"relativistic mass" still a no-no?

[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.

 

[Mister T]

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".

 

[Mister T]

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.

 

[Mister T]

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.

 

[Mister T]

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.

 

[PeterDonis]

I am closing the thread for moderation.

 

[PeterDonis]

A number of off topic posts have been deleted and the thread is reopened.

 

[rude man]

The whole thing could have been avoided had the old way of calling rest mass m₀ and the relativistic mass m = γm₀. Then E=mc² would have survived, also centripetal force mv²/r. Like Regietheater opera, dumbing-down prevailed however. I was happy to see that even in the new millennial edition of Dr. Feynman's Lectures on Physics the editors decided to stick with what did not need fixin' 'cause it wasn't broke.

 

[Mister T]

The whole thing could have been avoided had the old way of calling rest mass m₀ and the relativistic mass m = γm₀. Then E=mc² would have survived, also centripetal force mv²/r.

You cannot make the Newtonian expression $\frac{mv^2}{r}$ valid by replacing $m$ with the relativistic mass.

Like Regietheater opera, dumbing-down prevailed however. I was happy to see that even in the new millennial edition of Dr. Feynman's Lectures on Physics the editors decided to stick with what did not need fixin' 'cause it wasn't broke.

Feynman states that you can replace $m$ in the expressions of Newtonian physics with the relativistic mass and create relations that are valid. It's been discussed in the literature that such a notion is in general wrong. There are a few important and often-used relations where that can be done, but in the general case it's not valid.

In other words, it's an oversimplification.

The fact is, high energy physicists have never changed the practice of referring to only one kind of mass in their work and in their professional publications. Some of them, when authoring books and articles for the public, have used the concept of relativistic mass.

I'll leave it to you to decide which arrangement is a "dumbing down".

The main argument for doing away with it was indeed that it allowed survival of $E=mc^2$. To people trying to learn physics it was obscuring the true meaning of Einstein's mass-energy relation. A misconception that often persisted into the professional phase of a physicist's life.

 

[SiennaTheGr8]

The whole thing could have been avoided had the old way of calling rest mass m₀ and the relativistic mass m = γm₀. Then E=mc² would have survived, also centripetal force mv²/r. Like Regietheater opera, dumbing-down prevailed however. I was happy to see that even in the new millennial edition of Dr. Feynman's Lectures on Physics the editors decided to stick with what did not need fixin' 'cause it wasn't broke.

The whole thing could have been avoided if Einstein and his contemporaries had stuck with either "mass" or "energy" for all mass/energy terms. Instead they adopted a confusing mishmash of various $m$'s and $E$'s.

There's nothing that $m$ quantifies that $E$ doesn't. They're the same quantity in different units.

There's nothing that $m_0$ quantifies that $E_0$ doesn't. They're the same quantity in different units.

Why we've ended up with $E$ and $m_0$ (now usually just called $m$) is beyond me. I use $E$ and $E_0$. If an answer is needed in mass units, I convert.

 

[DrStupid]

You cannot make the Newtonian expression $\frac{mv^2}{r}$ valid by replacing $m$ with the relativistic mass.

You can. It's one of the special cases where it works. If you should it's another question.