Hecht's ResponseEugene Hecht

Citation: The Physics Teacher 48, 5 (2010); doi: 10.1119/1.3274347 View online: http://dx.doi.org/10.1119/1.3274347 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/48/1?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Whittaker and Special Relativity Phys. Teach. 48, 4 (2010); 10.1119/1.3274344 “Teaching Special Relativity Without Calculus” Revisited Phys. Teach. 47, L3 (2009); 10.1119/1.3196257 The Wondrous New World of Modern Particle Astrophysics Phys. Teach. 47, 274 (2009); 10.1119/1.3116835 Understanding the Fine Tuning in Our Universe Phys. Teach. 46, 285 (2008); 10.1119/1.2909746 Lorentz Transformation and Charge Conservation Phys. Teach. 45, 328 (2007); 10.1119/1.2768683

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4 The Physics Teacher ◆ Vol. 48, January 2010

letters to the editor

Fish in Space

The cover story about the goldfish (Oct. 2009)1 reminded me about a classroom discussion that we had in a college physics class. After I told the class that water would be formed into a ball by surface tension in an orbiting space capsule, a student asked me if we could put a goldfish in the ball of water! I replied that it probably could be done very carefully so that the ball of water would not be broken into many droplets in the cabin.“But would the fish just swim out of the water through the surface tension?” “Would the ball of water move in the opposite direction of the fish to conserve momen-tum?” “If the fish swam in a circle, would the ball rotate in the opposite direction?” “Could you actually insert the fish without breaking up the ball of water?”

The discussion prompted me to call a friend working as a contractor for NASA. I was told that goldfish had been in orbit but always in a closed container. You can find interesting phys-ics everywhere!

1. Yuhua Zhu and Fengliang Shi, “Why does the goldfish disap-pear in the fishbowl?” Phys. Teach. 47, 424 (Oct. 2009).

Terrence ToepkerDept. of Physics

Xavier University

Whittaker and Special RelativityI was pleased to see John F. Devlin’s article “A More Intuitive Version of the Lorentz Velocity Addition Formula” in the Oc-tober 2009 issue of The Physics Teacher.1 I looked immediately to see if my own paper2 on the same topic had been cited, but was disappointed to see that it had not. While Devlin and I probably both thought we had found a new way to look at this problem, we had been anticipated by many years in the work of Edmund Whittaker,3 something that had been kindly pointed out to me by Ariel Valladares4 soon after my paper appeared.

This form of the velocity addition formula has much more than mnemonic value in the teaching of special relativity. The formula can be derived directly from the postulates upon which the theory is based, and from that starting point Whit-taker goes on to derive Fitzgerald contraction and other famil-iar consequences. I recommend all teachers of special relativ-ity read this approach.

1. John F. Devlin, Phys. Teach. 47, 442 (Oct. 2009). 2. Leigh Hunt Palmer, “Simple form of the law of addition of par-

allel velocities from the special theory of relativity,” Am. J. Phys. 44, 702 (1976).

3. Edmund Whittaker, From Euclid to Eddington: A Study of Con-ceptions of the External World (Cambridge University Press,

London, 1949), pp. 49–50 (based on his 1947 Tarner Lectures in Trinity College, Cambridge).

4. Ariel A. Valladares, “Concerning the addition of parallel ve-locities in the special theory of relativity,” Am. J. Phys. 45, 578 (1977). Whittaker’s derivation of the relativistic velocity for-mula is reproduced here.

Leigh Hunt Palmer Dept. of Physics

Simon Fraser University leigh@sfu.ca

Recasting the Lorentz Velocity Addition FormulaIn his article “A More Intuitive Version of the Lorentz Veloc-ity Addition Formula,” John F. Devlin compares the standard form of the velocity addition formula,

,+=+ 21 /

AB BCAC

AB BC

V VV

V V c

with a proposed alternate form, = + − 2/ ,AC AB BC AC AB BCV V V V V V c

and argues that the latter form 1) has a more intuitive con-nection to the low-velocity, Galilean limit, 2) is more memo-rable, 3) reminds us that relativistic velocities “add” to less than the algebraic sum (at least when in the same direction), and 4) is easier to work with algebraically. I might argue that the standard formula better satisfies at least the first three objectives, but these are certainly matters of taste.

My purpose here is only to point out that one can perhaps best satisfy at least the goals of memorability, algebraic sim-plicity, and aesthetic symmetry by writing the formula as

β β β β β β+ + =−AB BC CA AB BC CA

with β≡ / .V c

John MallinckrodtProfessor Emeritus of Physics

Cal Poly Pomona

The Definition of MassI appreciated Eugene Hecht’s article “Einstein Never Approved of Relativistic Mass,” especially the historical insights.1 But I disagree with his insistence that the word “mass” must be defined as rest mass. For example, Hecht states, “After all, there is no such thing [as relativistic mass]; mass is Lorentz invariant” (my italics). At another point, Hecht refers to the “correct relativistic form of p” (my italics), namely p = gmv, implying that the correct meaning of “mass”

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The Physics Teacher ◆ Vol. 48, January 2010 5

Hecht’s ResponseProfessor Hobson makes a few interesting points regarding my article that deserve to be addressed. My paper seeks to establish that although Einstein embraced speed-dependent mass in his June ’05 paper on relativity, within a year or so he changed his mind and shunned the notion of relativistic mass when it was later devised (ca. 1908). Today there are two com-peting schools of thought when it comes to interpreting the special theory: those who take mass to be speed-dependent, and those who follow Einstein and take mass to be Lorentz invariant (i.e., speed-independent). Professor Hobson has astutely detected that I am a proud member of the “relativistic mass is utter nonsense” contingent and he uses this as a raison d’ être for a defense of that anachronistic concept.

Hobson writes, “My point is that the meaning of ‘mass’ is entirely a question of semantics, not physics.” This is an old and, alas, faulty argument, and I could not disagree with it more. Establishing the meaning of “mass,” or any other basic operating concept in physics, is fundamental to physics; it is a significant part of the very process of doing physics. Simply because it is hard to accomplish, establishing the meanings of the concepts we use, as best we can, is an ongoing process and should not be considered “a question of semantics,” especially not as a justification for a lack of rigor. Either mass is a func-tion of speed or it is not; there is nothing semantical about it.

Hobson goes on to suggest that “[w]e should choose the definition having the greatest conceptual clarity.” Again, re-spectfully, I disagree. We should choose the definition that is the most accurate in its congruence with Nature, the one that is in greatest harmony with experimental facts, and our communal understanding of those facts. We should choose the definition that best relates to our best theoretical formula-tions. If that definition is obscure, or conceptually complex, or incomplete, we have to live with it and work to do better. But to accept an idea like relativistic mass as an alternative because of its purported “conceptual clarity,” despite its obvious short-comings, is folly.

In physics, powerful formulations replace less powerful ones; we don’t hang on to weak limited ideas (e.g., Aristotle’s law of motion, the continuous universe, or the raisin pudding model of the atom) because there are circumstances for which they offer “conceptual clarity.” Wrong ideas have often offered “conceptual clarity.” Kepler’s “sweeping force” explained the motion of the planets around the Sun quite nicely, and that idea was much easier to understand than all that obscure talk about gravity and angular momentum.

Special relativity is essentially a space-time construct that, because of spatial contractions and temporal dilations, revised our notions of kinematics. By contrast, mass is a dynamical concept and its Lorentz invariance is a fundamental aspect of Nature that engenders the ultimate in “conceptual clarity.” It’s been argued that relativistic mass is easier for students to

is “rest mass.” My point is that the meaning of “mass” is entirely a ques-

tion of semantics, not physics. We should choose the defini-tion having the greatest conceptual clarity.2

Hecht shows persuasively that, in the context of the me-chanics of individual particles, Einstein used the word “mass” to mean “rest mass.” But there’s another context in which “mass” arises in special relativity, namely E = mc2, which Einstein considered the most important result of his special theory.3 There’s no doubt that, for Einstein, m in this equation meant “inertia,” or “resistance to acceleration.”2

One might assert that E = mc2 refers only to systems that are at rest, but this has curious implications when applied to N-body systems (N ≥ 2). For example, some of the mass of a nucleus arises from the kinetic energy of its protons and neutrons. It’s odd, then, that the kinetic energy of a proton (or neutron) does not contribute to the proton’s mass when the proton is outside the nucleus, but does contribute to the nucleus’ mass when the proton is inside the nucleus.2

The use of “mass” to mean “inertia” carries with it the stip-ulation that, if applied to a moving particle, the inertia (i.e., the ratio of force to acceleration) must be measured at con-stant speed, as in magnetic deflection. With this stipulation, we recover the simple relations p = mv and E = mc2, with m = gx (rest mass), even for moving particles and moving systems.

To summarize: It’s a matter of semantics. If single-particle mechanics and Lorentz transformations are the main consid-eration, there’s much to be said for defining “mass” as “rest mass.” But if many-body systems and mass-energy equiva-lence are the main consideration, there’s much to be said for defining “mass” as “inertial mass” even in the case of moving particles.

1. Eugene Hecht “Einstein never approved of relativistic mass,” Phys. Teach. 47, 336-341 (Sept. 2009).

2. Ralph Baierlein “Teaching E=mc2: An exploration of some is-sues,” Phys. Teach. 29, 170–175 (March 1991), and references therein.

3. Albert Einstein, “What Is the Theory of Relativity?” The Lon-don Times, Nov. 28, 1919, reprinted in The Collected Papers of Albert Einstein, Vol. 7, The Berlin Years: 1918-1921, English translation (Princeton University Press, Princeton, 2002), p. 100; see p. 103.

Art HobsonProfessor Emeritus of Physics

University of Arkansas, Fayetteville, AR 72701ahobson@uark.edu

letters

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6 The Physics Teacher ◆ Vol. 48, January 2010

letters

force to acceleration.” But of course F = ma is only a low-speed approximation. In general, F and a need not even be parallel. Thus it’s problematic to equate mass and inertia, especially since the former is a scalar and the latter a tensor.

In Hobson’s penultimate paragraph he resurrects p = mv, the equation whose conception dates back to Jean Buridan in medieval Paris. It’s a mystery to me why the followers of speed-dependent mass have no hesitation replacing classical kinetic energy with KE = mc2(g – 1), and yet cannot appreci-ate the fact that momentum is now simply p = g mv, which only approaches p = mv at low speeds; there seems so much simplicity and “conceptual clarity” to it.

In the end, given the demands of brevity, it has to suffice to say that there are powerful theoretical arguments, from gen-eral relativity to quantum field theory, that mass, however it’s defined, and whatever its ultimate cause, must be relativisti-cally invariant. In the last 50 years or so, the antique concept of relativistic mass has simply and unceremoniously been by-passed by modern theory. And that would have made Einstein smile in approval.

1. Eugene Hecht, “Einstein never approved of relativistic mass,” Phys. Teach. 47, 336–341 (Sept. 2009).

2. Lev B. Okun, Energy and Mass in Relativity Theory (World Sci-entific, New Jersey, 2009).

3. Ralph Baierlein, “Teaching E = mc2: An exploration of some is-sues,” Phys. Teach. 29, 170–175 (March 1991).

Eugene HechtPhysics Department

Adelphi Universitygenehecht@aol.com

Invariability of MassI wish to applaud the article by Eugene Hecht “Einstein Never Approved of Relativistic Mass” in the September issue. It was not until I had been a faculty member several years, and at-tended a Chatauqua Short Course by Robert Brehme, that I began to appreciate the concept of the invariability of mass in special relativity. It will help students and faculty to appreciate invariant quantities (those that don’t change with reference frames) like mass and the speed of light if they will discard the erroneous preconception of relativistic mass.

David A. CornellProfessor Emeritus of Physics

18624 Blue Ridge Drive Lynnwood, WA 98037

grasp, and since the definition of mass is a matter of semantics we might just as well keep using it. If that’s what Professor Hobson means by “conceptual clarity,” we still disagree. No concept that does not serve the purposes of physics (i.e., the furtherance of our understanding of the universe) should be retained, and/or taught, simply because its erroneous nature makes it easier to “comprehend.”

“But there’s another context” Professor Hobson informs us, “in which ‘mass’ arises in special relativity, namely E = mc2, which Einstein considered the most important result of his special theory.” Well, not quite! First, that equation does not even appear in Hobson’s reference to the London Times. Addi-tionally, Einstein certainly did not consider that equation, E = mc2, his most important result; he did not say so in the London Times, or anywhere else, ever. In fact, he cautioned that ex-pressing the “equivalence of mass and energy” by that equation was to do so “somewhat inexactly.”1

We have to be very careful here; the literature is full of these sorts of pronouncements that repeat attributions to Einstein that are simply false. Whenever Einstein wrote E = mc2, and he occasionally did so especially in later years when that equation had become emblematic of relativity, he made it clear that he was talking about an object at rest, whereupon the total energy E equaled the rest energy E0. More often he wrote his famous equation as E0 = mc2, wherein the m was always the invariant inertial mass; never ever was m relativistic mass.1 When he wrote the expression for total energy it was unambiguously E = gmc2, not E = mc2.

Hobson next raises the subject of the masses of N-body systems. Modern-day relativistic dynamics owes much to the four-dimensional space-time formulation of Minkowski. It gave us the following important equation that summarizes the relationship of E, p, and m for all real particles (as opposed to virtual ones) that are capable of being free (unlike quarks), and by extension to systems of particles: 2

E2 = m2c4 + p2c2 . (1)

This formula can be used to determine the invariant mass of any system, compound (whereupon the symbols represent summations) or otherwise. If you wish to compute mass, that is the expression to use, not E = mc2.

Hobson’s next paragraph starts “[t]he use of ‘mass’ to mean ‘inertia’ carries with it .…” I must confess I don’t really under-stand what he is getting at here. Yes, Einstein talked about “in-ertial mass,” particularly prior to 1907. But despite some mod-ern authors3 who insist that mass and inertia are completely synonymous, Einstein did not. In fact, he pointed out that the faster an object travels, the greater is its kinetic energy, and the harder it is to get it to go still faster. Consequently, the greater is its resistance to a further change in motion, even as its mass remains constant. That can best be envisioned as a time dilation effect rather than an increase in mass. In the same paragraph Professor Hobson refers to inertia as “the ratio of

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