Restless Mass
The energy of a particle with rest mass m0 and speed V is m0/(1-V2)1/2. As discussed in the note on Probabilities and Velocities, if the speed V is composed of orthogonal components vx and vy, we have
It follows that the total energy (neglecting stress and other forms of potential energy) of a ring of matter with a rest mass m0 spinning with an intrinsic circumferential speed u and translating with a speed v in axial direction is
A similar argument applies to translatory motions of the ring in any direction, not just the axial direction. For example, consider motions in the plane of the ring, and focus on the contributions of two diametrically opposed particles (each of rest mass m0/2) on the ring, as illustrated below.
If the circumferential motion of the two particles happens to be perpendicular to the translatory motion of the ring, as shown in the left-hand figure, then the preceding formula for E is applicable, and represents the total energy of the two particles. If, on the other hand, the circumferential motion of the two particles is parallel to the motion of the ring's center, as shown in the right-hand figure, then the two particles have the speeds (v+u)/(1+vu) and (v-u)/(1-vu) respectively, so the combined total energy (i.e., the relativistic mass) of the two particles is given by the sum
Thus each pair of diametrically opposed particles with equal and opposite intrinsic motions parallel to the extrinsic translatory motion contribute the same total amount of energy as if their intrinsic motions were both perpendicular to the extrinsic motion. Every bound system of particles can be decomposed into pairs of particles with equal and opposite intrinsic motions, and these motions are either parallel or perpendicular or some combination relative to the extrinsic motion of the system, so the preceding analysis shows that the relativistic mass of the bound system of particles is isotropic, and the system behaves just like an object whose rest mass equals the sum of the intrinsic relativistic masses of the constituent particles. (Note again that we are not considering internal stresses and other kinds of potential energy.)
This nicely illustrates how, if the spinning ring was mounted inside a box, we would simply regard the angular kinetic energy of the ring as part of the rest mass M0 of the box with speed v, i.e.,
where the "rest mass" of the box is now explicitly dependent on its energy content. This naturally leads to the idea that each original particle might also be regarded as a "box" whose contents are in an excited energy state via some kinetic mode (possibly rotational), and so the "rest mass" m0 of the particle is actually just the relativistic mass of a lesser amount of "true" rest mass, leading to an infinite regress, and the idea that perhaps all matter is really some form of energy.
But does it really make sense to imagine that all the mass (i.e., inertial resistance) is really just energy, and that there is no irreducible rest mass at all? If there is no original kernel of irreducible matter, then what ultimately possesses the energy? Clearly Maxwell's equations can't provide an explanation for any bound state of pure electro-magnetic energy, because they are purely linear, whereas some non-linear interaction is required in order to yield a bound state. This is one of the main reasons that we know Maxwell's equations, by themselves, are not sufficient to account for the structure of the elementary particles of matter. Today we can observe the collision of an electron and a positron, two massive particles of matter, resulting in their mutual annihilation and the release of (apparently) pure energy in the form of photons, seeming to support the view that there is no irreducible "rest mass", and all matter is actually just a manifestation of energy.
To picture how an aggregate of massless energy can have non-zero rest mass, first consider two identical massive particles connected by a massless spring, as illustrated below.
Suppose these particles are oscillating in a simple harmonic motion about their common center of mass, alternately expanding and compressing the spring. The total energy of the system is conserved, but part of the energy oscillates between kinetic energy of the moving particles and potential (stress) energy of the spring. At the point in the cycle when the spring has no tension, the speed of the particles (relative to their common center of mass) is a maximum. At this point the particles have equal and opposite speeds +u and -u, and we've seen that the combined rest mass of this configuration (corresponding to the amount of energy required to accelerate it to a given speed v) is m0/(1-u2)1/2. At other points in the cycle, the particles are at rest with respect to their common center of mass, but the total amount of energy in the system with respect to any given inertial frame is constant, so the effective rest mass of the configuration is constant over the entire cycle. Since the combined rest mass of the two particles themselves (at this point in the cycle) is just m0, the additional rest mass to bring the total configuration up to m0/(1-u2)1/2 must be contributed by the stress energy stored in the "massless" spring. This is one example of a massless entity acquiring rest mass by virtue of its stored energy.
Recall that the energy-momentum vector of a particle is defined as [E, px, py, pz] where E is the total energy and px, py, pz are the components of the momentum, all with respect to some fixed system of inertial coordinates t,x,y,z. The rest mass m0 of the particle is then defined as the Minkowskian "norm" of the energy-momentum vector, i.e.,
If the particle has rest mass m0, then the components of its energy-momentum vector are
If the object is moving with speed u, then dt/dt = g = 1/(1-u2)1/2, so the energy component is equal to the transverse relativistic mass. The rest mass of a configuration of arbitrarily moving particles is simply the norm of the sum of their individual energy-momentum vectors. The energy-momentum vectors of two particles with individual rest masses m0 moving with speeds dx/dt = u and dx/dt = -u are [gm0, gm0u, 0, 0] and [gm0, -gm0u, 0, 0], so the sum is [2gm0, 0, 0, 0], which has the norm 2gm0. This is consistent with the previous result, i.e., the rest mass of two particles in equal and opposite motion about the center of the configuration is simply the sum of their (transverse) relativistic masses, i.e., the sum of their energies.
A photon has no rest mass, which implies that the Minkowskian norm of its energy-momentum vector is zero. However, it does not follow that the components of its energy-momentum vector are all zero, because the Minkowskian norm is not positive-definite. For a photon we have E2 - px2 - py2 - pz2 = 0 (where E = hn), so the energy-momentum vectors of two photons, one moving in the positive x direction and the other moving in the negative x direction, are of the form [E, E, 0, 0] and [E, -E, 0, 0] respectively. The Minkowski norms of each of these vectors individually are zero, but the sum of these two vectors is [2E, 0, 0, 0], which has a Minkowski norm of 2E. This shows that the rest mass of two identical photons moving in opposite directions is m0 = 2E = 2hn, even though the individual photons have no rest mass.
If we could imagine a means of binding the two photons together, like the two particles attached to the massless spring, then we could conceive of a bound system with positive rest mass whose constituents have no rest mass. In normal circumstances photons do not interact with each other, because they are essentially linear disturbances, i.e., they can be superimposed without affecting each other (although there are non-linear effects in quantum theory). However, we can, in principle, imagine photons bound together by the gravitational field of their energy. (Wheeler refers to such configurations as "geons".) The ability of electrons and anti-electrons (positrons) to completely annihilate each other in a release of energy suggests that these actual massive particles are also, in some sense, bound states of pure energy, but the processes that hold an electron together, and that determine its characteristic mass, charge, etc., are not presently understood.
It's worth noting that the definition of "rest mass" is somewhat context-dependent when applied to complex accelerating configurations of entities, because the momentum of such entities depends on the space and time scales on which they are evaluated. For example, we may ask whether the rest mass of a spinning disk should include the kinetic energy associated with its spin. For another example, if the Earth is considered over just a small portion of its orbit around the Sun, we can say that it has linear momentum (with respect to the Sun's inertial rest frame), so the energy of its circumferential motion is excluded from the definition of its rest mass. However, if the Earth is considered as a bound particle during many complete orbits around the Sun, it has no net momentum with respect to the Sun's frame, and in this context the Earth's orbital kinetic energy is included in its "rest mass".
Similarly a "stationary" chunk of lead not microscopically stationary, but in the aggregate, averaged over the characteristic time scale of the mean free oscillation time of a particle, it is stationary, and is treated as such. The temperature of the lead actually represents changes in the states of motion of the constituent particles, but over a suitable length of time the particles are still stationary. We can continue to smaller scales, down to sub-atomic particles comprising individual atoms, and we find that the position and momentum of a particle cannot even be precisely stipulated simultaneously. In each case we must choose a context in order to apply the definition of rest mass.
Physical entities possess multiple modes of excitation (kinetic energy), and some of these modes we may choose (or be forced) to absorb into the definition of the object's "rest mass", because they do not vanish with respect to any inertial reference frame, whereas other modes we may choose (and be able) to exclude from the "rest mass". In order to assess the momentum of complex physical entities in various states of excitation, we must first decide how finely to decompose the entities, and the time intervals over which to make the assessment. The "rest mass" of an entity invariably includes some of what would be called energy or "relativistic mass" if we were working on a lower level of detail.