Inertial mass is the resistance to accleration. If a force of F is required to achieve an accleration of a for a physical object then the mass of the object is m=F/a. If an acceleration of ka is to be achieved it will take a force of kF.

If the centers of two spherical objects of masses m₁ and m₂ are separated by a distance s the force each experiences is given by

F = Gm₁m₂/s²

where G is a constant.

There appears to be no relationship between these two phenomena, but there must be something that unites them. This is one of the greatest puzzle in physics, if not the greatest one.

Albert Einstein pointed out that for occupants in a closed room experiencing an attraction toward the floor it is impossible for them to tell whether they are in an accelerating elevator or in a room resting on the surface of a massive planet. That establishes that there is an equivalency between inertial mass and gravitational mass but not why it has to be so.

There is a third aspect of mass. It can be converted into energy according to the Einstein equation E=mc².

Notions of Gravitation

Gravitation in Newtonian physics involves action at a distance taking place mysteriously. Later classical physics has the massive body establishing a gravitational field throughout space. Another massive body moves according to the gradient of the field of the first body.

More recent physical theory has one body moving as a result of the force carried by transmission particles called gravitons.

The notion of gravitation established by Einstein's General Theory of Relativity is that mass warps space. This indicated that light from the stars which passed near the Sun during an eclipse would deviate from straight lines and make it appear that the stars had changed their location compared to where their light is not passing close to the Sun. There was an eclipse in 1929 only observable from a location in Africa. British scientists went there and verified that Einstein's prediction was correct. So indeed mass warps space.

It is difficult to visualize the warping of three dimensional space, but it is easy to do so for two dimensional space. A warping is a dimple in a flat plane. So the dimple is an essential aspect of a massive object.

If space is not infinitely flexible, pushing a dimple through space requires force and gives the dimple energy. On the front side of its motion the grid lines of space would be pushed together. This distortion of the grid lines of space would be more severely distorted when the dimple is being accelerated. It is like the bow wave of a boat being acclerated through water. The resistance of the boat to acceleration depends upon the shape of the hull, its depth and width, as well as the mass of the boat. The mass of a particle determines the depth and width of its dimple and hence its resistance to acceleration.

If the dimple is eliminated energy would be released. The dimple image shows how there would be an anti-particle for a dimple. It would be a pimple and it would also have resistance to acceleration. A dimple and a pimple which are mirror images of each other with respect to the two dimensional flat plane would have the same mass; i.e., resistance to acceleration. Such a dimple/pimple pair when brought together would annihilate each other. These latter remarks would involve charge as well as mass.

So the relationship between gravitational mass and inertial mass stems from gravitational mass being a warping of space and inertial mass stemming from the dynamics of moving such warpings through space. For more detailed analysis some new mathematics must be introduced.

The Development of
the Soliton Concept

In 1834 the noted engineer John Scott Russell was riding a horse along a canal near Edinburgh in Scotland. He noticed that a mound of water was being pushed along with a boat. The boat stopped and the single peaked wave continued on. Russell rode along with the wave and followed it through the canal system. Here is his exact description.

I believe I shall best introduce this phenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped -- not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel without change of form or dimunition of speed. I followed it on horse back, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears.

When he told others of his observation some of them said that it could not be. They believed there was only only one particular equation for water waves

u_xx = v²u_tt

where u is the vertical displacement from equilibrium, t is time, x is horizontal distance and v is velocity. The subscripts denote partial derivatives.

The solution to this equation is a sinusoidal function extending to infinity in both directions; i.e.,

u(x, t) = A·sin(x−vt)

Korteweg
and de Vries

The issue of the explanation of Russell's wave remained unresolved until 1895. In that year two Dutch mathematical physicists published an article in which they showed that surface waves on a liquid should satisfy a particular nonlinear partial differential equation. In effect they incorporated the nonlinearity of the surface of the liquid into the analysis.

The equation which came to be known as the Korteweg-de-Vries (KdV) equation is of the following form

u_t = u_xxx + 6uu_x

This equation has a solution of the nature of the one witnessed by John Scott Russell.

u(x, t) = 2a²·sech²(a(x + 4a²t))

where sech(z) is the hyperbolic secant function and is equal to 2/(exp(z)+exp(−z)). Its square has the shape shown below

The parameter a determines not only the amplitude of the wave (2a²) but also the velocity of the wave (4a³). Thus the larger amplitude wave travels at a faster speed than a lower amplitude wave. For a positive value of a the wave travels to the left and for a negative value to the right. For more on the sech solution see Sech².

The KdV equation resolved the issue of Russell's wave but was otherwise of little significance.

The Discovery of Solitons

In the 1950's when mathematicians began to have access to computers they used them to solve equations. They applied the techniques they were developing to any equation they found. When applied to the KdV equation they found a form that moved as a unit.

Around 1965 N.J. Zabusky and M.D. Kruskal discovered a remarkable phenomenon. If a solution with a lower value of a is initially situated to the left of a solution with higher value of a the two solutions will crash into each other. In the interval of the crash the solution will fluctuate chaotically but subsequently the wave forms will reappear unaffected by the encounter. In other words, the wave forms behave as particles.

Zabusky and Kruskal coined the term soliton to describe the wave forms. However that naming is a bit misleading. The wave forms have no existence separate from the nonlinear partial differential equation for which they are solutions. It is the nonlinear partial differential equation that it the source of the phenomenon.

The research into the phenomenon found that there are conservation laws that are satisfied. Kruskal and Zabusky found two such conserved quantities and then their colleague R.M. Miura found another six. It was subsequently found that there are an infinite number of conserved quantities for the solition phenomenon for the KdV equation.

The researchers then explored other nonlinear partial differential equations for evidence of soliton phenomena. At first this exploration focused on minor modifications of the KdV equation. The Regularized Long Wave Equation (RLWE) was formulated by D.H. Peregrine (1966) as an alternative to KdV equation for studying soliton phenomenon. It was proposed because it would not have the same limitations for the size of the time step in numerical solution that the KdV has.

u_x + u_t − 6u·u_x − u_txx = 0

This modification of the KdV equation resulted in new phenomena. After collision of wave forms for the RLWE the wave forms reappeared but not quite of the same amplitude as the pre-collision forms. At first it was thought the descreptancy was due simply to numerical inaccuracies, but later it was established that changes occurred in the wave forms as a result of the collision. The term soliton was reserved for the cases in which the wave forms were exactly preserved. The soliton-like solutions to the RLWE were changed from their pre-collision form by the interaction of the collision. The name given to such wave forms was solitary waves. For more material on the solutions to the RLWE see RLWE and traveling wave.

It was recognized very early that the soliton phenomena could be a model for subatomic particles. Some particles such as protons emerge from interactions unscathed. Other particles such as mesons interact with other particles and new particles emerge. Therefore the behavior of the solitary waves of the RLWE were welcomed as something that corresponded to some particle interactions.

In 1972 the analysis was taken by V.E. Zakharov and A.B. Shabat beyond the KdV and its modified forms. They were able to solve the nonlinear Schödinger Equation (NLSE). Other researchers solved the so-called Sine-Gordon Equation, whose name was a play on the name of the Klein-Gordon Equation. The type of soliton found for some of these equations was a kink, such as

u(x, t) = a·tanh(bx+vt)

where tanh(z) is the hyperbolic tangent function and is equal to (exp(z)−exp(−z))/(exp(z)−exp(−z)).

Such solutions correspond to physical phenomena, such as tidal bores. Here is a picture of one tidal bore.

Multi-dimensional Solitons
and Solitary Waves

The wave John Scott Russell saw in the canal was essentially one dimensional and thus could maintain its magnitude. Waves on two dimensional surfaces such as that of a pond spread out and their amplitude diminishes with the distance from the wave source. It was therefore a real challenge to find a two dimensional version of soliton or solitary wave phenomena. In 1970 a two dimensional version of the KdV equation was published by B.B. Kadomtsev and V.I. Petviashvili and this became known as the Kadomtsev-Petviashvili Equation (KP).

In 1974 S. Maxon and J. Viecelli published derivations of a cylindrically and spherically symmetric versions of the KdV equation.

For some of the three dimensional cases examined it was found that no soliton or solitary wave solutions exist . There still might be solutions of an oscillatory nature, such as the one displayed below. These are called pulson solutions.

So the work on solitons revealed how something could be both a wave and a particle. Thus the particles of the physical world may be in the nature of solitons or the related concept called solitary waves introduced above. We just at this time do not know the nonlinear partial differential equations they satisfy.

References:

• Roger Dodd, J. C. Eilbeck, John D. Gibbon and Hedley C. Morris, Solitions and Nonlinear Wave Equations, Academic Press, London, 1982.
• Kadomtsev, B.B. and Petviashvili, V.I., 1970, The stability of solitary waves in weakly dispersive media, Dokl. Akad. Nauk. SSR, vol. 192, pp. 753-756.
• Maxon, S. and Viecelli, J. 1974 Cylindrical solitons Physics of Fluids, vol. 17, pp. 1614-1616.
• Miura, R.M. , 1968, Korteweg-de Vries equation and generlizations A remarkable explicit nonlinear transformation, Journal of Mathematiical Physics, vol. 9, pp. 1202-1204.
• Peregrine, D.H., 1966, Calculation of the development of an undular bore, Journal of Fluid Mechanics, vol. 25, pp. 321-330.
• Zabusky, N.J. and Kruskal, M.D., 1965 Interaction of solitons in a collisionless plasma and the recurrence of initial states, Physical Review Letters, vol. 15, pp. 240-243.
• Zakharov, V.E. and Shabat, A.B. 1972, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Physics-JETP, vol. 34, pp. 62-69.

(To be continued.)

Conclusions

Physical particles with mass are in the nature of solitons or solitary waves, solutions to nonlinear partial differential equations. As such they are warpings of space. These warpings are described as the effect of the gravitational mass of the particles but more fundamentally they are the particle and its mass.

The solutions to the unknown nonlinear partial differential equations have the property that there is a resistance to the acceleration of their motion because such acceleration produces more intense warping of space in the direction of the acceleration.