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THEORETICAL PHYSICS: ON CONSERVATION OF MASS

The following points are made by Frank Wilczek (Physics Today 2004 December):

1) Is the conservation of mass as used in classical mechanics a consequence of the conservation of energy in special relativity? Superficially, the case might appear straightforward. In special relativity we learn that the mass of a body is its energy at rest divided by the speed of light squared [m = E/c^(2)]; and for slowly moving bodies, it is approximately that. Since energy is a conserved quantity, this equation appears to supply an adequate candidate, E/c^(2), to fill the role of mass in the "culture of force".

2) However, that reasoning will not withstand scrutiny. The gap in its logic becomes evident when we consider how we routinely treat reactions or decays involving elementary particles. To determine the possible motions, we must explicitly specify the mass of each particle coming in and of each particle going out. Mass is a property of isolated particles, whose masses are intrinsic properties -- that is, all protons have one mass, all electrons have another, and so on. (For experts: "Mass" labels irreducible representations of the Poincare group.) There is no separate principle of mass conservation. Rather, the energies and momenta of such particles are given in terms of their masses and velocities, by well-known formulas, and we constrain the motion by imposing conservation of energy and momentum. In general, it is simply not true that the sum of the masses of what goes in is the same as the sum of the masses of what goes out.

3) Of course when everything is slowly moving, then mass does reduce to approximately E/c^(2). It might therefore appear as if the problem, that mass as such is not conserved, can be swept under the rug, for only inconspicuous (small and slowly moving) bulges betray it. The trouble is that as we develop mechanics, we want to focus on those bulges. That is, we want to use conservation of energy again, subtracting off the mass-energy exactly (or rather, in practice, ignoring it) and keeping only the kinetic part E - mc^(2) ~= mv^(2)/2. But you can't squeeze two conservation laws (for mass and nonrelativistic energy) out of one (for relativistic energy) honestly. Ascribing conservation of mass to its approximate equality with E/c^(2) begs an essential question: Why, in a wide variety of circumstances, is mass-energy accurately walled off, and not convertible into other forms of energy?

4) To explain why most of the energy of ordinary matter is accurately locked up as mass, we must first appeal to some basic properties of nuclei, where almost all the mass resides. The crucial properties of nuclei are persistence and dynamical isolation. The persistence of individual nuclei is a consequence of baryon number and electric charge conservation, and the properties of nuclear forces, which result in a spectrum of quasi-stable isotopes. The physical separation of nuclei and their mutual electrostatic repulsion -- Coulomb barriers --guarantee their approximate dynamical isolation. That approximate dynamical isolation is rendered completely effective by the substantial energy gaps between the ground state of a nucleus and its excited states. Since the internal energy of a nucleus cannot change by a little bit, in response to small perturbations it does not change at all.

5) Because the overwhelming bulk of the mass-energy of ordinary matter is concentrated in nuclei, the isolation and integrity of nuclei--their persistence and lack of effective internal structure--go most of the way toward justifying the zeroth law (the law of the conservation of mass). But note that to get this far, we needed to appeal to quantum theory and special aspects of nuclear phenomenology. For it is quantum theory that makes the concept of energy gaps available, and it is only particular aspects of nuclear forces that insure substantial gaps above the ground state. If it were possible for nuclei to be very much larger and less structured -- like blobs of liquid or gas -- the gaps would be small, and the mass-energy would not be locked up so completely.

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THEORETICAL PHYSICS: ON THE CONCEPT OF FORCE

The following points are made by Frank Wilczek (Physics Today 2004 October):

1) Newton's second law of motion, F = ma, is the soul of classical mechanics. Like other souls, it is insubstantial. The right-hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left-hand side, on the other hand, has no independent meaning. Yet clearly Newton's second law is full of meaning, by the highest standard: It proves itself useful in demanding situations. Splendid, unlikely looking bridges, like the Erasmus Bridge (known as the Swan of Rotterdam), do bear their loads; spacecraft do reach Saturn.

2) The paradox deepens when we consider force from the perspective of modern physics. In fact, the concept of force is conspicuously absent from our most advanced formulations of the basic laws. It doesn't appear in Schroedinger's equation, or in any reasonable formulation of quantum field theory, or in the foundations of general relativity. Astute observers commented on this trend to eliminate force even before the emergence of relativity and quantum mechanics.

3) In his 1895 Dynamics, the prominent physicist Peter G. Tait, who was a close friend and collaborator of Lord Kelvin (1824-1907) and James Clerk Maxwell (1831-1879), wrote

"In all methods and systems which involve the idea of force there is a leaven of artificiality.... there is no necessity for the introduction of the word "force" nor of the sense-suggested ideas on which it was originally based."(1)

4) Particularly striking, since it is so characteristic and so over-the-top, is what Bertrand Russell (1872=1970) had to say in his 1925 popularization of relativity for serious intellectuals, /The ABC of Relativity/:

"If people were to learn to conceive the world in the new way, without the old notion of "force," it would alter not only their physical imagination, but probably also their morals and politics.... In the Newtonian theory of the solar system, the sun seems like a monarch whose behests the planets have to obey. In the Einsteinian world there is more individualism and less government than in the Newtonian."(2)

The 14th chapter of Russell's book is entitled "The Abolition of Force." (3,4)

References (abridged):

1. P. G. Tait, Dynamics, Adam & Charles Black, London (1895)

2. B. Russell, The ABC of Relativity, 5th rev. ed., Routledge, London (1997)

3. I. Newton, The Principia, I. B. Cohen, A. Whitman, trans., U. of Calif. Press, Berkeley (1999)

4. S. Vogel, Prime Mover: A Natural History of Muscle, Norton, New York (2001), p. 79

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