A Usefulness of relativistic mass

[vanhees71]

As I said before, this is just an issue of pedagogy and communication/semantics. If one knows what one is doing and prefers to use "relativistic mass" in lieu of "total energy," then there's no problem. Likewise, there's no problem using "rest energy" in lieu of "invariant mass," which I tend to do.

How can this undercut the logical structure of the theory? It's like making a choice between "timelike interval" $ds$ and "proper time" $d\tau$—they're the same quantity measured in different units.

Well, then you can as well say didactics doesn't matter at all, and I can explain any physics in as complicated way I want. I doubt, whether this point of view is appreciated by students.

 

[stevendaryl]

But there is another pedagogical point. I see no reason to shield students against the phenomenon of changes of convention and notation. Such transitions are necessary whenever one switches from one field of research to another. They better get used to it.

I think that's a wonderful point. A lot of physics professionals get very upset with particular pedagogical choices, and say: "Don't teach things that way! It will only make the student more confused when he gets to more advanced topics!" People also get upset at the heuristic (or simplistic) explanations for things given in pop-science books. "It's misleading! They'll just have to unlearn that when they actually study the topic rigorously!"

I actually don't feel that way, at all. People who make it past a certain level in science generally learn that there is more than one way to look at a topic. Learning that your understanding is incomplete, and that some of your beliefs are misconceptions that must be corrected is really what growth in scientific maturity is all about. Perfecting pedagogy so that the student never needs to learn new conventions or never needs to fix misconceptions is both impossible and maybe not desirable, since it means giving the student a false impression of how orderly scientific progress is.

And I also have found that a lot of scientists were first inspired to become scientists by reading pop science books of questionable pedagogy. In a recent BackReaction blog post, Sabine Hossenfelder says that she was inspired to become a physicist by reading Hawking's "A Brief History of Time", even though she now considers it a bad book, from a pedagogical point of view.

 

[SiennaTheGr8]

Well, then you can as well say didactics doesn't matter at all, and I can explain any physics in as complicated way I want. I doubt, whether this point of view is appreciated by students.

I think we're actually in complete agreement, and just talking past each other a bit. As I've said, I oppose the use of relativistic mass in textbooks and teaching, precisely because it's confusing and makes things more complicated than they need to be.

 

[Herman Trivilino]

In SR, the relativistic mass is the inertial mass,

The real advantage in using only one kind of mass is pedagogical. If you look at introductory textbooks written in the last few decades you find that prior to 1990 the vast majority of them used more than one kind of mass. After 1990 more and more of them stopped. Now the vast majority use only one kind of mass.

 

[PAllen]

Or to neither: $\vec p = \gamma m \vec v$. Both the celerity $\gamma \vec v$ and the plain old 3-velocity are useful quantities, I think, and even if you choose celerity for this equation you'll want to break it down into $\gamma$ and $\vec v$ when differentiating with respect to time (to derive the relativistic 3-force).

Of course, the advantage to attaching the nonlinearity to mass is that it makes the expression applicable to [rest-]massless things like light. The disadvantage is that it uses "relativistic mass." The best of both worlds is $\vec p c = E \vec \beta$.
To be clear: are you referring here to the kinetic energy of a system's constituents (as measured in the system's rest frame)?

Answering your questions in order:

I wouldn't normally bother with celerity. gamma(v) * v is a nonlinear function of v that is normally best manipulated in that form (unless you go the hyperbolic function route).

Attaching nonlinearity to mass doesn't help much because m*gamma is undefined for m=0. What does help, after noting momentum as m*gamma*v, is to consider modification to Newtonian energy, arriving at the altogether new notion of total energy that includes mass as well as kinetic and potential energy. Then to note that momentum can now be written E*v, from which you have a consistent path to massless particles. Of course, I would make the transition 4-vectors before addressing generalization to things like massless particles.

As to your last question, yes, except that using covariant quantities, you compute total 4-momentum in any frame, just adding/integrating covariant quanitities. Then, the invariant mass is simply the norm.

 

[PAllen]

This is very strange! To the contrary I find the cleanest bridge from Newtonian to SR point mechanics is to keept the one and only mass known in Newton's theory, namely the invariant mass. The only bridge needed from Newton to SR, and this is admittedly a pretty difficult bridge to be built for beginners but it has to be built anyway, the relativity of simultaneity and thus the transformation properties of space and time coordinates. Also it's pretty clear that Newtonian mechanics should be approximately valid for particles moving with a speed much less than the speed of light and that thus in fact Newtonian time in the differential laws has to be substituted by the time in an instantaneous inertial rest frame, which is proper time.

This then leads to the definition of four-momentum
$$p^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=m c u^{\mu},$$
where $m$ is the invariant mass, which is the one and only mass appearing also in Newtonian point-particle mechanics.

Also the Newtonian equation of motion stays the same:
$$\frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=K^{\mu},$$
where $K^{\mu}$ is the Minkowski four-force.

I don't know what you are disagreeing with. What you write is completely consistent with what I wrote, as far as I understand.

 

[vanhees71]

I don't know what you are disagreeing with. What you write is completely consistent with what I wrote, as far as I understand.

Hm, then I misread somehow your posting. I thought you argued in favor of relativistic mass, as some modern textbook writers even do today.

 

[atyy]

What is "ADM mass/energy"?

The ADM energy is sometimes called the ADM mass (see the references in post #26), so there are analogous issues to how the energy is sometimes called the relativistic mass.

 

[atyy]

Argh ;-((. This is very bitter since 't Hooft is one of my heroes of physics, getting a Nobel prize for his PhD thesis :-((.

Don't worry, he's now degenerated to superdeterminism. Maybe he'll go to Mars soon https://www.mars-one.com/about-mars-one/ambassadors/prof.-dr.-gerard-t-hooft :)

One way trip :P

 

[vanhees71]

The ADM energy is sometimes called the ADM mass (see the references in post #26), so there are analogous issues to how the energy is sometimes called the relativistic mass.

I see, ok. That's about the problem of mass in GR, which is of course much more subtle than in SR. If you start in GR with non-covariant quantities for sure you are lost to begin with. That's why I don't know any idea to use something like "relativistic mass" in GR. How to define the mass of a self-gravitating body (like a neutron star), is of course another issue and much more complicated than the cure for the confusion in SR, where you simply use the notion of "invariant mass".

 

[atyy]

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.

Personally, before I went to university I was largely influenced by WGV Rosser's special relativity text which explicitly said it would not use the relativistic mass, only the invariant mass, since that would be simpler. I did of course know Feynman's treatment too, and thought it was very insightful. However, I usually worked my problems without relativistic mass following Rosser which I had studied more. Then at university, the textbook was Purcell, which I indeed never understood, but there are 2 things in there I like - the explanation of E and B field and the relativistic transforms, and the derivation of the Lamor formula. Years later, I was pleased to find that Schroeder agreed with me: http://physics.weber.edu/schroeder/mrr/mrrtalk.html

Anyway, I told my physics lecturers that I preferred to use only the invariant mass, and they (following Purcell), in their broad minded wisdom, told me to learn both.

Purcell has been dead a long time now, and he's been betrayed by Jackson and his new editors who've switched to SI units. Perhaps they'll remove the relativistic mass in the 4th edition.

 

[vanhees71]

Hm, yes, but isn't this very complicated too? You have to teach Maxwell's equations anyway in the introductory E&M theory lecture (usually they are already introduced before in the experimental lecture too) and in the traditional approach at the end ("if time permits" ;-)) you also teach the relativistic formulation. So given the Maxwell equations and an introduction to relativistic kinematics in terms of Minkowski four-vectors (if possible, avoid the $\mathrm{i} c t$ convention, which also spoils the beauty of relativity a lot, but I'm not as strictly against this as I'm against relativistic mass), why not simply analyzing the Maxwell equations in the usual way and formulate them in covariant form.

Then you have the right mathematical tools, making the physics most transparent, not being hindered by awful notation as Purcell, only because you think that math is some disease to be avoided as much as possible (which seems to me to be the reason, why Purcell failed in his Berkeley volume with his good idea to teach E&M right away as relatvistic theory). Then Purcell's example can be even more simplified to a single uniformly moving particle. We can then first use the rest frame of this particle, where the four-potential in Lorenz gauge is of course given by (I use Heaviside Lorentz units with $c=1$)
$$A^{0}=\Phi(\vec{x}), \quad \vec{A}=0, \quad \Phi(\vec{x})=\frac{Q}{4 \pi r}.$$
Now we can easily bring this in manifestly covariant form by introducing the four-velocity of the particle, $u^{\mu}$. Since it's $(1,0,0,0)$ in the particle rest frame and thus the invariant restframe distance to the particle is given by $r^2=(u \cdot x)^2-x^2$, you get the invariant vector field
$$\boldsymbol{A}=\Phi(\sqrt{(u \cdot x)^2-x^2}) \boldsymbol{u}.$$
With these few lines it is very clear that the field of moving charges has both electric and magnetic field components.

Of course, also the example in Purcell's book as well as in Schroeder's talk, is much more lucent using manifestly covariant notation and the fact that $(j^{\mu})=(\rho,\vec{j})$ is a four-vector field.

What I'm not so sure about concerning didactics is whether one should really teach E&M in the intro lecture really as relativistic theory from the beginning, because to get the necessary intuition for the subject you need to work out a lot of examples in the (1+3) formalism in a fixed reference frame anyway. The best books using a SRT-first approach I know of are

M. Schwartz, Principles of Electrodynamics, Dover
Landau&Lifshitz, vol. II

I think the former book's approach is doable in the intro theory lecture, particularly since nowadays SR is already introduced in the mechanics lecture before.

 

[sweet springs]

Hmm, it feels like it's question about semantics of the word "system". Whether "system" includes binding energy or not.

Say there are two similar both plus charged balls in system #1 and there are two similar plus and minus balls in system #2 in the same positions. Rest mass of system #1 > system #2 due to binding Coulomb energy. Does this example satisfy you?

 

[stevendaryl]

Here's a way to think about invariant mass that applies in both relativistic and nonrelativistic physics.

An object's trajectory (a slower-than-light object, anyway) is a set of moments/events that can be characterized by 4 coordinates, $(x,y,z,t)$. Those are external to the object itself. We can also talk about a fourth, internal coordinate, which I'll call $s$, which measures the elapsed time experienced by the object. (If the object has a built-in clock, then it would be the elapsed time on the clock. If it doesn't, then we can pretend it does.) Putting these 5 quantities together, we can come up with a 4-velocity $V$ characterizing the object's progress through spacetime as a function of its internal time:

$V$ is a vector with components $(\frac{dx}{ds}, \frac{dy}{ds}, \frac{dz}{ds}, \frac{dt}{ds})$.

This 4-vector $V$ associated with a moving object exists for both relativistic and nonrelativistic physics, even though nobody ever talks about the component $\frac{dt}{ds}$ in nonrelativistic physics, since it's always equal to 1 (in inertial Cartesian coordinates, anyway). But I'm going to use in both cases to illustrate the similarities between relativistic and nonrelativistic physics.

For simplicity, I'm going to assume no long-range interactions between objects. If an object is far away from other objects, then (both relativistically and nonrelativistically) its 4-velocity $V$ will be constant (its components in an inertial Cartesian coordinate system will be constants).

Now, let's consider interactions between moving objects. As I said, I'm ignoring long-range forces, which leave us with contact forces. Basically, objects can collide into each other. The collisions can change the course (the 4-velocity) of an object, they can break an object into two or more smaller objects, they can cause two or more small objects to merge into larger objects.

So the number of objects and their 4-velocities can be changed by a collision. But in both relativistic and non-relativistic physics, there is a cumulative, composite 4-velocity associated with the whole collection of objects involved in a collision which is left unchanged by the collision. The assumption is that the composite 4-velocity is linearly related to the 4-velocities of the objects that make it up. That is, if objects $O_1, O_2, ...$ with 4-velocities $V_1, V_2, ...$ collide, and produce objects $O_1', O_2', ...$ with 4-velocities $V_1', V_2', ...$, then there are real numbers $m_1, m_2, ..., m_1', m_2', ..., M$ and a 4-velocity $V_{composite}$ such that

$M V_{composite} = m_1 V_1 + m_2 V_2 + ...$
$M V_{composite} = m_1' V_1' + m_2 V_2' + ...$

(where vector addition for inertial cartesian coordinates just means componentwise)

So the masses of the objects are the weights (no pun intended) with which that object's 4-contributes to the composite 4-velocity. The weighted 4-velocities, $m_i V_i$ are the 4-momenta of the objects. So the sum of the 4-momenta gives a composite 4-momentum, which is conserved in collisions.

In nonrelativistic physics, we assume that the 4-th component of the 4-velocity is always equal to 1: $\frac{dt}{ds} = 1$. This implies that the 4-component of the equation for composite velocity satisfies:

$M = m_1 + m_2 + ...$

So mass is conserved, nonrelativistically. Relativistically, we don't assume that $\frac{dt}{ds} = 1$, so mass may not be conserved, but there will still be a conserved quantity for the composite system: the 4th component of the 4-momentum, which we identify with the total energy.

An interesting feature of this connection between relativistic and nonrelativistic kinematics: The 4th component of the 4-momentum gives rise to conservation of energy in the relativistic case, but to conservation of mass in the nonrelativistic case. There is no direct correspondence between the nonrelativistic kinetic energy and anything in the relativistic case. Kinetic energy is not conserved in collisions nonrelativistically, and is not the 4th component of a 4-momentum.

 

[stevendaryl]

Here's a way to think about invariant mass that applies in both relativistic and nonrelativistic physics.

What I think is nice about this treatment is that the way of defining mass as the invariant for 4-momentum: $m = \frac{1}{c^2} \sqrt{E^2 - p^2 c^2}$, has no counterpart in nonrelativistic physic. But defining mass as the weighting of an object's velocity to produce a conserved momentum works both relativistically and nonrelativistically.

 

[vanhees71]

The reason why this has no counterpart in nonrelatistic physics is worth thinking about. The reason is a subtle difference between the Lie groups describing the symmetries of the spacetime manifolds in special relativity (Minkowski space, an affine Lorentz manifold) and non-relativistic spacetime (a fiber bundle). The group-theoretical difference becomes fully appreciable in quantum theory, where the observable algebra is determined (or at least motivated) by group theory with the physics meaning given via Noether's theorems, according to which any one-parameter Lie symmetry implies the conservation of a quantity, which is the generator of the group action on the states (in classical mechanics phase-space distributions, in QT statistical operators on Hilbert space) and vice versa. That makes the 10 main observables of all theories based on these spacetimes quite uniquely defined (energy, momentum, angular momentum, generators for boosts).

Now in QT what you need are unitary ray representations. Now comes the difference: In SR the symmetry group is the proper orthochronous Lorentz group whose Lie algebra has no non-trivial central charges, i.e., any unitary ray representation can be lifted to a unitary representation on Hilbert space. This is not the case for the proper orthochronous Galileo group, which underlies non-relativistic Newtonian spacetime. It turns out that there is one non-trivial central charge, which turns out to be the mass in terms of physical quantities, and the massless case, leading to a proper unitary representation of the Galileo group, does not lead to any useful dynamics, which is a famous result by Eönü and Wigner.