"relativistic mass" still a no-no?

[Herman Trivilino]

I provided you with Newton's explicit definition for quantity of matter. It's not my problem if it doesn't satisfied your requirement. I don't need it.

Apparently, if you want to make yourself understood you do need it. You may not need it for other purposes, but our purpose here is to have others understand us. Otherwise, why post?

 

[pervect]

This is a long thread, and maybe I'm confused on people's positions. Are the people who are arguing for mass as a "quantity of matter" arguing for, or against, the use of relativistic mass? My perspective is that if one favors the idea that mass is, in some sense, a "quantity of matter" one would logically favor a formulation of mass that doesn't change when it's velocity changes. Otherwise one is left in the unfortunate position of claiming that when observer A is moving relative to observer B, an object C, representing some isolated system, has a different quantity of matter for observer A than for observer B. Which seems to me to be rather against the whole spirit of the idea of mass as "a quantity of matter".

For an isolated system, invariant mass is the sort of mass that is best suited to be called a "quantity of matter", because observers A and B will agree on the quantity of matter in C using this definition.

But is seems to me that the people who are arguing for mass as a "quantity of matter" are the same ones who are favoring relativistic mass. This has me scratching my head, and thinking that perhaps I've missed something in this long thread. (Which wouldn't be the first time).

 

[Herman Trivilino]

This is a long thread, and maybe I'm confused on people's positions.

It started out being a question about why there was a recommendation in a Wikipedia article against the use of relativistic mass. There were many responses but mine included a reference to "Newtonian mass" that created a kerfuffle. That was then followed by DrStupid's claim that Newton's quantity of matter is, well, something. Ever since then we've been trying to figure out what that something is. He's the only one who seems to be promoting the use of relativistic mass. But his justification seems to be that it's a match to what Newton meant by quantity of matter. One thing everyone seems to agree upon is that no one can make sense of his arguments.

In Newton's time quantity of matter was a big deal because people were, at that time, able to establish it as a standard for purposes of trade. Using a balance to weigh something was, and still is, what merchants do to determine the quantity of matter. They measure what we call the mass.

No merchant would want to buy and sell something using a measure that varied with speed, temperature, or location.

 

[Herman Trivilino]

For an isolated system, invariant mass is the sort of mass that is best suited to be called a "quantity of matter", because observers A and B will agree on the quantity of matter in C using this definition.

For that reason it is better suited than a mass that's different for observers A, B, and C.

But even then (invariant) mass is only as good as the Newtonian approximation, because the energies of the constituents of a composite body make contributions to the (invariant) mass of that body. The Newtonian approximation is valid only because those contributions are negligible. For example, if the contributions due to the increased thermal energies associated with an increase in temperature weren't negligible, we'd have a quantity of matter that varies with temperature.

 

[vanhees71]

Well, there is some progress in our understanding of the fundamental properties since Newton's times, and nowadays the basic notions are defined via mathematical structures in the theories, mostly symmetry principles, which have proved to be very successful in analyzing the mathematical description of matter. For me that's the main merit of Einstein's famous paper on Special Relativity. The first sentence, can be read as a working program of theoretical physics for the next centennium: "It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena." Cited from the English translation here:

http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

From this point of view mass has a different meaning in Newtonian and relativistic physics. The basic description of matter is in terms of elementary particles, which are defined as the quantum systems that are described by irreducible ray representations of the symmetry group (or its Lie algebra) of space and time. In Newtonian physics that's the orthochronous Galileo group and for special relativistic physics the proper orthochronous Poincare group. In the former it turns out that if you try to lift the ray representations of the classical Galileo group to unitary representations in Hilbert space, you don't obtain a QT with sensible dynamics. Thus you have to use a non-trivial central extension of the Galileo group which introduces mass as a central charge of the corresponding Lie algebra, and this defines the usual non-relativistic QT (e.g., realized by the Schrödinger wave-mechanics formulation). For the Poincare group there are no non-trivial central extensions, and the mass is a Casimir operator of the Lie algebra, which leads to the energy-momentum relation ("on-shell condition") for free particles.

That's why it is very clear that mass is what's called "invariant mass", while what in the older days (before Minkowski's work in 1908) was called "relativistic mass" is just the energy of a free particle (divided by $c^2$) and as such the temporal component of four-momentum.

 

[stevendaryl]

Maybe somebody else remembers the details better than I do, but there was a Physics Forums poster from a good number of years back who argued that a good way to compare relativistic and nonrelativistic physics was by starting with the Galilei group of transformations and then looking at how they relate to the Poincare group. He claimed that relativistic mass was helpful in understanding the relationship between these two groups, but I don't remember anything about the details.

Maybe the author was Mark Hopkins, or something like that?

 

[stevendaryl]

Maybe somebody else remembers the details better than I do, but there was a Physics Forums poster from a good number of years back who argued that a good way to compare relativistic and nonrelativistic physics was by starting with the Galilei group of transformations and then looking at how they relate to the Poincare group. He claimed that relativistic mass was helpful in understanding the relationship between these two groups, but I don't remember anything about the details.

Maybe the author was Mark Hopkins, or something like that?

He used an 11-parameter Galilei group, although Google searches show only a 10-parameter group. I assume those were: Time translation (1 parameter), spatial translations (3 parameters), rotations (3 parameters), boosts (3 parameters). I don't know what the 11th parameter was supposed to be.

 

[dextercioby]

He used an 11-parameter Galilei group, although Google searches show only a 10-parameter group. I assume those were: Time translation (1 parameter), spatial translations (3 parameters), rotations (3 parameters), boosts (3 parameters). I don't know what the 11th parameter was supposed to be.

The only 11-parameter extension of the (proper) Galilei group which has physical meaning is the central extension by the (unit operator times the Newtonian, invariant) mass. It's the group of symmetry of non-relativistic physics (mentioned by vanHees71 above in post# 65) whose universal covering group is of utmost importance in Quantum Mechanics (as per the work of Wigner/Bargmann/Mackey/Levy-Leblond).

//

My saying related to the topic at hand. Back in college my Classical Mechanics teacher told us in the first lecture that in Newton's formulation of classical (particle, as hydrodynamics had not been discovered before 1700) mechanics the mass of a body had the meaning of "quantity of substance of that body with the sense used in chemistry"(1). Ironically, in 1687 (when "Principia..." was published) chemistry was not a NUMERICAL science, meaning that in an (al)chemist lab there was no balance (scale) to weigh whatever substances were there to react. The balance was first used in chemistry by A-L. Lavoisier around 1770 who came up (in 1774) with the clear statement of (mass = quantity of substance entering chemical reactions) mass conservation. I could go on to tell you the story with Dalton, Avogadro, Mendeleev, J.J. Thomson, Mulliken, etc., but I'd rather leave the famous historian Max Jammer tell it: https://www.amazon.com/dp/0486299988/?tag=pfamazon01-20

(1) was my teacher's take of the subject. Upon reading since then, I believe the technical definition in terms of the non-trivial central extension of the proper Galilei group/1st Casimir of the proper Poincare group is the best we have.

 

[harrylin]

If one chooses to use the term "observer" to refer to a reference system in which that observer is at rest, as it is clear that @Mister T does then the kinetic energy, velocity and momentum of an object change as a function of the observer's velocity relative to that object precisely because those things depend on the choice of reference system.

Once more, that's a misapplication of the laws of physics - they are not valid between reference systems. Newton's first law, energy conservation etc. all don't work, it's just nonsense.

 

[harrylin]

Its history is not relevant, however physical it might have been. It makes no difference if at sometime in its past it accelerated.

I was talking about invalid physics, not history...

 

[jbriggs444]

Once more, that's a misapplication of the laws of physics - they are not valid between reference systems. Newton's first law, energy conservation etc. all don't work, it's just nonsense.

Non-inertial frames are not automatically nonsense. You need to re-think that claim.

 

[stevendaryl]

Non-inertial frames are not automatically nonsense. You need to re-think that claim.

There is an ambiguity about what "Newton's laws" are, in the context of noninertial coordinates. The second law of motion is written in inertial Cartesian coordinates as:

F^j = m \frac{d^2 x^j}{dt^2}

That form for the second law only works for inertial Cartesian coordinates. Using other kinds of coordinates, there are additional terms appearing on the right-hand side (due to nonzero connection coefficients, which give rise to terms such as the "Coriolis force" and "Centrifugal force"). You can try to fix things by moving those terms to the left side, and calling them forces, but that contradicts Newton's third law (because there is no equal and opposite force corresponding to the Centrifugal force).

However, you can generalize Newton's laws of motion so that they have the same form in every coordinate system, inertial or not.

 

[vanhees71]

He used an 11-parameter Galilei group, although Google searches show only a 10-parameter group. I assume those were: Time translation (1 parameter), spatial translations (3 parameters), rotations (3 parameters), boosts (3 parameters). I don't know what the 11th parameter was supposed to be.

The 11th parameter is mass, and it's a non-trivial central charge, and that's it's role in non-relativistic quantum theory. For details, see, e.g., Ballentine, Quantum Mechanics.

 

[vanhees71]

There is an ambiguity about what "Newton's laws" are, in the context of noninertial coordinates. The second law of motion is written in inertial Cartesian coordinates as:

F^j = m \frac{d^2 x^j}{dt^2}

That form for the second law only works for inertial Cartesian coordinates. Using other kinds of coordinates, there are additional terms appearing on the right-hand side (due to nonzero connection coefficients, which give rise to terms such as the "Coriolis force" and "Centrifugal force"). You can try to fix things by moving those terms to the left side, and calling them forces, but that contradicts Newton's third law (because there is no equal and opposite force corresponding to the Centrifugal force).

However, you can generalize Newton's laws of motion so that they have the same form in every coordinate system, inertial or not.

Of course, the "fictitious forces" (I like to call the "inertial forces") are no forces at all but belong to the left-hand side of the equation and are just parts of the components of acceleration in a non-inertial frame of reference. It's nothing mysterious about them. It's of course easily possible to extent this study to special relativity, where you can as well use (local) non-inertial reference frames. In GR there's no need to distinguish inertial and non-inertial frames since it's a theory that's generally covariant anyway. In GR it's the other way around: The local inertial frames are the special cases, but the (weak) equivalence principle tells you that at any (regular) point of space-time there's a class of local inertial frames.

 

[stevendaryl]

Of course, the "fictitious forces" (I like to call the "inertial forces") are no forces at all but belong to the left-hand side of the equation

Well, in F = ma, they belong on the right side.

and are just parts of the components of acceleration in a non-inertial frame of reference. It's nothing mysterious about them.

Yes, I agree. My point is that there are two unfortunate responses to the presence of the extra terms:

1. To say that Newton's laws only apply in an inertial frame.
2. To say that F = m \frac{d^2 x}{dt^2} applies in every frame, which means that the extra terms are noninertial forces.

The correct (in my opinion) approach is that F=ma is true in every frame, but it's only in inertial frames that the acceleration is equal to \frac{d^2 x}{dt^2}.

 

[vanhees71]

In a manifest covariant formulation they are automatically all on the left-hand side, but never mind, that's again just an empty semantics discussion.

 

[Herman Trivilino]

I was talking about invalid physics, not history...

Let's say you have two identical cars, $\mathrm{A}$ and $\mathrm{B}$, on a freeway moving in opposite directions, and in the rest frame of the freeway each has $0.45 \ \mathrm{MJ}$ of kinetic energy.

Applying the definition of kinetic energy we reach the following conclusion:

In the rest frame of car $\mathrm{A}$, car $\mathrm{B}$ has a kinetic energy of $1.80 \ \mathrm{MJ}$.

But we already know that in the rest frame of the freeway car $\mathrm{B}$ has a kinetic energy of $0.45 \ \mathrm{MJ}$.

The velocity and kinetic energy of a fast particle cannot change due to your relative speed to it

The notion of a "fast particle" has to be determined from measurements taken using a reference frame. Unless that reference frame is known the notion is meaningless.

For example, in a different reference frame measurements indicate that it's a "slow particle".

Such values are relative in that they depend on your choice of reference system, which does not mean that they can fluctuate as function of your velocity relative to it.

In the example I gave above an observer at rest in the freeway's rest frame will observe that car $\mathrm{B}$ has a speed of $30 \ \mathrm{m/s}$.

If that observer changes rest frames so that he is at rest in car $\mathrm{A}$'s rest frame he will observe that car $\mathrm{B}$ has a speed of $60 \ \mathrm{m/s}$.

Once more, that's a misapplication of the laws of physics - they are not valid between reference systems. Newton's first law, energy conservation etc. all don't work, it's just nonsense.

Can you show us how to apply the laws of physics in such a way that you get values that are different from mine?

Consider the fact that those two identical cars could have been manufactured six months apart in the same factory. One was made in March when Earth, in its orbit around the sun, was moving at a speed of $\mathrm{30\ km/s}$ in one direction, as measured in the sun's rest frame. Six months later it's September and Earth is moving in the opposite direction at a speed of $\mathrm{30\ km/s}$.

 

[DrStupid]

Using other kinds of coordinates, there are additional terms appearing on the right-hand side (due to nonzero connection coefficients, which give rise to terms such as the "Coriolis force" and "Centrifugal force"). You can try to fix things by moving those terms to the left side, and calling them forces, but that contradicts Newton's third law (because there is no equal and opposite force corresponding to the Centrifugal force).

In his personal copy of the Principia Newton changed that with a http://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/49 . But that didn't become generally accepted.

 

[stevendaryl]

In a manifest covariant formulation they are automatically all on the left-hand side, but never mind, that's again just an empty semantics discussion.

I'm just saying that whether they are on the left-hand side or the right-hand side depends on whether you're writing F=ma or ma=F. It's not deep. I was writing F=ma.

 

[harrylin]

Let's say you have two identical cars, $\mathrm{A}$ and $\mathrm{B}$, on a freeway moving in opposite directions, and in the rest frame of the freeway each has $0.45 \ \mathrm{MJ}$ of kinetic energy.

Applying the definition of kinetic energy we reach the following conclusion:

In the rest frame of car $\mathrm{A}$, car $\mathrm{B}$ has a kinetic energy of $1.80 \ \mathrm{MJ}$.

But we already know that in the rest frame of the freeway car $\mathrm{B}$ has a kinetic energy of $0.45 \ \mathrm{MJ}$.

[..]

Can you show us how to apply the laws of physics in such a way that you get values that are different from mine? [..].

That's a misunderstanding of what I said. Thus, maybe it's just a matter of semantics. You and jbriggs ignored my clarification that physically it is different if the car changes velocity or if you change velocity. According to you, the kinetic energy of a system changes (and thus is not conserved) when someone chooses to change the reference frame with which he determines the energy. I said that that is nonsensical, it's similar to saying that I can make you run in circles by changing my reference system. You cannot affect that system, instead the kinetic energy is different according to different reference systems.

Anyway, I still don't know if this has any relevance to the topic, as you forgot to clarify what your point was!

 

[harrylin]

[..] No merchant would want to buy and sell something using a measure that varied with speed, temperature, or location.

However, that's exactly what most merchants do - already length and volume depend on temperature - and even on speed and location. In real life one has to work with "standard" conditions. Why would a merchant prefer a convention rule for "mass" that is not consistent with that for "volume"?

 

[jbriggs444]

That's a misunderstanding of what I said. Thus, maybe it's just a matter of semantics. You and jbriggs ignored my clarification that physically it is different if the car changes velocity or if you change velocity. According to you, the kinetic energy of a system changes (and thus is not conserved) when someone chooses to change the reference frame with which he determines the energy. I said that that is nonsensical, it's similar to saying that I can make you run in circles by changing my reference system. You cannot affect that system, instead the kinetic energy is different according to different reference systems.

We seen to be arguing about terminology, not physics. The energy of a system is different when you use different reference frames. If one chooses to say that it "changes" when one "changes" reference frames instead, that seems to be an acceptable use of prose and not "nonsensical".

 

[Herman Trivilino]

You and jbriggs ignored my clarification that physically it is different if the car changes velocity or if you change velocity.

It doesn't appear to me that either of us ignored it. What I said is it's part of the history, and is therefore not relevant to what is happening now. There is no experiment you can do to distinguish whether it was the observer or the object that was accelerated in the past. Indeed, it's very possible that neither was acclerated in its past. The two may have been born in relative motion.

According to you, the kinetic energy of a system changes (and thus is not conserved) when someone chooses to change the reference frame with which he determines the energy.

Except for the part in parentheses, that is correct.

I said that that is nonsensical,

I must have missed that. I was responding to your claim that it's a misapplication of the laws of physics. If it doesn't make sense to you, that's a different issue.

Anyway, I still don't know if this has any relevance to the topic, as you forgot to clarify what your point was!

I applied the laws of physics, showing by example, that a change in the observer's rest frame does indeed change the kinetic energy. Hopefully that clarifies my point.

 

[Herman Trivilino]

However, that's exactly what most merchants do -

When merchants determine the quantity of matter they use a balance to determine what physicists call the mass. That's a requirement imposed by law.

 

[jbriggs444]

When merchants determine the quantity of matter they use a balance to determine what physicists call the mass. That's a requirement imposed by law.

In general, it will vary by jurisdiction. However, my understanding is that most commercial scales are based on electronic load cells rather than balances. The legal requirement has to do with periodic calibration -- against a set of known masses.

I like to think that a balance compares a test mass against a known mass using the assumption that gravity is uniform across the relevant distance and that a load cell compares a test mass against a known mass using the assumption that gravity is uniform across the relevant time.

 

[Herman Trivilino]

In general, it will vary by jurisdiction. However, my understanding is that most commercial scales are based on electronic load cells rather than balances. The legal requirement has to do with periodic calibration -- against a set of known masses.

In practice, yes, they will use load cells. But as you say, they must be calibrated against standards, and ultimately it's a balance that's used to certify those standards. My understanding that a balance is still more precise than a load cell.

 

[harrylin]

When merchants determine the quantity of matter they use a balance to determine what physicists call the mass. That's a requirement imposed by law.

Exactly - but it seems that you missed the point that I made. Length, volume and mass are usually measured at 1 bar at room temperature and in rest and locally. There is no reason to treat "mass" differently from "volume" and the issue is effortlessly taken care of by the merchants.

 

[jbriggs444]

Exactly - but it seems that you missed the point that I made. Length, volume and mass are usually measured at 1 bar at room temperature and in rest and locally. There is no reason to treat "mass" differently from "volume" and the issue is effortlessly taken care of by the merchants.

Agreed. We have no serious problem with gold merchants labeling their product with [relativistic] masses computed based on the rest frame of some particle at CERN. And it is not worth the energy expenditure to heat the bars before weighing, thereby increasing their [invariant] mass.

Which seems to make your point -- we cannot look to merchants to definitively disambiguate such distinctions about the meaning of "quantity of matter".

 

[Fausto Vezzaro]

In my opinion, relativistic mass is useful in general only in the context defined by these 7 assumption:

1. speed of light is isotropic only in one inertial frame (that we can call absolute space, presumably anchored to cosmic masses).
2. when an object is in absolute motion, its dimensions are contracted along the directions of motion by γ factor.
3. when an object is in absolute motion, all his processes are affected by a slowdown by a γ factor.
4. observers in different inertial frames synchronize clock using Einstein convention.
5. For observers in absolute rest, classical physics works.
6. When a point mass is in absolute motion, its mass is increased by a γ factor, while charge remain unchanged.
7. If a force $\mathbf{F}$ is applied on a body then, in virtue of its absolute motion, is exerted an additive force $$- ( \mathbf{F} \cdot \mathbf{u} ) \frac{\mathbf{u}}{c^2}$$where $\mathbf{u}$ is the absolute speed.

First 4 assumptions are equivalent to relativistic kinematics. Last 3 ones gives relativistic electrodynamics: every inertial observer can use classical electromagnetism (i.e. Maxwell equations and Lorentz force) finding results of relativistic electrodynamics and having no way to detect absolute motion. You can be forced to consider last 3 assumption by considering that
$$
\mathbf{F}=\frac{d}{dt}(\gamma m \mathbf{u})
$$
can be written in this way
$$
\gamma m \mathbf{a} = \mathbf{F} - (\mathbf{F} \cdot \mathbf{u}) \frac{\mathbf{u}}{c^2}
$$

 

[Nugatory]

In my opinion, relativistic mass is useful in general only in the context defined by these 7 assumption:

1. speed of light is isotropic only in one inertial frame (that we can call absolute space, presumably anchored to cosmic masses).
2. ...

If these are the only conditions under which relativistic mass is useful, then it appears that it is not useful at all - #1 is inconsistent with experimental results.
(If you, not unreasonably, believe that this argument demands a response, start a new thread - it's a digression here. But do review our https://www.physicsforums.com/insights/pfs-policy-on-lorentz-ether-theory-and-block-universe/ first.