Usefulness of relativistic mass

[atyy]

The other fundamental quantities which can be derived from space-time symmetry in SR are via Noether's theorem the 10 conserved quantities, forming the generators of the Poincare group (to be precise the proper orthochronous Poincare group). The relation ##p_{\mu} p^{\mu}=m^2 c^2## follows from these symmetry considerations, and this also shows that the most useful notion of total energy of a closed system is the one including "rest energy", because then total energy forms together with total momentum a four-vector. Everything becomes much simpler in physics, when the symmetry principles are taken seriously.

Of course you can as well express everything in terms of other quantities, but everything gets much more complicated, and physics is complicated enough not to make it even more complicated by using inconvenient quantities!

I suppose I don't understand how using "relativistic mass" instead of "total energy" undercuts the logical structure of the theory. They're just words and symbols (and units, if you don't choose ##c=1##).

Rather than just criticizing any use of the relativistic mass for teaching, I think it would be more productive to say why it is useful as a link to the Newtonian conception of dynamics using the concept of 3-force and inertial mass, and then to say that it turns out that in quantum relativity and general relativity, the concept of force (neither 3-force nor 4-force) is no longer fundamental, and only useful in special limits (like the Newtonian limit). Rather we have fields interacting with fields.

One has to remember that students don't just learn one theory. They have to learn many theories, and the relationship between the theories, and there may be multiple different limits of the theories with different emergent concepts.

 

[PAllen]

Rather than just criticizing any use of the relativistic mass for teaching, I think it would be more productive to say why it is useful as a link to the Newtonian conception of dynamics using the concept of 3-force and inertial mass, and then to say that it turns out that in quantum relativity and general relativity, the concept of force (neither 3-force nor 4-force) is no longer fundamental, and only useful in special limits (like the Newtonian limit). Rather we have fields interacting with fields.

One has to remember that students don't just learn one theory. They have to learn many theories, and the relationship between the theories, and there may be multiple different limits of the theories with different emergent concepts.

Except that I feel, pedagogically, that invariant mass is the cleanest bridge from Newtonian physics to relativity. I see no value relativistic mass. In Newtonian physics, mass is frame invariant and changes only via flow of something. Energy is frame variant, observer dependent. Momentum is linear in mass and velocity. In SR, it is not linear, and you can choose to attach the nonlinearity to the mass or the velocity. But we already know velocity itself no longer adds linearly, so it makes more sense to say momentum is linear in mass and nonlinear in velocity. We also have to say mass can change within a boundary by flow of radiation or matter, but remains invariant without flow of something. This preserves the most essential intuitions from Newtonian physics. To turn around and give another name to frame dependent energy serves no purpose. The role of kinetic energy in inertia is most correctly handled in SR via invariant mass. To have any notion of a scalar inertia (resistance to force), you have to go 4 vectors, finding that invariant mass is, indeed, a scalar inertia that incorporates kinetic energy.

 

[PAllen]

Another advantage to invariant mass pedagogy, in going to GR is the frequent question of why an a baseball moving ultrarelaativistically relative to me does not turn into a BH. Using invariant mass, one would never expect this to be so, nor would you expect two comoving baseballs to become a BH no matter what their speed relative to me as a distant observer. However, you would legitimately wonder about two baseballs approaching each other at a close flyby, ultra relativistically. And for this latter case, BH may indeed form. Thus, invariant mass leads to far superior initial intuitions in GR

 

[atyy]

Except that I feel, pedagogically, that invariant mass is the cleanest bridge from Newtonian physics to relativity. I see no value relativistic mass. In Newtonian physics, mass is frame invariant and changes only via flow of something. Energy is frame variant, observer dependent. Momentum is linear in mass and velocity. In SR, it is not linear, and you can choose to attach the nonlinearity to the mass or the velocity. But we already know velocity itself no longer adds linearly, so it makes more sense to say momentum is linear in mass and nonlinear in velocity. We also have to say mass can change within a boundary by flow of radiation or matter, but remains invariant without flow of something. This preserves the most essential intuitions from Newtonian physics. To turn around and give another name to frame dependent energy serves no purpose. The role of kinetic energy in inertia is most correctly handled in SR via invariant mass. To have any notion of a scalar inertia (resistance to force), you have to go 4 vectors, finding that invariant mass is, indeed, a scalar inertia that incorporates kinetic energy.

Another advantage to invariant mass pedagogy, in going to GR is the frequent question of why an a baseball moving ultrarelaativistically relative to me does not turn into a BH. Using invariant mass, one would never expect this to be so, nor would you expect two comoving baseballs to become a BH no matter what their speed relative to me as a distant observer. However, you would legitimately wonder about two baseballs approaching each other at a close flyby, ultra relativistically. And for this latter case, BH may indeed form. Thus, invariant mass leads to far superior initial intuitions in GR

My comments are in the context that the teaching of relativistic mass is necessarily bad. Do you agree with this? Do you agree we should say Purcell is bad, don't read it. Feynman is bad, don't read it. The whole literature which uses ADM mass for ADM energy is bad, don't read it? Science teachers who use relativistic mass are teaching bad physics, they are harming the field?

 

[PAllen]

My comments are in the context that the teaching of relativistic mass is necessarily bad. Do you agree with this? Do you agree we should say Purcell is bad, don't read it. Feynman is bad, don't read it. The whole literature which uses ADM mass for ADM energy is bad, don't read it? Science teachers who use relativistic mass are teaching bad physics, they are harming the field?

Well, I have a much higher opinion of Purcell’s pedagogy than @vanhees71 , and a very high opinion of Feynman, however each ones use of relativistic mass is a negative on their balance sheet. Similar to recognizing the brilliance of Minkowslki while still saying the tradition of imaginary time is unfortunate.

The literature on ADM mass/energy has a major feature you are ignoring. In practice, it is almost always computed in coordinates in which ADM momentum spatial components are zero, in which case there is no difference. In the very rare cases when this is not so, you are more likely to see ADM energy distinguished from ADM mass, with the latter being invariant.

 

[atyy]

Well, I have a much higher opinion of Purcell’s pedagogy than @vanhees71 , and a very high opinion of Feynman, however each ones use of relativistic mass is a negative on their balance sheet. Similar to recognizing the brilliance of Minkowslki while still saying the tradition of imaginary time is unfortunate.

The literature on ADM mass/energy has a major feature you are ignoring. In practice, it is almost always computed in coordinates in which ADM momentum spatial components are zero, in which case there is no difference. In the very rare cases when this is not so, you are more likely to see ADM energy distinguished from ADM mass, with the latter being invariant.

WelI, I can't agree. I do agree that it is ok to teach relativity using only the invariant mass, which is how I usually do my calculations.

I do not agree that it is necessarily bad and ignorant teaching to use the relativistic mass. Similarly, if imaginary time is used, that is ok too. Neither is incorrect, and neither precludes also teaching the formalism without relativistic mass, and without imaginary time.

 

[pervect]

Well, I have a much higher opinion of Purcell’s pedagogy than @vanhees71 , and a very high opinion of Feynman, however each ones use of relativistic mass is a negative on their balance sheet. Similar to recognizing the brilliance of Minkowslki while still saying the tradition of imaginary time is unfortunate.

The literature on ADM mass/energy has a major feature you are ignoring. In practice, it is almost always computed in coordinates in which ADM momentum spatial components are zero, in which case there is no difference. In the very rare cases when this is not so, you are more likely to see ADM energy distinguished from ADM mass, with the latter being invariant.

Yes, as I recall Wald gives the formula for the ADM momentum of a system along with the ADM mass. I've always thought of the ADM formalism as giving a 4-vector in asymptotically flat space-time, though I've never seen it written in exactly these words.

 

[atyy]

Similar to recognizing the brilliance of Minkowslki while still saying the tradition of imaginary time is unfortunate.

BTW, you may find this amusing.
http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf
"Some readers expressed their irritation over the fact that after 12 pages I switch notation: the i in the time components of vectors disappears, and the metric becomes the − + + + metric. Why this “inconsistency” in the notation?

There were two reasons for this. The transition is made where we proceed from special relativity to general relativity. In special relativity, the i has a considerable practical advantage: Lorentz transformations are orthogonal, and all inner products only come with + signs. No confusion over signs remain. The use of a − + + + metric, or worse even, a + − − − metric, inevitably leads to sign errors. In general relativity, however, the i is superfluous. Here, we need to work with the quantity g00 anyway. Choosing it to be negative rarely leads to sign errors or other problems.

But there is another pedagogical point. I see no reason to shield students against the phenomenon of changes of convention and notation. Such transitions are necessary whenever one switches from one field of research to another. They better get used to it."

 

[m4r35n357]

Except that I feel, pedagogically, that invariant mass is the cleanest bridge from Newtonian physics to relativity.

Beware bridges that go nowhere and have holes in them ;)

Doing Newtonian physics with gamma? No thanks!

 

[vanhees71]

I suppose I don't understand how using "relativistic mass" instead of "total energy" undercuts the logical structure of the theory. They're just words and symbols (and units, if you don't choose ##c=1##).

No, since "energy" implies that in an arbitrary inertial frame it is the temporal component of the invariant energy-momentum four-vector, while relativistic mass is in no sense a covariant object, while the invariant mass is a scalar!

 

[vanhees71]

My comments are in the context that the teaching of relativistic mass is necessarily bad. Do you agree with this? Do you agree we should say Purcell is bad, don't read it. Feynman is bad, don't read it. The whole literature which uses ADM mass for ADM energy is bad, don't read it? Science teachers who use relativistic mass are teaching bad physics, they are harming the field?

I agree that teaching relativistic mass is bad if one does not state its drawbacks and how to formulate the theory covariantly. You save so much confusion working covariantly that you can't justify the use of non-covariant quantities if you can formulate the theory much simpler covariantly (that particularly holds for electromagnetism, where you have so much confusion just due to the non-relativistic treatment of matter in the traditional approach). The Feynman Lectures are an exception in the sense that there most of these unnecessary paradoxes are brilliantly resolved, like his masterful treatment of the homopolar generator and other quibbles related to the incomplete treatment of Faraday's Law in integral form.

I agree with "Purcell is bad, don't read it", while the Feynman Lectures are too good to disfavor students from reading it since it provides so much intuitive and "original Feynman" insights into physics that the use of non-covariant objects like relativistic mass becomes almost a forgivable sin ;-)). Of course, any advice about a textbook is highly subjective. Maybe, some people get something good out of Purcell. I found it confusing as a student when checking it out as an additional read for the E&M theory lecture (which was taught in the 4th semester; the theory course started in the 3rd semester then) and I still find it confusing when reading it again today. The only difference is that nowadays I know, what he wanted to say, because I've learned some SRT in the meantime.

What is "ADM mass/energy"?

 

[vanhees71]

BTW, you may find this amusing.
http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf
"Some readers expressed their irritation over the fact that after 12 pages I switch notation: the i in the time components of vectors disappears, and the metric becomes the − + + + metric. Why this “inconsistency” in the notation?

There were two reasons for this. The transition is made where we proceed from special relativity to general relativity. In special relativity, the i has a considerable practical advantage: Lorentz transformations are orthogonal, and all inner products only come with + signs. No confusion over signs remain. The use of a − + + + metric, or worse even, a + − − − metric, inevitably leads to sign errors. In general relativity, however, the i is superfluous. Here, we need to work with the quantity g00 anyway. Choosing it to be negative rarely leads to sign errors or other problems.

But there is another pedagogical point. I see no reason to shield students against the phenomenon of changes of convention and notation. Such transitions are necessary whenever one switches from one field of research to another. They better get used to it."

Argh ;-((. This is very bitter since 't Hooft is one of my heroes of physics, getting a Nobel prize for his PhD thesis :-((.

 

[SiennaTheGr8]

Momentum is linear in mass and velocity. In SR, it is not linear, and you can choose to attach the nonlinearity to the mass or the velocity.

Or to neither: ##\vec p = \gamma m \vec v##. Both the celerity ##\gamma \vec v## and the plain old 3-velocity are useful quantities, I think, and even if you choose celerity for this equation you'll want to break it down into ##\gamma## and ##\vec v## when differentiating with respect to time (to derive the relativistic 3-force).

Of course, the advantage to attaching the nonlinearity to mass is that it makes the expression applicable to [rest-]massless things like light. The disadvantage is that it uses "relativistic mass." The best of both worlds is ##\vec p c = E \vec \beta##.

The role of kinetic energy in inertia is most correctly handled in SR via invariant mass. To have any notion of a scalar inertia (resistance to force), you have to go 4 vectors, finding that invariant mass is, indeed, a scalar inertia that incorporates kinetic energy.

To be clear: are you referring here to the kinetic energy of a system's constituents (as measured in the system's rest frame)?

 

[SiennaTheGr8]

No, since "energy" implies that in an arbitrary inertial frame it is the temporal component of the invariant energy-momentum four-vector, while relativistic mass is in no sense a covariant object, while the invariant mass is a scalar!

As I said before, this is just an issue of pedagogy and communication/semantics. If one knows what one is doing and prefers to use "relativistic mass" in lieu of "total energy," then there's no problem. Likewise, there's no problem using "rest energy" in lieu of "invariant mass," which I tend to do.

How can this undercut the logical structure of the theory? It's like making a choice between "timelike interval" ##ds## and "proper time" ##d\tau##—they're the same quantity measured in different units.

 

[vanhees71]

Except that I feel, pedagogically, that invariant mass is the cleanest bridge from Newtonian physics to relativity. I see no value relativistic mass. In Newtonian physics, mass is frame invariant and changes only via flow of something. Energy is frame variant, observer dependent. Momentum is linear in mass and velocity. In SR, it is not linear, and you can choose to attach the nonlinearity to the mass or the velocity. But we already know velocity itself no longer adds linearly, so it makes more sense to say momentum is linear in mass and nonlinear in velocity. We also have to say mass can change within a boundary by flow of radiation or matter, but remains invariant without flow of something. This preserves the most essential intuitions from Newtonian physics. To turn around and give another name to frame dependent energy serves no purpose. The role of kinetic energy in inertia is most correctly handled in SR via invariant mass. To have any notion of a scalar inertia (resistance to force), you have to go 4 vectors, finding that invariant mass is, indeed, a scalar inertia that incorporates kinetic energy.

This is very strange! To the contrary I find the cleanest bridge from Newtonian to SR point mechanics is to keept the one and only mass known in Newton's theory, namely the invariant mass. The only bridge needed from Newton to SR, and this is admittedly a pretty difficult bridge to be built for beginners but it has to be built anyway, the relativity of simultaneity and thus the transformation properties of space and time coordinates. Also it's pretty clear that Newtonian mechanics should be approximately valid for particles moving with a speed much less than the speed of light and that thus in fact Newtonian time in the differential laws has to be substituted by the time in an instantaneous inertial rest frame, which is proper time.

This then leads to the definition of four-momentum
$$p^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=m c u^{\mu},$$
where ##m## is the invariant mass, which is the one and only mass appearing also in Newtonian point-particle mechanics.

Also the Newtonian equation of motion stays the same:
$$\frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=K^{\mu},$$
where ##K^{\mu}## is the Minkowski four-force.

 

[vanhees71]

As I said before, this is just an issue of pedagogy and communication/semantics. If one knows what one is doing and prefers to use "relativistic mass" in lieu of "total energy," then there's no problem. Likewise, there's no problem using "rest energy" in lieu of "invariant mass," which I tend to do.

How can this undercut the logical structure of the theory? It's like making a choice between "timelike interval" ##ds## and "proper time" ##d\tau##—they're the same quantity measured in different units.

Well, then you can as well say didactics doesn't matter at all, and I can explain any physics in as complicated way I want. I doubt, whether this point of view is appreciated by students.

 

[stevendaryl]

But there is another pedagogical point. I see no reason to shield students against the phenomenon of changes of convention and notation. Such transitions are necessary whenever one switches from one field of research to another. They better get used to it.

I think that's a wonderful point. A lot of physics professionals get very upset with particular pedagogical choices, and say: "Don't teach things that way! It will only make the student more confused when he gets to more advanced topics!" People also get upset at the heuristic (or simplistic) explanations for things given in pop-science books. "It's misleading! They'll just have to unlearn that when they actually study the topic rigorously!"

I actually don't feel that way, at all. People who make it past a certain level in science generally learn that there is more than one way to look at a topic. Learning that your understanding is incomplete, and that some of your beliefs are misconceptions that must be corrected is really what growth in scientific maturity is all about. Perfecting pedagogy so that the student never needs to learn new conventions or never needs to fix misconceptions is both impossible and maybe not desirable, since it means giving the student a false impression of how orderly scientific progress is.

And I also have found that a lot of scientists were first inspired to become scientists by reading pop science books of questionable pedagogy. In a recent BackReaction blog post, Sabine Hossenfelder says that she was inspired to become a physicist by reading Hawking's "A Brief History of Time", even though she now considers it a bad book, from a pedagogical point of view.

 

[SiennaTheGr8]

Well, then you can as well say didactics doesn't matter at all, and I can explain any physics in as complicated way I want. I doubt, whether this point of view is appreciated by students.

I think we're actually in complete agreement, and just talking past each other a bit. As I've said, I oppose the use of relativistic mass in textbooks and teaching, precisely because it's confusing and makes things more complicated than they need to be.

 

[Mister T]

In SR, the relativistic mass is the inertial mass,

The real advantage in using only one kind of mass is pedagogical. If you look at introductory textbooks written in the last few decades you find that prior to 1990 the vast majority of them used more than one kind of mass. After 1990 more and more of them stopped. Now the vast majority use only one kind of mass.

 

[PAllen]

Or to neither: ##\vec p = \gamma m \vec v##. Both the celerity ##\gamma \vec v## and the plain old 3-velocity are useful quantities, I think, and even if you choose celerity for this equation you'll want to break it down into ##\gamma## and ##\vec v## when differentiating with respect to time (to derive the relativistic 3-force).

Of course, the advantage to attaching the nonlinearity to mass is that it makes the expression applicable to [rest-]massless things like light. The disadvantage is that it uses "relativistic mass." The best of both worlds is ##\vec p c = E \vec \beta##.
To be clear: are you referring here to the kinetic energy of a system's constituents (as measured in the system's rest frame)?

Answering your questions in order:

I wouldn't normally bother with celerity. gamma(v) * v is a nonlinear function of v that is normally best manipulated in that form (unless you go the hyperbolic function route).

Attaching nonlinearity to mass doesn't help much because m*gamma is undefined for m=0. What does help, after noting momentum as m*gamma*v, is to consider modification to Newtonian energy, arriving at the altogether new notion of total energy that includes mass as well as kinetic and potential energy. Then to note that momentum can now be written E*v, from which you have a consistent path to massless particles. Of course, I would make the transition 4-vectors before addressing generalization to things like massless particles.

As to your last question, yes, except that using covariant quantities, you compute total 4-momentum in any frame, just adding/integrating covariant quanitities. Then, the invariant mass is simply the norm.

 

[PAllen]

This is very strange! To the contrary I find the cleanest bridge from Newtonian to SR point mechanics is to keept the one and only mass known in Newton's theory, namely the invariant mass. The only bridge needed from Newton to SR, and this is admittedly a pretty difficult bridge to be built for beginners but it has to be built anyway, the relativity of simultaneity and thus the transformation properties of space and time coordinates. Also it's pretty clear that Newtonian mechanics should be approximately valid for particles moving with a speed much less than the speed of light and that thus in fact Newtonian time in the differential laws has to be substituted by the time in an instantaneous inertial rest frame, which is proper time.

This then leads to the definition of four-momentum
$$p^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=m c u^{\mu},$$
where ##m## is the invariant mass, which is the one and only mass appearing also in Newtonian point-particle mechanics.

Also the Newtonian equation of motion stays the same:
$$\frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=K^{\mu},$$
where ##K^{\mu}## is the Minkowski four-force.

I don't know what you are disagreeing with. What you write is completely consistent with what I wrote, as far as I understand.

 

[vanhees71]

I don't know what you are disagreeing with. What you write is completely consistent with what I wrote, as far as I understand.

Hm, then I misread somehow your posting. I thought you argued in favor of relativistic mass, as some modern textbook writers even do today.

 

[atyy]

What is "ADM mass/energy"?

The ADM energy is sometimes called the ADM mass (see the references in post #26), so there are analogous issues to how the energy is sometimes called the relativistic mass.

 

[atyy]

Argh ;-((. This is very bitter since 't Hooft is one of my heroes of physics, getting a Nobel prize for his PhD thesis :-((.

Don't worry, he's now degenerated to superdeterminism. Maybe he'll go to Mars soon https://www.mars-one.com/about-mars-one/ambassadors/prof.-dr.-gerard-t-hooft :)

One way trip :P

 

[vanhees71]

The ADM energy is sometimes called the ADM mass (see the references in post #26), so there are analogous issues to how the energy is sometimes called the relativistic mass.

I see, ok. That's about the problem of mass in GR, which is of course much more subtle than in SR. If you start in GR with non-covariant quantities for sure you are lost to begin with. That's why I don't know any idea to use something like "relativistic mass" in GR. How to define the mass of a self-gravitating body (like a neutron star), is of course another issue and much more complicated than the cure for the confusion in SR, where you simply use the notion of "invariant mass".

 

[atyy]

I agree with "Purcell is bad, don't read it", while the Feynman Lectures are too good to disfavor students from reading it since it provides so much intuitive and "original Feynman" insights into physics that the use of non-covariant objects like relativistic mass becomes almost a forgivable sin ;-)). Of course, any advice about a textbook is highly subjective. Maybe, some people get something good out of Purcell. I found it confusing as a student when checking it out as an additional read for the E&M theory lecture (which was taught in the 4th semester; the theory course started in the 3rd semester then) and I still find it confusing when reading it again today. The only difference is that nowadays I know, what he wanted to say, because I've learned some SRT in the meantime.

Personally, before I went to university I was largely influenced by WGV Rosser's special relativity text which explicitly said it would not use the relativistic mass, only the invariant mass, since that would be simpler. I did of course know Feynman's treatment too, and thought it was very insightful. However, I usually worked my problems without relativistic mass following Rosser which I had studied more. Then at university, the textbook was Purcell, which I indeed never understood, but there are 2 things in there I like - the explanation of E and B field and the relativistic transforms, and the derivation of the Lamor formula. Years later, I was pleased to find that Schroeder agreed with me: http://physics.weber.edu/schroeder/mrr/mrrtalk.html

Anyway, I told my physics lecturers that I preferred to use only the invariant mass, and they (following Purcell), in their broad minded wisdom, told me to learn both.

Purcell has been dead a long time now, and he's been betrayed by Jackson and his new editors who've switched to SI units. Perhaps they'll remove the relativistic mass in the 4th edition.

 

[vanhees71]

Hm, yes, but isn't this very complicated too? You have to teach Maxwell's equations anyway in the introductory E&M theory lecture (usually they are already introduced before in the experimental lecture too) and in the traditional approach at the end ("if time permits" ;-)) you also teach the relativistic formulation. So given the Maxwell equations and an introduction to relativistic kinematics in terms of Minkowski four-vectors (if possible, avoid the ##\mathrm{i} c t## convention, which also spoils the beauty of relativity a lot, but I'm not as strictly against this as I'm against relativistic mass), why not simply analyzing the Maxwell equations in the usual way and formulate them in covariant form.

Then you have the right mathematical tools, making the physics most transparent, not being hindered by awful notation as Purcell, only because you think that math is some disease to be avoided as much as possible (which seems to me to be the reason, why Purcell failed in his Berkeley volume with his good idea to teach E&M right away as relatvistic theory). Then Purcell's example can be even more simplified to a single uniformly moving particle. We can then first use the rest frame of this particle, where the four-potential in Lorenz gauge is of course given by (I use Heaviside Lorentz units with ##c=1##)
$$A^{0}=\Phi(\vec{x}), \quad \vec{A}=0, \quad \Phi(\vec{x})=\frac{Q}{4 \pi r}.$$
Now we can easily bring this in manifestly covariant form by introducing the four-velocity of the particle, ##u^{\mu}##. Since it's ##(1,0,0,0)## in the particle rest frame and thus the invariant restframe distance to the particle is given by ##r^2=(u \cdot x)^2-x^2##, you get the invariant vector field
$$\boldsymbol{A}=\Phi(\sqrt{(u \cdot x)^2-x^2}) \boldsymbol{u}.$$
With these few lines it is very clear that the field of moving charges has both electric and magnetic field components.

Of course, also the example in Purcell's book as well as in Schroeder's talk, is much more lucent using manifestly covariant notation and the fact that ##(j^{\mu})=(\rho,\vec{j})## is a four-vector field.

What I'm not so sure about concerning didactics is whether one should really teach E&M in the intro lecture really as relativistic theory from the beginning, because to get the necessary intuition for the subject you need to work out a lot of examples in the (1+3) formalism in a fixed reference frame anyway. The best books using a SRT-first approach I know of are

M. Schwartz, Principles of Electrodynamics, Dover
Landau&Lifshitz, vol. II

I think the former book's approach is doable in the intro theory lecture, particularly since nowadays SR is already introduced in the mechanics lecture before.

 

[sweet springs]

Hmm, it feels like it's question about semantics of the word "system". Whether "system" includes binding energy or not.

Say there are two similar both plus charged balls in system #1 and there are two similar plus and minus balls in system #2 in the same positions. Rest mass of system #1 > system #2 due to binding Coulomb energy. Does this example satisfy you?

 

[stevendaryl]

Here's a way to think about invariant mass that applies in both relativistic and nonrelativistic physics.

An object's trajectory (a slower-than-light object, anyway) is a set of moments/events that can be characterized by 4 coordinates, ##(x,y,z,t)##. Those are external to the object itself. We can also talk about a fourth, internal coordinate, which I'll call ##s##, which measures the elapsed time experienced by the object. (If the object has a built-in clock, then it would be the elapsed time on the clock. If it doesn't, then we can pretend it does.) Putting these 5 quantities together, we can come up with a 4-velocity ##V## characterizing the object's progress through spacetime as a function of its internal time:

##V## is a vector with components ##(\frac{dx}{ds}, \frac{dy}{ds}, \frac{dz}{ds}, \frac{dt}{ds})##.

This 4-vector ##V## associated with a moving object exists for both relativistic and nonrelativistic physics, even though nobody ever talks about the component ##\frac{dt}{ds}## in nonrelativistic physics, since it's always equal to 1 (in inertial Cartesian coordinates, anyway). But I'm going to use in both cases to illustrate the similarities between relativistic and nonrelativistic physics.

For simplicity, I'm going to assume no long-range interactions between objects. If an object is far away from other objects, then (both relativistically and nonrelativistically) its 4-velocity ##V## will be constant (its components in an inertial Cartesian coordinate system will be constants).

Now, let's consider interactions between moving objects. As I said, I'm ignoring long-range forces, which leave us with contact forces. Basically, objects can collide into each other. The collisions can change the course (the 4-velocity) of an object, they can break an object into two or more smaller objects, they can cause two or more small objects to merge into larger objects.

So the number of objects and their 4-velocities can be changed by a collision. But in both relativistic and non-relativistic physics, there is a cumulative, composite 4-velocity associated with the whole collection of objects involved in a collision which is left unchanged by the collision. The assumption is that the composite 4-velocity is linearly related to the 4-velocities of the objects that make it up. That is, if objects ##O_1, O_2, ...## with 4-velocities ##V_1, V_2, ...## collide, and produce objects ##O_1', O_2', ...## with 4-velocities ##V_1', V_2', ...##, then there are real numbers ##m_1, m_2, ..., m_1', m_2', ..., M## and a 4-velocity ##V_{composite}## such that

##M V_{composite} = m_1 V_1 + m_2 V_2 + ...##
##M V_{composite} = m_1' V_1' + m_2 V_2' + ...##

(where vector addition for inertial cartesian coordinates just means componentwise)

So the masses of the objects are the weights (no pun intended) with which that object's 4-contributes to the composite 4-velocity. The weighted 4-velocities, ##m_i V_i## are the 4-momenta of the objects. So the sum of the 4-momenta gives a composite 4-momentum, which is conserved in collisions.

In nonrelativistic physics, we assume that the 4-th component of the 4-velocity is always equal to 1: ##\frac{dt}{ds} = 1##. This implies that the 4-component of the equation for composite velocity satisfies:

##M = m_1 + m_2 + ...##

So mass is conserved, nonrelativistically. Relativistically, we don't assume that ##\frac{dt}{ds} = 1##, so mass may not be conserved, but there will still be a conserved quantity for the composite system: the 4th component of the 4-momentum, which we identify with the total energy.

An interesting feature of this connection between relativistic and nonrelativistic kinematics: The 4th component of the 4-momentum gives rise to conservation of energy in the relativistic case, but to conservation of mass in the nonrelativistic case. There is no direct correspondence between the nonrelativistic kinetic energy and anything in the relativistic case. Kinetic energy is not conserved in collisions nonrelativistically, and is not the 4th component of a 4-momentum.

 

[stevendaryl]

Here's a way to think about invariant mass that applies in both relativistic and nonrelativistic physics.

What I think is nice about this treatment is that the way of defining mass as the invariant for 4-momentum: ##m = \frac{1}{c^2} \sqrt{E^2 - p^2 c^2}##, has no counterpart in nonrelativistic physic. But defining mass as the weighting of an object's velocity to produce a conserved momentum works both relativistically and nonrelativistically.

 

[vanhees71]

The reason why this has no counterpart in nonrelatistic physics is worth thinking about. The reason is a subtle difference between the Lie groups describing the symmetries of the spacetime manifolds in special relativity (Minkowski space, an affine Lorentz manifold) and non-relativistic spacetime (a fiber bundle). The group-theoretical difference becomes fully appreciable in quantum theory, where the observable algebra is determined (or at least motivated) by group theory with the physics meaning given via Noether's theorems, according to which any one-parameter Lie symmetry implies the conservation of a quantity, which is the generator of the group action on the states (in classical mechanics phase-space distributions, in QT statistical operators on Hilbert space) and vice versa. That makes the 10 main observables of all theories based on these spacetimes quite uniquely defined (energy, momentum, angular momentum, generators for boosts).

Now in QT what you need are unitary ray representations. Now comes the difference: In SR the symmetry group is the proper orthochronous Lorentz group whose Lie algebra has no non-trivial central charges, i.e., any unitary ray representation can be lifted to a unitary representation on Hilbert space. This is not the case for the proper orthochronous Galileo group, which underlies non-relativistic Newtonian spacetime. It turns out that there is one non-trivial central charge, which turns out to be the mass in terms of physical quantities, and the massless case, leading to a proper unitary representation of the Galileo group, does not lead to any useful dynamics, which is a famous result by Eönü and Wigner.