A Usefulness of relativistic mass

[vanhees71]

Thanks for the recommendation! I'll have to read that to see whether they mention relativistic mass :P

Edit: Are these it http://www.ita.uni-heidelberg.de/research/bartelmann/previous.shtml?lang=en ? Looks like no relativistic mass (that's ok), and they do mention that Copenhagen is the default interpretation, and that there is a measurement problem (which is good assuming, I understood the German correctly).
http://www.ita.uni-heidelberg.de/research/bartelmann/files/theorie4.pdf
"Obwohl die Kopenhagener Deutung eine Anwendung der Quantenmechanik erlaubt, die mit den Ergebnissen von Experimenten hervorragend übereinstimmt und die solche auch zuverlässig vorherzusagen erlaubt, erscheint sie bis heute vor allem deswegen als nicht völlig befriedigend, weil in ihrem Rahmen ungeklärt bleibt, was genau passiert, wenn Eigenschaften quantenmechanischer Systeme gemessen werden. Ungeachtet der fortlaufenden Diskussionen zu diesem und damit verwandten Themen legen wir in dieser Vorlesung, die einen ersten Durchgang durch die Quantenmechanik bieten soll, die Kopenhagener Deutung zugrunde und berühren ihre offenen Fragen nur am Rand. Um mit der Theorie und ihren wesentlichen Aussagen vertraut zu werden, ist das vielleicht der geeignetste Zugang. Bleiben Sie sich aber dessen bewusst, dass auch in der formal und experimentell hervorragend etablierten Quantenmechanik wichtige Fragen zu klären bleiben."

That's a good example, why I like this book so much. They carefully show, where problems are and then take a pragmatic decision. They mention that there are some "not completely satisfying features" and then take the standard interpretation. I also don't like Feynman's use of the relativistic mass (which in this new book of course is not used, because at least Bartelmann (astrophysicist) and Rebhan (relativistic TFT expert) are applying relativity in their research, but that doesn't mean that the Feynman Lectures are not an excellent book, which is also the case for all of the other books by Feynman I know of: Lectures on Gravitation (a very good read to get an alternative view to the usual emphasis of the geometric-only interpretation, Statistical Physics, Path Integrals (with Hibbs). The book (I mean the real textbook not the popular one) on QED is, to my surprise, pretty conservative (no path integrals!).

The links to Bartelmann's websites are not identical with the book, but these are manuscripts of his lectures in Heidelberg, which are excellent too.

 

[vanhees71]

In any case, if one knows what one is doing, then there's obviously nothing wrong with using "relativistic mass" or even "kinetic mass." You might get funny looks if you use them when communicating with other people, but otherwise the "debate" is purely pedagogical, ain't it?

Well, in some sense yes, but my main reason is the logical structure of relativity, which now is much better known than in 1905 (and Einstein came to the conclusion not to use relativistic mass anymore pretty soon). The most fundamental theory we have today is the Standard Model of elementary particle physics (+GR of course for the gravitational interaction), and there it's very clear that invariant mass is fundamental as it is one of the intrinsic properties of any closed system: It's a Casimir operator of the Poincare group. 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!

 

[SiennaTheGr8]

Well, in some sense yes, but my main reason is the logical structure of relativity, which now is much better known than in 1905 (and Einstein came to the conclusion not to use relativistic mass anymore pretty soon). The most fundamental theory we have today is the Standard Model of elementary particle physics (+GR of course for the gravitational interaction), and there it's very clear that invariant mass is fundamental as it is one of the intrinsic properties of any closed system: It's a Casimir operator of the Poincare group. 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 $c=1$).

 

[ZapperZ]

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 $c=1$).

I think we have covered this quite a while back:

https://www.physicsforums.com/threads/relativistic-mass.642188/#post-4106101

Zz.

 

[SiennaTheGr8]

I think we have covered this quite a while back:

https://www.physicsforums.com/threads/relativistic-mass.642188/#post-4106101

Zz.

Good link, though it doesn't really address my point (see my post #30 up-thread, and @vanhees71's response to it—I'm quite aware of the mainstream view and what Okun wrote on this).

 

[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.

 

[Herman Trivilino]

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".