A Usefulness of relativistic 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.