"relativistic mass" still a no-no?

[DrStupid]

So, for you, "quantity of matter" is the the m in $p=m \gamma v$.

No, according to definition 2 quantity of matter is the $m \gamma$ in $p=m \gamma v$. Of course that makes no difference in classical mechanics due to

$\mathop {\lim }\limits_{c \to \infty } \frac{m}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} = m$

 

[Mister T]

I use it as a definition of quantity of matter (see above).

What then is your definition of mass?

 

[Mister T]

The SI unit for the "amount of substance" is not kilogram, which is the unit for mass, but mol.

Right. Since we measure mass in kilograms and amount of substance in moles we don't measure amount of substance in kilograms.

I guess I don't understand your point.

 

[DrStupid]

What then is your definition of mass?

Mass can be defined in many different ways. One of the most popular definitions is based on the Minkowski norm of the four-momentum:

$m = \frac{{\sqrt {E^2 - p^2 c^2 } }}{{c^2 }}$

In order to keep the link to classical mechanics it can also be defined as the quantity of matter (aka relativistic mass) in the rest frame of the system.

 

[PeterDonis]

That answers your question for the explicit definition.

Yes, but now we need an explicit definition of "volume" and "density". Are those explicitly defined in the Principia?

And since both of those things behave differently in relativity than they do in Newtonian mechanics, I don't see how any of this argues against the point I have been making.

 

[Mister T]

In order to keep the link to classical mechanics it can also be defined as the quantity of matter.

Then your original claim that the two are equal is, as I already told you, a tautology.

 

[DrStupid]

Yes, but now we need an explicit definition of "volume" and "density".

For which purpose?

I don't see how any of this argues against the point I have been making.

You need to be more specific. Which point are you talking about?

 

[DrStupid]

Then your original claim that the two are equal is, as I already told you, a tautology.

Learn to quote correctly.

 

[PeterDonis]

For which purpose?

Because without such explicit definitions, your definition of "quantity of matter" is not explicit either. It just pushes back the implicitness one step, so to speak.

Which point are you talking about?

See post #5 of this thread.

 

[Mister T]

I don't define "quantity of matter" to be the mass

[mass] can also be defined as the quantity of matter

When one term is used to define another, then a claim that the two terms are equivalent is a mere tautology, devoid of any meaning.

When we're told the quantity of matter contained in a body equals $\gamma m$ it implies that the quantity of matter contained in that body depends on the relative motion of an observer. Hence it changes according to the observer's speed relative to it.

When we're told the quantity of matter contained in a body is $m$ it implies that the quantity of matter depends on the energies of the body's constituents relative to its rest frame. Hence it changes when those energies change.

 

[harrylin]

[..] When we're told the quantity of matter contained in a body equals $\gamma m$ it implies that the quantity of matter contained in that body depends on the relative motion of an observer. Hence it changes according to the observer's speed relative to it.[..]

I'm not sure what you try to argue there, but it's simply wrong, due to a misapplication of the laws of physics. Physically it's not the same if you accelerate or if that body accelerates. The velocity and kinetic energy of a fast particle cannot change due to your relative speed to it - that would be magical action at a distance. 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.

 

[jbriggs444]

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.

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.

 

[Mister T]

Physically it's not the same if you accelerate or if that body accelerates.

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

 

[DrStupid]

Because without such explicit definitions, your definition of "quantity of matter" is not explicit either.

Of course not. I already told you in #26 that I use an implicite definition. What's your point?

See post #5 of this thread.

OK, let me see:

No, it isn't, and that's a big part of the problem. In Newtonian physics, the concept of "mass" conflated several different things that, in relativistic physics, turn out to be different. One of those things (roughly, "quantity of matter") turns out to correspond with $m$; another (roughly, "amount of inertia") turns out to correspond with $\gamma m$ (with some caveats, since the relationship between force and acceleration is direction-dependent in SR).

My calculation shows that "quantity of matter" also corresponds to $\gamma m$ and the resulting relationship between force and acceleration is also direction-dependent in SR.

And yet a third (gravitational mass) turns out to correspond with neither, since in GR the source of gravity is not "mass" but the stress-energy tensor.

That's not a problem of quantity of matter but of the law of gravitation.

 

[DrStupid]

When one term [quantity of matter] is used to define another [mass], then a claim that the two terms are equivalent is a mere tautology, devoid of any meaning.

I don't claim that the two terms are equivalent and I do not use one of them to define the other in the way that your truncated quote suggests. This is what I actually wrote:

And it [relativistic mass] is identical with Newton's quantity of matter.

I don't define "quantity of matter" to be the mass and that wouldn't be true.

In order to keep the link to classical mechanics it [mass] can also be defined as the quantity of matter (aka relativistic mass) in the rest frame of the system.

 

[jbriggs444]

My calculation shows that "quantity of matter" also corresponds to $\gamma m$

You cannot successfully calculate "quantity of matter" without having a definition for "quantity of matter". So... what is your definition for "quantity of matter"?

 

[DrStupid]

You cannot successfully calculate "quantity of matter" without having a definition for "quantity of matter".

I actually did it in #33.

 

[jbriggs444]

I actually did it in #33.

Without a definition for density, that's a little pointless, don't you think?

 

[PeterDonis]

I already told you in #26 that I use an implicite definition.

And then in #29 I asked you for an explicit one, and in #33 you gave a definition that you claimed satisfied my requirement for an explicit definition. But that's not the case unless you can also give an explicit definition of "volume" (Newton used the term "bulk") and "density". Which you haven't.

 

[DrStupid]

Without a definition for density, that's a little pointless, don't you think?

Did you even read #33?

 

[jbriggs444]

Did you even read #33?

Yes, I did. The relevant definition for "quantity of matter" was as the product of volume and density. The rest of the post went on without providing any definitions to ground either of those terms.

 

[DrStupid]

The relevant definition for "quantity of matter" was as the product of volume and density.

What makes this definition relevant? I didn't used it in my calculation.

 

[DrStupid]

And then in #29 I asked you for an explicit one, and in #33 you gave a definition that you claimed satisfied my requirement for an explicit definition. But that's not the case unless you can also give an explicit definition of "volume" (Newton used the term "bulk") and "density". Which you haven't.

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.

 

[jbriggs444]

What makes this definition relevant? I didn't used it in my calculation.

It seems that you want to work backwards. You are taking momentum as primitive and using the assertion that $p=q \cdot v$ as the defining property for quantity of matter q.

That's fine, but if you are going to do that, it would be good to discard the other definition (or be clear that you are interpreting Newton to be defining density in terms of momentum).

 

[DrStupid]

That's fine, but if you are going to do that, it would be good to discard the other definition (or be clear that you are interpreting Newton to be defining density in terms of momentum).

1. Post #33 was an answer to PeterDonis who asked me for an explicite definition for quantity of matter. I did him this favour but explained why this definition is neither helpful nor required to derive the properties of quantity of matter. For this purpose I needed to put it toghether with my calculation into the same post.

2. There is no reason to discard definition 1. It might not helful for this topic but that doesn't make it wrong. I already wrote in #26 that we today use it to define density.

 

[Mister T]

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).

 

[Mister T]

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.

 

[Mister T]

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...