physicists, the c=1 system is useful because it simplifies equations but I still don't see at all how this can possibly help people who will not do a career in physics understand better relativity.

Yes, I spent some time thinking about that. I decided I'd see if I could get by without ever introducing x' and t'. It may not work, but if I want to meet Goal 2 (spend one week) I have to sacrifice something. I won't introduce invariance of the interval except insofar as to use proper time as the length of timelike intervals. Spacelike intervals are another thing that I sacrifice.

I personally feel that it is absolutely necessary to explain that $\Delta x' =0$. I have had students learning the time dilation or length contraction formula without understanding that these formula are not always valid! Then, whenever they see a $\Delta t'$ given in a problem, they assume that $\Delta t = \gamma \Delta t'$ which if of course not always correct. And when this is pointed out to them, they get all confused if they haven't learned the proper use of the equations the first time around.
If you introduce the concept of spacetime interval, why not calculate it in S' and then use the fact that it must be the same in S? It seems to me that all the work that went into introducing the spacetime interval is lost if it is not fully used in all problems.

But again, that's just my humble opinion.

 

[Orodruin]

I personally feel that it is absolutely necessary to explain that Δx′=0. I have had students learning the time dilation or length contraction formula without understanding that these formula are not always valid!

The sad part here is that even in higher relativity courses, students get this wrong because they are used to it always applying in the problems they have seen so far in earlier courses. Even if you are discussing the derivation of the length contraction formula in detail and warn about its use for objects which are not moving at constant velocity, you can safely bet on that someone (most) will try to apply it when you ask them to figure out what is going on in Bell's spaceship paradox.

The $\Delta x' = 0$ part is really difficult to get out of students preconceptions. In particular, this is not made easier by some textbooks in modern physics making statements such as "the muon travels 600 m in its rest frame" ...

 

[nrqed]

The sad part here is that even in higher relativity courses, students get this wrong because they are used to it always applying in the problems they have seen so far in earlier courses. Even if you are discussing the derivation of the length contraction formula in detail and warn about its use for objects which are not moving at constant velocity, you can safely bet on that someone (most) will try to apply it when you ask them to figure out what is going on in Bell's spaceship paradox.

The $\Delta x' = 0$ part is really difficult to get out of students preconceptions. In particular, this is not made easier by some textbooks in modern physics making statements such as "the muon travels 600 m in its rest frame" ...

Yes, this last quote is indeed terrible!

I always do a few examples where neither of the $\Delta x, \Delta x', \Delta t, \Delta t'$ are zero (for example, Joe throws a ball to Jack, with both of them in S', and I have y students find the time and space intervals in S, and make them check that the ball's velocity in S is indeed given by the relativistic velocity addition law. I think it is not pedagogically sound to only do examples where at least one of the intervals is zero as it gives the impression that some special formula are always valid (like $\Delta t = \gamma \Delta t'$).

 

[Herman Trivilino]

Thanks to all for taking the time to contribute. I really appreciate it, and I'm still actively working on this project. Finals week and its approach have kept me busy with less desirable tasks (I love teaching, hate grading, but you can't have one without the other).

You've given me lots of stuff to think about and read about.

My concern about this approach is that in a one-week introduction to SR you are going to leave students confused unless you relate the space-time geometry to the underlying physics. The essential element is the constancy of the speed of light for all inertial observers.

That's a very good point. The textbook reading assignments and my first lecture specify how the 2nd Postulate fits into the topics of Goal 1, but I would like to find a way to fold it into the lesson on spacetime geometry. Any suggestions on how to do this given the constraints of Goal 2?

The reason $(\Delta t_0)^2 = (\Delta t)^2 + (\Delta x)^2$ (as you choose to write it with units of x being units of time rather than distance) is not easy to see. Yet you make it look so simple that the students are going to think they are stupid if they don't see it. And that is not a good thing from a pedagogical point of view.

I'm sure you meant to write $(\Delta t_0)^2 = (\Delta t)^2 - (\Delta x)^2$. Your point is very well put. I need to address this issue. But how to do it? I'm thinking ...

I do not plan to introduce a primed coordinate system. I realize the importance of stressing that $\Delta x'=0$ is a necessary condition for $\Delta t'$ to be a proper time. But the point has been well made in this thread that the only way to address that issue is to effectively communicate situations where there is no proper time in either of the frames being considered (for a timelike separation of two events). The way I'm doing it is to simply stress that the reason it's a proper time in this particular frame is because the one clock being used to measure the time interval is present at both events.

The vast majority of students taking introductory physics in colleges in the US are enrolled in a sequence of courses that are labelled either "calculus-based" or "trigonometry-based". The lesson being discussed here is for the latter. I have been reviewing textbooks used in both sequences and almost all of them never consider two frames where there is neither a proper time nor a proper length in one of them. The section in the chapter devoted to "relativistic addition of velocities" is the only situation that's the exception, and even then they don't explicitly refer to a primed coordinate system.

 

[Herman Trivilino]

But all terms have the same units, of course, even if we keep c not equal to 1. Then there is $(c \Delta t)^2$ in the square of the spacetime interval which clearly has the same units as $(\Delta x)^2$. I still don't see the advantage of having c=1.

Every time I have taught the course I've regretted not teaching the students about the simplicity of a system where c=1. There are many encounters in this course where it's an advantage to analyze a situation using units like light years and years. Almost every textbook I looked at works at least one example this way.

N. David Mermin uses the phoot and the nanosecond. (The phoot is the distance light travels in a nanosecond, equal to 0.299 792 458 m, which is less than 2% different from the foot, which is 0.3048 m). As has already been mentioned in this thread Taylor and Wheeler use meters of time.

I do agree that the discussion of minutes of distance in my lesson may be confusing. I'll see what I can do to clean it up. The issue I ran into when writing it is the following. I want students to more easily be able to switch back and forth between units where c=1 and SI units. One way to do that is to use $\beta$ instead of $v$ in the equations. But if you do that you can't use, for example, years and light years. $v$ would be measured in light years per year, $c$ would be 1 light year per year, and you have to use $v=\beta c$ to get the units to come out right. Maybe this concern is overly pedantic, and if I can convince myself of that it may help me write that discussion of minutes of distance in a way that's less confusing.

 

[nrqed]

Every time I have taught the course I've regretted not teaching the students about the simplicity of a system where c=1. There are many encounters in this course where it's an advantage to analyze a situation using units like light years and years. Almost every textbook I looked at works at least one example this way.

N. David Mermin uses the phoot and the nanosecond. (The phoot is the distance light travels in a nanosecond, equal to 0.299 792 458 m, which is less than 2% different from the foot, which is 0.3048 m). As has already been mentioned in this thread Taylor and Wheeler use meters of time.

I do agree that the discussion of minutes of distance in my lesson may be confusing. I'll see what I can do to clean it up. The issue I ran into when writing it is the following. I want students to more easily be able to switch back and forth between units where c=1 and SI units. One way to do that is to use $\beta$ instead of $v$ in the equations. But if you do that you can't use, for example, years and light years. $v$ would be measured in light years per year, $c$ would be 1 light year per year, and you have to use $v=\beta c$ to get the units to come out right. Maybe this concern is overly pedantic, and if I can convince myself of that it may help me write that discussion of minutes of distance in a way that's less confusing.

I also think that working in light years (and in years) is good for many problems (when talking about traveling to other solar systems for example). But I still don't see the need for setting c=1 (sorry if I sound like a broken record). All I do is to tell my students that by definition, 1 ly is the distance traveled at the speed of light in one year, so that although we could in principle convert it to meters, it is much simpler to write instead 1 ly = 1 year x c . So, for example if a ratio of a distance over the speed of light enters a calculation, we directly get that 1 ly/c = 1 yearc/c = 1 year. If we divide, say, 20 ly/40 years, we directly get half the speed of light, c/2.

Everything works well and the units confirm that the calculation was done correctly, so the presence of units is helpful as a mean to reassure the students that their manipulations are probably correct. We don't care about units normally but when encountering these new strange formula with weird physical results, I think that having the units shown explicitly is an added help.

I will stop arguing my point since I am the only one thinking this way but one final thought: with c=1, sometimes results that have no dimensions are really meant to be a speed whereas other numbers that are dimensionless are truly dimensionless. I know that for us, the teachers, it is trivial to tell when a factor of 1/2 somewhere truly means 1/2 c and a factor of 1/2 somewhere else truly means 1/2, we just see what is supposed to be a speed and what is supposed to be a pure number. I don't think that students should have to even think about this while they are struggling to understand something as counterintuitive as SR, but that's just me.

 

[Orodruin]

But I still don't see the need for setting c=1 (sorry if I sound like a broken record).

You do not need to set c = 1, but it highlights the structure and symmetry of the theory. Furthermore, as space-time is a manifold, it is somewhat artificial to measure different directions in different units, in particular when doing coordinate transformations. Imagine living on the sea. There are then two directions you probably wish to measure in relatively large units, such as kilometers, and one direction you will probably use smaller units for. This is of course not wrong, but it does not highlight the fact that the spatial directions are equivalent (simply because for you the presence of the sea seemingly breaks the equivalence). Let us say we measure lengths parallel to the surface in km and depth in feet (to underline the fact that these units a priori are independent). The trajectory along which something moves underwater then has a rate of descent measured in the unit 1 foot/km which is a universal constant. With the introduction of Minkowski space-time which underlines the symmetry between space and time, c is very similar to the 1 foot/km in the above example. Sure, it may be convenient to work in such units, but it is somewhat artificial.

 

[Herman Trivilino]

I also think that working in light years (and in years) is good for many problems (when talking about traveling to other solar systems for example). But I still don't see the need for setting c=1 (sorry if I sound like a broken record).

I appreciate you highlighting the issue. So you see c = 1 to be different from c = 1 ly/y, which of course they are. This is the issue I referred to above that I was thinking about when writing the lesson.

I know that for us, the teachers, it is trivial to tell when a factor of 1/2 somewhere truly means 1/2 c and a factor of 1/2 somewhere else truly means 1/2, we just see what is supposed to be a speed and what is supposed to be a pure number.

This is analogous to saying the slope of a line in Euclidean 2-space is a dimensionless number. If you were in the habit of using a system where x and y were measured in different units you'd run into the same issue. The foundation of spacetime geometry is that the relationship between space and time is a slope.

 

[nrqed]

I appreciate you highlighting the issue. So you see c = 1 to be different from c = 1 ly/y, which of course they are. This is the issue I referred to above that I was thinking about when writing the lesson.
This is analogous to saying the slope of a line in Euclidean 2-space is a dimensionless number. If you were in the habit of using a system where x and y were measured in different units you'd run into the same issue. The foundation of spacetime geometry is that the relationship between space and time is a slope.

But I think of the coordinates as being $\Delta x$ and $c \Delta t$, which have the same units. It is because you insist on using $\Delta x$ and $\Delta t$ that you find yourself forces to introduce strange units (from the point of view of students) with c=1. But I won't insist :-). Good luck with your classes, it is a challenging topic to teach!

 

[Andrew Mason]

I appreciate you highlighting the issue. So you see c = 1 to be different from c = 1 ly/y, which of course they are. This is the issue I referred to above that I was thinking about when writing the lesson.

I don't follow this. c can't be equal to a dimensionless "1". It has units of distance/time. So if the magnitude of c is equal to 1, it is implicit that the unit of distance must be equal to the distance that light travels in a unit of time. So c = 1 ly/y or c = 1 phoot/nanosecond.

This is analogous to saying the slope of a line in Euclidean 2-space is a dimensionless number. If you were in the habit of using a system where x and y were measured in different units you'd run into the same issue. The foundation of spacetime geometry is that the relationship between space and time is a slope.

This suggests that time and space are physically the same. They aren't. Time is measured in units of time and space is measured in units of distance.

But I think of the coordinates as being $\Delta x$ and $c \Delta t$, which have the same units. It is because you insist on using $\Delta x$ and $\Delta t$ that you find yourself forces to introduce strange units (from the point of view of students) with c=1.

I agree.

AM

 

[Orodruin]

This suggests that time and space are physically the same.

Which they are! This is a very fundamental insight in relativity! One observer's time direction is a time and space direction (although time-like) for another!

 

[nrqed]

Which they are! This is a very fundamental insight in relativity! One observer's time direction is a time and space direction (although time-like) for another!

Then what do we get if we add 5 minutes to 2 meters? :-)

I would rather say that a fundamental insight of relativity is that the speed of light is an invariant, which then implies that $c\Delta t$ is closely related to $\Delta x$, and therefore a value of $c \Delta t$ in one frame shows up as a combination of $c \Delta t'$ and $\Delta x'$ in another frame. I really don't see why it is necessary to have c=1 to be able to appreciate all this. I understand that it simplifies calculations but setting c=1 is only a special choice of units (and it is actually cheating to say that c=1 since what we really mean is that we choose units of distance "uod" and of time "uot" such that c= 1uod/uot and we lazily drop the units). And a choice of units has absolutely no fundamental meaning! I think that here it just obscures the essential role of c in making it disappear from the equations.

 

[Orodruin]

You would not add something so small as 5 m to 2 s. One of them is about 8 orders of magnitude larger than the other ...

I really don't see why it is necessary to have c=1 to be able to appreciate all this.

The introduction of c only acts to obscure the symmetry of the theory and the fact that space and time are really only two sides of the same coin.

and it is actually cheating to say that c=1 since what we really mean is that we choose units of distance "uod" and of time "uot" such that c= 1uod/uot and we lazily drop the units

No, this is just plain wrong. It is as valid to consider units of time and units of distance to be equivalent, just as units of depth and units along the ocean surface. There is absolutely no difference here. In both cases, the directions are related by a coordinate transformation and all you are doing when introducing c is using different units for different coordinates. This does not affect the manifold itself. It has absolutely nothing to do with laziness and everything to do with treating space and time on an equal footing - which should be done because they are really only different directions on the same manifold. It is not only that what shows up as time for one observer shows up as time and space for another. It is that they are fundamentally intertwined and that there is absolutely no way of separating them in a physically unambiguous way.

Regardless of what you think of it, this is the preferred modus operandi for the large majority of physicists. If you dislike the use of t when doing this, introduce the new time coordinate $x^0 = ct$. The c does not even help you in dimensional analysis as it always comes along with the t (or dt). The only thing you can get out of it is whether or not you introduced enough powers of c, which is irrelevant if c = 1.

 

[Andrew Mason]

Which they are! This is a very fundamental insight in relativity! One observer's time direction is a time and space direction (although time-like) for another!

The fundamental insight into relativity is that space and time are related. While an interval of time in one inertial reference frame may appear to be an interval of both space and time in other inertial frames of reference, space and time are fundamentally different physical concepts. An interval of time in one inertial reference frame cannot be seen as an interval of space in another. An interval of space (i.e distance between two simultaneous events) measured by one observer cannot be seen as an interval of time by another.

This same kind of confusion arises when talking about mass and energy 'equivalence'. Mass and energy are very different concepts. Mass may be related to energy content of a body, but mass and energy are very different concepts. with very different units and very different physical attributes.

AM

 

[Herman Trivilino]

I don't follow this. c can't be equal to a dimensionless "1". It has units of distance/time. So if the magnitude of c is equal to 1, it is implicit that the unit of distance must be equal to the distance that light travels in a unit of time. So c = 1 ly/y or c = 1 phoot/nanosecond.

If you have a speed of 0.5 ly/y, then $v=0.5 \ \mathrm{ly/y}$, and $\beta=\frac{v}{c}=\frac{0.5 \ \mathrm{ly/y}}{1 \ \mathrm{ly/y}}=0.5$. Thus $\beta$ is dimensionless but $v$ is not.

If you have a speed of 0.5 m/m, then $v=0.5$, and $\beta=\frac{v}{c}=\frac{0.5}{1}=0.5$ in which case both $\beta$ and $v$ are dimensionless.

This is what I meant when I mentioned being (possibly) overly pedantic and why the wording in the lesson gets bogged down over the definition of, e.g. minutes of distance.

This suggests that time and space are physically the same.

In Euclidean geometry it would because all dimensions are dimensions of space. But in spacetime geometry it doesn't.

 

[Herman Trivilino]

An interval of time in one inertial reference frame cannot be seen as an interval of space in another. An interval of space (i.e distance between two simultaneous events) measured by one observer cannot be seen as an interval of time by another.

In Euclidean geometry a vertical distance measured by one observer can never be seen as a horizontal distance by another observer, but they are nevertheless measured in the same units.

This same kind of confusion arises when talking about mass and energy 'equivalence'. Mass and energy are very different concepts. Mass may be related to energy content of a body, but mass and energy are very different concepts. with very different units and very different physical attributes.

Mass is just a form of energy, called rest energy. There is no need, other than convention and the utility associated with the fact that they have separate conservation laws in the Newtonian approximation, for example, to measure them in different units.

 

[PeroK]

It seems to me that space and time are inextricably linked in classical physics, in the sense that you can't have one without the other. The original spacetime diagram was when someone first plotted $x$ against $t$ on a graph to describe the motion of a particle! The invariant between inertial reference frames would be $\Delta x^2 + \Delta t^2$. And, in a way, you could interpret this as space and time being "the same": just different coordinates in a 4D Euclidean spacetime. But, this was so simple that no one noticed. Or argued about it!

One could argue that SR made time and space more different from each other, because time is the odd man out in the non-Euclidean flat spacetime metric.

 

[nrqed]

In Euclidean geometry a vertical distance measured by one observer can never be seen as a horizontal distance by another observer, but they are nevertheless measured in the same units.

.

Of course a vertical distance measured by one observer may be seen as a horizontal distance by another observer, they just have to use coordinates systems rotated relative to one another!

 

[nrqed]

You would not add something so small as 5 m to 2 s. One of them is about 8 orders of magnitude larger than the other ...

The introduction of c only acts to obscure the symmetry of the theory and the fact that space and time are really only two sides of the same coin.

No, this is just plain wrong.

Then explain something to me: someone comes along and chooses to pick units such that c=2. You will say that this person is wrong, I guess. But based on what?? Based on the fact that then you do not find the equations pretty enough?

For me, the key point is that space and time are intimately linked through the fact that there is a spacetime invariant that mixes $\Delta t$ and $\Delta x$. I don't see how it matters what coefficient appears in the relation. You say that writing $\Delta s^2 = \Delta x^2 - \Delta t^2$ is the only correct way to write the equation and that someone who would use instead units such that $\Delta s^2 = \Delta x^2 - 2 \Delta t^2$ would of course be absolutely wrong. As far as I can tell, the difference is just that the number 1 is supposed to be prettier than the number 2. That has nothing to do with physics, as far as I am concerned but maybe I misunderstand who profound the number "1" is. But since it seems that I must surely be completely wrong, I will not argue anymore.

 

[Herman Trivilino]

Of course a vertical distance measured by one observer may be seen as a horizontal distance by another observer, they just have to use coordinates systems rotated relative to one another!

That won't work because the two directions are not equivalent. In one direction a particle accelerates, in the other it doesn't. You'll never a get a plumb bob, for example, to hang in a horizontal direction. If you were to use different units to measure distances in these two directions it would emphasize this difference, but that's not necessarily a good reason for doing so.

Likewise, you'll never get a clock to measure distance and you'll never get a meter stick to measure time. If you were to use different units to measure intervals in these two quantities it would emphasize this difference, but that's not necessarily a good reason for doing so.

 

[nrqed]

That won't work because the two directions are not equivalent. In one direction a particle accelerates, in the other it doesn't. You'll never a get a plumb bob, for example, to hang in a horizontal direction. If you were to use different units to measure distances in these two directions it would emphasize this difference, but that's not necessarily a good reason for doing so.

Likewise, you'll never get a clock to measure distance and you'll never get a meter stick to measure time. If you were to use different units to measure intervals in these two quantities it would emphasize this difference, but that's not necessarily a good reason for doing so.

Ah you are not working in an isolated system. I was talking about an isolated system, you are assuming we are near the surface of the Earth, in which case of course there is a distinction. The post I was responding to was saying that in Euclidian geometry, a horizontal distance in one frame could never be seen as a vertical distance in another frame, by horizontal and vertical I though he/she meant x vs y, I did not realize what was meant was in a non isolated system.

But, in special relativity, a clock *can* in principle be used to measure distances and a clock can be used to measure time, because of the invariance of the speed of light. I agree with this and I thought this was everyone's point behind saying that "time and space are on equal footing". My only objection is that the value c=1 is just a random choice and has no deep physical meaning.

 

[nrqed]

No, this is just plain wrong. .

So you said that I am wrong when I say that there is a choice of units made when wet set c=1 (my point was that we really mean c = 1 unit of distance/1 unit of time).

Well, I must then have to relearn everything about physics. I thought that someone could use two synchronized clock at rest in an inertial frame, send a light signal from one to the other and use that to measure the speed of light. I thought that one would define the speed of light as being the distance between the two clocks divided by the time interval measured by the clocks. Since, I thought, the choice of units used for the distance is arbitrary and since the choice of unit of time is, I believed, arbitrary, I thought that the speed of light obtained could take any possible value (well, larger than zero), depending on the choice of units of distance and of time.

What I learned here is that either it is physically inconsistent to define the speed of light as distance over time (because the only correct value of c is 1!) OR one is not allowed to pick distance or time units as one desires.

I will have to go back to basics!

 

[Herman Trivilino]

Ah you are not working in an isolated system. I was talking about an isolated system, you are assuming we are near the surface of the Earth, in which case of course there is a distinction. The post I was responding to was saying that in Euclidian geometry, a horizontal distance in one frame could never be seen as a vertical distance in another frame, by horizontal and vertical I though he/she meant x vs y, I did not realize what was meant was in a non isolated system.

There is no way to define vertical and horizontal in such an isolated system. You need the presence of gravity in one direction (vertical) and an absence of gravity in the other (horizontal) to define the directions.

Consider any vertical plane and draw in it a diagonal line segment. That line has a vertical component and a horizontal component that lie in that plane. You can increase one at the expense of decreasing the other, and vice-versa, by rotating the line segment in that plane. Moreover you can show that the vertical and horizontal components can be combined in such a way that the length of the line segment remains the same regardless of how it's oriented in that plane. You can see how much harder it would be to demonstrate this invariance of length if distances in the vertical direction were measured in units that differ from the units used to measure distance in the horizontal direction.

Choosing different units for lengths in each direction wouldn't be "wrong" but it would "obscure" the geometrical relationship.

 

[Orodruin]

I thought that one would define the speed of light as being the distance between the two clocks divided by the time interval measured by the clocks.

This is wrong. The meter is defined as 1/299792458 s (which happens to be a very suitable unit for measuring the spatial size of many things), you never measure the speed of light.

What I learned here is that either it is physically inconsistent to define the speed of light as distance over time (because the only correct value of c is 1!) OR one is not allowed to pick distance or time units as one desires.

We never said it is wrong, it is just obscuring the geometry. By introducing and using ct everywhere you are essentially doing the same, you are just calling your time variable with a longer and more cumbersome name. And of course you can pick any units you like, just as you can chose to measure one spatial direction in feet and the other in light years. It just obstructs the symmetry and make the coordinate transformations awkward.

Then explain something to me: someone comes along and chooses to pick units such that c=2. You will say that this person is wrong, I guess. But based on what?? Based on the fact that then you do not find the equations pretty enough?

This is the same thing as selecting a non-normalised coordinate system in a Euclidean space. Of course you can do that, but introductory courses generally only deal with Minkowski coordinates, which are the equivalent of using a normalised Cartesian coordinate system in the Euclidean space.

This same kind of confusion arises when talking about mass and energy 'equivalence'. Mass and energy are very different concepts. Mass may be related to energy content of a body, but mass and energy are very different concepts. with very different units and very different physical attributes.

This is again a confusion. Mass and energy are very similar concepts, with mass simply being a measure of an objects rest energy. In relativity, it is very easy to see that the rest energy is also the inertia of the object in its rest frame. There is no other concept of mass in SR, the inertia of a moving object is a quantity that depends on the direction of acceleration.

 

[Andrew Mason]

This is wrong. The meter is defined as 1/299792458 s (which happens to be a very suitable unit for measuring the spatial size of many things), you never measure the speed of light.

1 m = ct where c = speed of light in a vacuum and t = (1/299792458) seconds. Since a second is defined as 9,192,631,770 periods of the radiation (one period = T) emitted by caesium-133 in the transition between the two hyperfine levels of the atom's ground state, with the atom at Earth sea-level and at rest (ie. at 0K), the definition of 1 m is not time, but is really a distance:

1 m = cT(9,192,631,770/299,792,458) = 30.66331898849837 times the wavelength of this radiation from the caesium-133 atom.

We never said it is wrong, it is just obscuring the geometry. By introducing and using ct everywhere you are essentially doing the same, you are just calling your time variable with a longer and more cumbersome name. And of course you can pick any units you like, just as you can chose to measure one spatial direction in feet and the other in light years. It just obstructs the symmetry and make the coordinate transformations awkward.

One could argue that by making the geometry pretty you are obscuring the physics. Every inertial observer has his own coordinate system where time and space are distinct physical quantities. Time is measured by clocks and distance is measured by the separation of the end points of sticks. The distinction between time and space is always maintained for each inertial observer.

It is just that they are not absolute: different inertial observers will disagree on time and space measurements between events because they disagree on simultanaeity of events.

This is again a confusion. Mass and energy are very similar concepts, with mass simply being a measure of an objects rest energy. In relativity, it is very easy to see that the rest energy is also the inertia of the object in its rest frame.

$m = E/c^2$ means that $m \propto E$. By setting units for c = 1 that still does not make m and E the same physical phenomena. That fact that mass or inertia can be converted into energy does not make them equal. Otherwise there would be no meaning to the "conversion".

There is no other concept of mass in SR, the inertia of a moving object is a quantity that depends on the direction of acceleration

?? This again is confusing. There is the concept of inertia and the concept of rest mass. Rest mass is constant for all observers. Inertia, or the ratio of $\vec{F}/\vec{a}$ is another concept of "mass", although it is generally frowned upon now because of the confusion with 'rest mass'.

AM

 

[Herman Trivilino]

I thought that one would define the speed of light as being the distance between the two clocks divided by the time interval measured by the clocks.

Actually, what that process does is define the distance as being one meter when the time interval is $\frac{1}{299\ 792\ 458}\ \mathrm{s}$. You don't measure the speed of light, you calibrate your meter stick. But this seems a tangential issue.

 

[Herman Trivilino]

$m = E/c^2$ means that $m \propto E$. By setting units for c = 1 that still does not make m and E the same physical phenomena.

Setting $c=1$ doesn't make them equivalent, nature does that, or at least as far as we can tell it does. Although it's the rest energy $E_o$ not the total energy $E$ that's proportional to $m$. More precisely, the thing that we measure and call $m$ is not distinguishable from the thing that we measure and call $E_o$. Measuring them in the same units is a matter of preference.

That fact that mass or inertia can be converted into energy does not make them equal. Otherwise there would be no meaning to the "conversion".
?? This again is confusing.

If you look at any of the processes that are called "conversions" what you see is that before the process what one is choosing to call $m$ is afterwards called $E_o$, or vice-versa.

There is the concept of inertia and the concept of rest mass. Rest mass is constant for all observers. Inertia, or the ratio of $\vec{F}/\vec{a}$ is another concept of "mass", although it is generally frowned upon now because of the confusion with 'rest mass'.

Calling the ratio $F/a$ the mass $m$ is valid only in the Newtonian approximation. The more general relation between $\vec{F}$, $m$, and $\vec{a}$ is $$\vec{a}=\frac{\vec{F}-(\vec{F}\cdot\vec{\beta})\vec{\beta}}{\gamma m}.$$ Note that the ratio $F/a$ is equal to $\gamma m$ only when $\vec{F}\cdot\vec{\beta}$ is zero. Identifying $m$ as the inertia has a meaning in the Newtonian approximation that I don't know how to generalize. (Certainly it's not $\gamma m$). Because of that, and the opinion that the concept of inertia clouds the true meaning of Newton's First Law, I try to avoid the term inertia when teaching Newtonian physics. And therefore when teaching relativity, too.

 

[Orodruin]

One could argue that by making the geometry pretty you are obscuring the physics.

I would argue that it is exactly the other way around. You are obstructing the actual physics by selecting a system of units that in relativity is obscure and not very natural. Physics do not depend on your choice of units.

By setting units for c = 1 that still does not make m and E the same physical phenomena.

Physics does not care if it makes sense to you or not. The fact of the matter is that the only mass you talk of in relativity is the rest energy, which in the non-relativistic limit is just the inertia of the object, there is no other mass concept.

There is the concept of inertia and the concept of rest mass.

But the point is that this is the only concept of mass and it is directly the same as the rest energy. Why do you want to introduce two quantities to describe the same thing? This is superfluous and confusing.

 

[nrqed]

I have given some more thoughts about how to explain my point about the choice c=1 and I will post a final comment...final, I promise;-)

The special rôle of the speed of light links space and time, I agree completely. But that does not assign a specific value to c, this is completely arbitrary.

There are three ways to think about this:A) One fixes (completely arbitrarily) a unit of distance.
and one fixes (completely arbitrarily) a unit of time. Then the speed of light is fixed (by a measurement) to some value in those units. The value of c can therefore take any value possible, it simply depends on the choice of units of time and distance, which is not fixed by physics!ORB) One fixes (completely arbitrarily) a unit of distance and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of time is: One unit of time is by definition how long it takes for light to cover 3 units of distance. Nothing forces one to use c=1 here, it is absolutely arbitrary to pick one value of c over another one. ORC)One fixes (completely arbitrarily) a unit own time and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of distance is: One unit of distance is by definition the distance traveled in one third of the unit of time.
My point is that in both B and C, the value chosen for c here is completely arbitrary! Nothing forces it to be 1. What I mean by this is that if two persons use different values of c (and the corresponding two different sets founts), they will agree on any physical result obtained by calculations.

I don’t know how to explain my point more clearly (sorry!) so I will definitely zip it :-)

 

[Herman Trivilino]

The special rôle of the speed of light links space and time, I agree completely. But that does not assign a specific value to c, this is completely arbitrary.

Right. I'm not sure how it's come about in this thread that anyone would think anyone is saying otherwise. The point under discussion, I thought, was whether or not it was of pedagogical value. Not whether it was right or wrong to choose $c=1$.

 

[vela]

A) One fixes (completely arbitrarily) a unit of distance.
and one fixes (completely arbitrarily) a unit of time. Then the speed of light is fixed (by a measurement) to some value in those units. The value of c can therefore take any value possible, it simply depends on the choice of units of time and distance, which is not fixed by physics!

OR

B) One fixes (completely arbitrarily) a unit of distance and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of time is: One unit of time is by definition how long it takes for light to cover 3 units of distance. Nothing forces one to use c=1 here, it is absolutely arbitrary to pick one value of c over another one.

OR

C)One fixes (completely arbitrarily) a unit own time and one fixes (completely arbitrarily!) a value for the speed of light, for example c = 3 units of distance/unit of time. Then, using a light beam this fixes completely what the unit of distance is: One unit of distance is by definition the distance traveled in one third of the unit of time.

My point is that in both B and C, the value chosen for c here is completely arbitrary! Nothing forces it to be 1. What I mean by this is that if two persons use different values of c (and the corresponding two different sets founts), they will agree on any physical result obtained by calculations.

I think the point you're missing is that it's completely unnecessary as far as the physics goes to have separate units for time and space. You can, of course, but at the cost of introducing a conversion factor, namely $c$, into the equations.

 

[FactChecker]

I wouldn't go into the Twins Paradox. It will only confuse things. Better to use that time on fundamentals, the experiments that forced the theory, and the relativity of simultaneity as the motivation of all that followed. (Maybe a brief mention of General Relativity at the end?)

 

[nrqed]

I think the point you're missing is that it's completely unnecessary as far as the physics goes to have separate units for time and space. You can, of course, but at the cost of introducing a conversion factor, namely $c$, into the equations.

I think you are missing my point. Yes, one can define

I think the point you're missing is that it's completely unnecessary as far as the physics goes to have separate units for time and space. You can, of course, but at the cost of introducing a conversion factor, namely $c$, into the equations.

A distance is still fundamentally different from a time. We can call a distance a "light-second" and conveniently drop the "light" and just call it a second but it is still a light-second. A distance cannot be measured with only a clock (and I do mean only a clock...not a clock plus a ruler or a clock plus a mirror etc etc).
So the speed of light can be given as one light-second per second, but to say that c = 1 (pure 1) is wrong. That was my whole point.

 

[nrqed]

Right. I'm not sure how it's come about in this thread that anyone would think anyone is saying otherwise. The point under discussion, I thought, was whether or not it was of pedagogical value. Not whether it was right or wrong to choose $c=1$.

Well, my point has always been that it is just a special choice of units that allows c=1, and that even with those units, we should say c=1unit off distance / 1 unit of time. We could equally well set c=2 or any other value. So pedagogically, as I said I think it is not helpful at all to students who are already struggling with understanding time dilation, etc. Good luck!

 

[vela]

A distance is still fundamentally different from a time.

I think this is the root of the disagreement. Sure, we perceive the dimensions differently, so for practical reasons, we measure them differently. From a mathematical and geometrical point of view, however, they're not so different, and that was the primary revelation of special relativity!

(Maybe a brief mention of General Relativity at the end?)

Another advantage of the geometrical approach to SR is that it sets you up for GR.

 

[nrqed]

Right. I'm not sure how it's come about in this thread that anyone would think anyone is saying otherwise. The point under discussion, I thought, was whether or not it was of pedagogical value. Not whether it was right or wrong to choose $c=1$.

Like I have mentioned before, you are right that it is useful to work with light-years or light-seconds or light-minutes. But instead of setting c=1 and calling these "years, seconds,minutes" and risking confusion when it can easily be avoided, what I do, and I think it is much simpler and pedagogically better (in my opinion), is to tell the student, to use 1 light-second = 1 second x c (which makes sense to them, it is the distance traveled by light in one second), 1 ly = 1 year x c and so on. Plugging these expressions in the equations works out nicely because the factors of c cancel out where they must,leaving time in years (or second or whatever) and leaving speeds in fractions of c. It works very nicely and I can focus on the physics of time dilation, etc.

 

[Herman Trivilino]

A distance cannot be measured with only a clock (and I do mean only a clock...not a clock plus a ruler or a clock plus a mirror etc etc).

Yes, it can. If I have a clock and know what time you're going to send me a light signal, I can use that clock to determine the distance between you and me.

 

[Herman Trivilino]

I wouldn't go into the Twins Paradox. It will only confuse things. Better to use that time on fundamentals, the experiments that forced the theory, and the relativity of simultaneity as the motivation of all that followed. (Maybe a brief mention of General Relativity at the end?)

I appreciate that. What would your reasons be for doing it? This is likely the last physics course these students will ever take. The fundamentals are covered before this lesson is presented, and the twin paradox is already a part of that textbook reading assignment. I would like for them to see this very interesting application of the theory, but perhaps that's just my personal preference.

 

[Andrew Mason]

I would argue that it is exactly the other way around. You are obstructing the actual physics by selecting a system of units that in relativity is obscure and not very natural. Physics do not depend on your choice of units.

I think this is the root of the disagreement. Sure, we perceive the dimensions differently, so for practical reasons, we measure them differently. From a mathematical and geometrical point of view, however, they're not so different, and that was the primary revelation of special relativity!

There is a reason that we use different units for space and time. Using the same units obscures the physical difference. A time interval in a given inertial reference frame is always a time interval to all observers in that reference frame. That same time interval may appear as an interval of space and time in another reference frame. But all observers make the distinction between time and space. If time and space were equal, it would be just a matter of changing the axes to make a time interval a space interval. But that cannot be done.

Physics does not care if it makes sense to you or not. The fact of the matter is that the only mass you talk of in relativity is the rest energy, which in the non-relativistic limit is just the inertia of the object, there is no other mass concept.

But the point is that this is the only concept of mass and it is directly the same as the rest energy. Why do you want to introduce two quantities to describe the same thing? This is superfluous and confusing.

I agree that matter contains potential energy and that the potential energy a body contains is proportional to its rest mass. I agree that a compressed spring has slightly more inertia or rest mass than an uncompressed spring, the difference in mass being $\Delta m = \frac{1}{2}kx^2/c^2$.

It is then a matter of semantics whether one wishes to say that the potential energy contained in a body and its mass are related by E = mc^2, or by choosing units of E and m such that E/m = 1, that potential energy and mass are equal. The problem with the latter is that the expression E = m is true ONLY if c=1 whereas E = mc^2 is always true.

Particle physicists often express the rest mass of particles in units of energy, such as meV, but it is always understood that this is just a short-hand. The units of mass are meV/c²

AM