A Usefulness of relativistic mass

[vanhees71]

[Moderator's note: This thread was spun off from a previous thread in the Quantum Physics forum. It was moved here since this specific subject is more on topic in this forum.]

I didn't say that somebody who uses the relativistic mass doesn't understand relativity. It's just an unneeded complication in presenting the theory, particularly there's no way to define it properly in GR. You know what I think about Purcell's didactics. Of course Feynman's didactics is brillant (the 2nd-best textbook writer I know of), but even a brillant mind can err!

 

[weirdoguy]

Still, as far as I know most physicists using relativity in their work see no point in using velcity-dependent mass. So why teach it? Because "The Big Feynman" used it in his book? I don't care that he used it, because I see no point in doing so. Physics didactics is a big fat mess...

 

[Dadface]

It's not just Feynman who favoured the concept of relativistic mass and from what I have gathered by searching around is that the pros and cons of the concept are still being debated in the wider physics community. Personally I don't see any problems with the concept of relativistic mass provided that it's realized that what can be defined as the total mass is equal to two parts, the (invariant) mass plus the kinetic energy expressed in the same mass units eg Kg.

 

[atyy]

I didn't say that somebody who uses the relativistic mass doesn't understand relativity. It's just an unneeded complication in presenting the theory, particularly there's no way to define it properly in GR. You know what I think about Purcell's didactics. Of course Feynman's didactics is brillant (the 2nd-best textbook writer I know of), but even a brillant mind can err!

So your complaining about relativistic mass is just your personal preference then. There is nothing wrong with it. Isn't it better to teach students that there are many right ways to the subject? Also, what do you make of the didactics of https://arxiv.org/abs/gr-qc/9909014 ?

 

[Dadface]

So your complaining about relativistic mass is just your personal preference then. There is nothing wrong with it. Isn't it better to teach students that there are many right ways to the subject? Also, what do you make of the didactics of https://arxiv.org/abs/gr-qc/9909014 ?

In my searches I have been trying to find relevant observational evidence which seems to be rather thin on the ground. The Carlip paper you referenced looks promising.

 

[vanhees71]

So your complaining about relativistic mass is just your personal preference then. There is nothing wrong with it. Isn't it better to teach students that there are many right ways to the subject? Also, what do you make of the didactics of https://arxiv.org/abs/gr-qc/9909014 ?

Well, it's a very nice paper. Indeed it shows that in GR there's no other mass than invariant mass or where do you see in this paper the notion of relativistic mass. In GR it's not conceivable to me, how to make sense of a relativistic mass at all. You can make some sense of this idea in SR, but why should one? In GR it's clear that the source of the gravitational field is the energy-momentum-stress tensor of the matter fields.

The factor-two problem with the Newtonian limit of free-falling test particles in gravitational fields (usually in the Schwarzschild field in standard textbooks) is simply that textbook authors are not careful enogh in deriving the non-relativistic limit, where one has to be careful with the spatial components of the metric-tensor components. It has nothing to do with the idea of relativistic mass at all.

 

[atyy]

Well, it's a very nice paper. Indeed it shows that in GR there's no other mass than invariant mass or where do you see in this paper the notion of relativistic mass. In GR it's not conceivable to me, how to make sense of a relativistic mass at all. You can make some sense of this idea in SR, but why should one? In GR it's clear that the source of the gravitational field is the energy-momentum-stress tensor of the matter fields.

The factor-two problem with the Newtonian limit of free-falling test particles in gravitational fields (usually in the Schwarzschild field in standard textbooks) is simply that textbook authors are not careful enogh in deriving the non-relativistic limit, where one has to be careful with the spatial components of the metric-tensor components. It has nothing to do with the idea of relativistic mass at all.

Well, it helps to motivate why energy is the source of gravity in GR.

 

[atyy]

Well, it's a very nice paper. Indeed it shows that in GR there's no other mass than invariant mass or where do you see in this paper the notion of relativistic mass. In GR it's not conceivable to me, how to make sense of a relativistic mass at all. You can make some sense of this idea in SR, but why should one? In GR it's clear that the source of the gravitational field is the energy-momentum-stress tensor of the matter fields.

The factor-two problem with the Newtonian limit of free-falling test particles in gravitational fields (usually in the Schwarzschild field in standard textbooks) is simply that textbook authors are not careful enogh in deriving the non-relativistic limit, where one has to be careful with the spatial components of the metric-tensor components. It has nothing to do with the idea of relativistic mass at all.

In post 82, I said that it help to motivate why energy is the source of gravity in GR.

Here are examples:

Blau, Lecture notes on gravity http://www.blau.itp.unibe.ch/newlecturesGR.pdf, gives heuristic motivation for relativistic gravity (p20) with statements like: "We already know (from Special Relativity) that ρ is not a scalar but rather the 00-component of a tensor, the energy-momentum tensor".

Schutz, Gravity from the Ground up http://www.gravityfromthegroundup.org/ also makes use of the notion of relativistic mass.http://www.gravityfromthegroundup.org/
p190 "As an object moves faster, its of an object increases with its speed. We noted above that no force, inertial mass increases, so it is harder to accelerate it. This enforces the speed of light as a limiting speed: as the object gets closer to the speed of light, its mass increases without bound"

On why rest mass is not the correct generalization for the source of gravity in GR:
p242 "What would happen to a gravitational field created by rest-mass when rest-mass is turned into energy by nuclear reactions? Would gravity disappear? This seems unreasonable. Rest mass is a dead end."

p242 "the active gravitational mass generates the curvature of time, which is the most important part of the geometry of gravity. Its density is defined as the density of ordinary mass-energy, plus three times the average pressure divided by c2."

 

[zonde]

On why rest mass is not the correct generalization for the source of gravity in GR:
p242 "What would happen to a gravitational field created by rest-mass when rest-mass is turned into energy by nuclear reactions? Would gravity disappear? This seems unreasonable. Rest mass is a dead end."

Binding energy and associated mass difference sort of indicates that there is something wrong with concept of rest-mass as physical quantity (not only in GR but in general).
So yes, "rest mass is a dead end" seems valid conclusion.

 

[jbriggs444]

Binding energy and associated mass difference sort of indicates that there is something wrong with concept of rest-mass as physical quantity (not only in GR but in general).
So yes, "rest mass is a dead end" seems valid conclusion.

You lost me there. Rest mass (aka invariant mass) is conserved in interactions in which binding energy is increased or decreased. [Energy is conserved, momentum is conserved, it follows that the norm of the energy-momentum four-vector is conserved]

 

[PAllen]

So your complaining about relativistic mass is just your personal preference then. There is nothing wrong with it. Isn't it better to teach students that there are many right ways to the subject? Also, what do you make of the didactics of https://arxiv.org/abs/gr-qc/9909014 ?

The author does not use relativistic mass and this paper is all about the contribution of kinetic energy to invariant mass. I would say the paper makes a very good argument against the usage relativistic mass.

 

[PAllen]

In post 82, I said that it help to motivate why energy is the source of gravity in GR.

Here are examples:

Blau, Lecture notes on gravity http://www.blau.itp.unibe.ch/newlecturesGR.pdf, gives heuristic motivation for relativistic gravity (p20) with statements like: "We already know (from Special Relativity) that ρ is not a scalar but rather the 00-component of a tensor, the energy-momentum tensor".

Schutz, Gravity from the Ground up http://www.gravityfromthegroundup.org/ also makes use of the notion of relativistic mass.
p190 "As an object moves faster, its of an object increases with its speed. We noted above that no force, inertial mass increases, so it is harder to accelerate it. This enforces the speed of light as a limiting speed: as the object gets closer to the speed of light, its mass increases without bound"

On why rest mass is not the correct generalization for the source of gravity in GR:
p242 "What would happen to a gravitational field created by rest-mass when rest-mass is turned into energy by nuclear reactions? Would gravity disappear? This seems unreasonable. Rest mass is a dead end."

p242 "the active gravitational mass generates the curvature of time, which is the most important part of the geometry of gravity. Its density is defined as the density of ordinary mass-energy, plus three times the average pressure divided by c2."

The first does not quite use relativistic mass, IMO, but comes close. The second does, but I find its discussion of rest mass in your p.242 quote exceedingly badly formulated in that it is NOT referring to invariant mass but apparently to the nonsensical quantity 'sum of rest masses'. System rest mass of annihilating particles does not change, thus this whole discussion I would tend to call actually wrong, despite the source.

 

[PAllen]

I would like to point out how old the ‘modern’ point of view on mass is. Besides the well known fact that Einstein apologized for ever introducing relativistic mass within a few years after his initial papers, Bergmann’s well known book from 1942 (which happens to be the first book I learned both SR and GR from) never ever refers to mass outside its invariant meaning. He also introduces (already in 1942) 4 vectors before addressing mass point mechanics, so he ends up immediateely, with no detours, with the extremely natural generalizations of Newton’s laws from 3 vectors to 4 vectors.

p = m U
And
F = dp/dτ = mA for case of no flow from a body.

 

[atyy]

The author does not use relativistic mass and this paper is all about the contribution of kinetic energy to invariant mass. I would say the paper makes a very good argument against the usage relativistic mass.

The first does not quite use relativistic mass, IMO, but comes close. The second does, but I find its discussion of rest mass in your p.242 quote exceedingly badly formulated in that it is NOT referring to invariant mass but apparently to the nonsensical quantity 'sum of rest masses'. System rest mass of annihilating particles does not change, thus this whole discussion I would tend to call actually wrong, despite the source.

Well, the problem with all the arguments against relativistic mass is that

(1) In SR, in certain contexts, there is nothing wrong with it - Purcell and Feynman are examples. If you reject relativistic mass, you reject these presentations of SR, and basically makes it seem like a well understood subject is still up for debate.
(2) In SR, the relativistic mass is the inertial mass, which together with certain formulations of the equivalence in Newtonian gravity does lead to the heuruistic for why energy is the source of gravity in GR. This is of course not strictly correct, but this goes in line with all the people who don't want to teach old quantum physics.

I would like to point out how old the ‘modern’ point of view on mass is. Besides the well known fact that Einstein apologized for ever introducing relativistic mass within a few years after his initial papers, Bergmann’s well known book from 1942 (which happens to be the first book I learned both SR and GR from) never ever refers to mass outside its invariant meaning. He also introduces (already in 1942) 4 vectors before addressing mass point mechanics, so he ends up immediateely, with no detours, with the extremely natural generalizations of Newton’s laws from 3 vectors to 4 vectors.

p = m U
And
F = dp/dτ = mA for case of no flow from a body.

Indeed, among my first introductions to SR did not use relativistic mass, and explicitly said it would be more convenient to use the invariant mass. This was WGV Rosser's book. But then my teachers in university did use relativistic mass - and one can still see this in the public outreach materials of CERN. To add to my list above, here is Tegmark's notes from a course at MIT: https://ocw.mit.edu/courses/physics/8-033-relativity-fall-2006/readings/dynamics.pdf which includes the use of relativistic mass in "Mass-energy unification". The argument against relativistic mass is purely semantic, and it stresses that in physics semantics is more important than actual content - a very deeply wrong view, in my opinion.

 

[pervect]

Invariant mass is a scalar, the simplest form of a tensor. Relativistic mass is not. Relativistic mass is part of various tensor quantities, the simplest being the energy-momentum tensor of a point particle in special relativity. Relativistic mass also has a perfectly fine alternative name, energy, so we don't even need the name.

Thus once one has made the decision to use tensors, I can't see that it makes any sense at all to teach "relativistic mass". One instead teaches the tensor quantities "invariant mass" and "energy momentum tensor". When one moves beyond point particles, one might introduce the stress-energy tensor in addition to the energy-momentum tensor.

Some people, who are probably never going to use tensors, insist strongly on using relativistic mass for reasons that I don't really understand. If they manage to get the right answer, I don't see any reason to argue with them about using the term. If the paper or post is well-written, one can convert it into whatever form is needed. If a paper or post is badly written, I tend to lose patience nowadays, though sometimes I'll try to wade through it anyway.

The idea of relativistic mass has taken a firm root in popularizations, though it's not terribly popular in the professional community from what I can tell. I'm rather surprised at the sporadic claims that things are otherwise. Typically, when I used to actually track down and read a paper that is being offered as evidence for the popularity of the usage of the term "relativistic mass", I would conclude that the paper didn't do any such thing. (An example of that in this thread is the Carlip paper).

I'd agree that there still are a few holdouts for "relativistic mass" in the professional community. Eventually they'll probably die of old age, and if we haven't blown ourselves up in the meantime, I expect invariant mass will eventually become the professional standard. (Assuming that it isn't already). It will take much longer (if it ever happens at all) for "relativistic mass" to be phased out of the popular literature, however.

 

[zonde]

You lost me there. Rest mass (aka invariant mass) is conserved in interactions in which binding energy is increased or decreased.

You mean that rest mass of the system plus mass equivalent of binding energy is conserved, right?
Well, the rest mass of components plus mass equivalent of kinetic and binding energy is not conserved as you never consider mass equivalent of potential energy. Mass equivalent of converted potential energy comes from nowhere.

 

[jbriggs444]

You mean that rest mass of the system plus mass equivalent of binding energy is conserved, right?

No. I mean the invariant mass of the system, period. Binding energy is not an addition to this. It is a part of this.

 

[zonde]

No. I mean the invariant mass of the system, period. Binding energy is not an addition to this. It is a part of this.

Now you lost me. Binding energy has to be removed from system for it become bound. When you remove binding energy the rest mass of the system is reduced. What do you mean with "It is part of this."?

 

[jbriggs444]

Now you lost me. Binding energy has to be removed from system for it become bound. When you remove binding energy the rest mass of the system is reduced. What do you mean with "It is part of this."?

It is part of it in that it is a deduction from the total energy that goes into computing the invariant mass.

You do not have to remove energy from a system in order to bind the system. You can extend the system boundaries to include the energy that has been "removed". If, for instance, you fuse two atoms of deuterium to form one atom of helium, mass is conserved when you close the system to include the emitted byproducts, such as gamma photons.

 

[zonde]

It is part of it in that it is a deduction from the total energy that goes into computing the invariant mass.

Yes. So why do you said "No"? I think I said exactly that in the sentence you quoted.
Hmm, it feels like it's question about semantics of the word "system". Whether "system" includes binding energy or not.

 

[jbriggs444]

Yes. So why do you said "No"? I think I said exactly that in the sentence you quoted.
Hmm, it feels like it's question about semantics of the word "system". Whether "system" includes binding energy or not.

Indeed. The sort of "system" for which invariant mass is conserved is a closed system. Importing or exporting energy is cheating.

 

[zonde]

Indeed. The sort of "system" for which invariant mass is conserved is a closed system. Importing or exporting energy is cheating.

Let me ask you. Do you agree that invariant mass of weakly bound system is equivalent to the sum of invariant mass of strongly bound (the same) system plus respective mass equivalent of removed binding energy?

 

[jbriggs444]

Let me ask you. Do you agree that invariant mass of weakly bound system is equivalent to the sum of invariant mass of strongly bound (the same) system plus respective mass equivalent of removed binding energy?

Yes.

[With the caveat that the energy removal be done so as to leave momentum unchanged]

 

[stevendaryl]

There was an article that I read years ago by somebody who knew what he was talking about---it might have been by a PhysicsForums regular---that showed that the relationship between the relativistic Poincare group and the nonrelativistic Galilean group was made clearer by introducing relativistic mass, but I can't remember the details.

 

[SiennaTheGr8]

Schutz, Gravity from the Ground up http://www.gravityfromthegroundup.org/ also makes use of the notion of relativistic mass.
p190 "As an object moves faster, its of an object increases with its speed. We noted above that no force, inertial mass increases, so it is harder to accelerate it. This enforces the speed of light as a limiting speed: as the object gets closer to the speed of light, its mass increases without bound"

At one point in that book (p. 190), Schutz himself misuses the relativistic mass concept in precisely one of the ways that beginners often accidentally do:

We noted above that no force, no matter how strong, could accelerate a particle to the speed of light. Does this mean that Newton's second law, $F = ma$, is wrong? After all, if I take F to be large enough, I should be able to make the acceleration large enough to beat the speed limit of $c$. No: relativity has a way out of this potential contradiction. The mass $m$ in Newton's equation gets unboundedly large as the particle gets near to the speed of light, again by the ubiquitous $\gamma$ factor[.]

Source: https://books.google.com/books?id=P_T0xxhDcsIC&pg=PA190

The equation $\vec F = \gamma m \vec a$ only holds if the force and velocity vectors are perpendicular, and Schutz gives no indication that he's referring to this special case. Almost certainly he's oversimplifying for the layman reader.

 

[atyy]

Here is another example from Jaramillo and Gourgoulhon:
https://arxiv.org/abs/1001.5429
"Remark 5. In the literature, references are found where the term ADM mass actually refers to this length of the ADM 4-momentum and other references where it refers to its time component, that we have named here as the ADM energy. These differences somehow reflect traditional usages in Special Relativity where the term mass is sometimes reserved to refer to the Poincare invariant (rest-mass) quantity, and in other occasions is used to denote the boost-dependent time component of the energy-momentum."

As an example, compare Gourgoulhon's https://arxiv.org/abs/gr-qc/0703035 Eq 7.14 with Jaramillo and Gourgoulhon's https://arxiv.org/abs/1001.5429 Eq 30.

 

[atyy]

I didn't say that somebody who uses the relativistic mass doesn't understand relativity. It's just an unneeded complication in presenting the theory, particularly there's no way to define it properly in GR. You know what I think about Purcell's didactics. Of course Feynman's didactics is brillant (the 2nd-best textbook writer I know of), but even a brillant mind can err!

So you put Sommerfeld > Feynman > Landau & Lifshitz?

 

[vanhees71]

Hm, that's a hard question... Sommerfeld>Feynman is ok, but who is 3rd? Landau & Lifshitz I like pretty much (except vol. IV on QED), but I'd not put it on the 3rd place. If it's about general theory textbooks, for me the hottest candidate at the moment is the German textbook covering the BSc-level subjects (classical mechanics, classical electromagnetism, non-relativistic quantum mechanics, and thermodynamics including (classical and quantum) statistical physics)

M. Bartelmann, B. Feuerbacher, T. Krüger, D. Lüst, A. Rebhan, A. Wiph, Theoretische Physik, Springer Spektrum 2015

 

[atyy]

M. Bartelmann, B. Feuerbacher, T. Krüger, D. Lüst, A. Rebhan, A. Wiph, Theoretische Physik, Springer Spektrum 2015

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."

 

[SiennaTheGr8]

Here's my long-winded and unasked-for take on the "relativistic mass" business.

An immediate consequence of discovering that our universe has an invariant speed was that it became possible to express physical quantities in units that traditionally belonged to certain other physical quantities with other dimensions (in the dimensional-analysis sense of the word). The "method" is simple: multiply by powers of $c$. So one could express time in "distance units" by multiplying by $c$, and one could express momentum in "energy units" by doing the same.

It's worth remembering that this method of unit conversion (really "dimension conversion") via multiplication by a physical constant wasn't new. We already had Newton's gravitational constant $G$. Here's a silly little example I just came up with: one can express linear mass density (dimensions [M]/[L]) in "units of squared-velocity" by multiplying by $G$. There's nothing deep here (as far as I can tell), but the point is that dimensioned physical constants allow for "dimension conversion."

Now, in the cases I mentioned of time/distance and momentum/energy, there is something "deep." It turns out that there's very good reason to express time and distance in the same unit by default. Same for momentum and energy. The mathematical elegance achieved in doing so suggests that we were doing something "artificial" by regarding the quantities as having different dimensions in the first place. But it doesn't mean that time is distance, or that momentum is energy. They aren't. The quantities and concepts remain distinct (though SR shows they're more closely related than we thought). Dimension-convertibility (even when useful!) doesn't mean equivalence.

Enter mass.

Dimensional analysis shows that mass can be expressed in "energy units" by multiplying by $c^2$ (or that energy can be expressed in "mass units" by dividing by $c^2$). If that were the end of the story, there'd be little more to say. What makes this situation different is the strict equivalence of mass and rest energy $E_0$. Two seemingly distinct physical quantities—the energy of a resting body and its mass—are actually one and the same, just expressed in different units (dimensions).

So on the one hand, we've got equations like $E = E_0 + E_k$ and $E = \gamma E_0$ (where $E$ is total energy and $E_k$ is kinetic energy). But on the other hand we have $E_0 = mc^2$, so that $E = mc^2 + E_k$ and $E = \gamma mc^2$. Where should we go from here?

In hindsight, the parsimonious answer (in my opinion) is to ditch the $m$ altogether. It's conceptually equivalent to $E_0$, so there's no need to keep them both around, and $E_0$ is already in the same unit as two other related quantities we're using ($E$ and $E_k$)—three if you count momentum $pc$. Alternatively, one could abandon $E_0$, $E_k$, and $E$, and replace them with their divided-by-$c^2$ equivalents $m$ ("rest mass"), $m_k$ ("kinetic mass"), and $m_r$ ("relativistic mass," though wouldn't "total mass" be preferable in some sense?), and express momentum in "mass units" by dividing by $c$. This "mass unit" alternative is not so good linguistically or psychologically, I think: Newtonian experience makes "kinetic mass" seem weird, and it also makes "total mass" $m_r$ seem like it should be the sum of rest masses, rather than the sum of $m$ and $m_k$ (I imagine this is why "relativistic mass" was always used instead of "total mass"; "total energy," by contrast, fits nicely with our Newtonian/Maxwellian experience).

Unfortunately, the "solution" that was adopted was to keep both $m$ and $E_0$, which is redundant, and further to introduce $m_r$ as the "mass unit" equivalent of total energy $E$ and redundantly keep both of these, too. From what I've seen, nobody used $m_k$. All this must have been very confusing to learners. How can one be expected to fully grasp the mass–energy equivalence when one is told to keep using conceptually equivalent physical quantities side by side? And the absence of $m_k$ couldn't have helped.

Over time it became clear that the "relativistic mass" concept was indeed leading beginners astray in all sorts of ways, so it was gradually abandoned in most textbooks and classrooms. Now we have $E$, $E_k$, $E_0$, and $m$. The latter two are still redundant, and it seems that $m$ is generally preferred over $E_0$, which means that equations like $E = mc^2 + E_k$ and $E^2 = (mc^2)^2 + (pc)^2$ are more common than the friendlier $E = E_0 + E_k$ and $E^2 = E_0^2 + (pc)^2$. I imagine that some beginners find this remaining redundancy confusing, but I don't think $m$ is going anywhere.

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?

 

[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.