A local model of the inertia of mass.
Galileo was the first to realize that it was not a body’s velocity but its acceleration that indicated that forces were acting on it.
A body in motion that is not subject to any force is normally slowed by friction, but if the friction is reduced to a minimum, as is the case for example with a smooth body sliding on a sheet of ice, its velocity remains constant for a long time.
If we push this reasoning to the limit and consider for example the case of a body moving through outer space and subject to little friction, we can appreciate how intuitive Galileo’s discovery was.
The fact that a force must be exerted to increase or decrease the velocity of a body is usually expressed by saying that the body possesses inertia. We state this to mean that matter “resists” attempts to modify its velocity in any way and that this resistance must be overcome by the application of force.
We can quantify inertia by measuring the force required to modify the velocity of a body.
It seems perfectly obvious, for example, that it takes more force to stop a locomotive than an automobile traveling at the same velocity. We say therefore that the locomotive possesses greater inertia than the automobile and can calculate how much in terms of the ratio of the forces that must be exerted to stop each of the two masses.
It has been asked whether a body’ inertia is always the same or if it changes as it approaches other bodies. As no change of the kind has ever been perceived, Newton claimed that inertia was a property specific to material bodies regardless of the presence of other bodies in their environments.
Newton’s theory of inertia was based on the second principle of dynamics, which stated that the force exerted on a body that was not restrained tended to make it achieve a particular degree of acceleration proportional to the amount of force applied.
This proportional constant is called the inertia or inertial mass of the body.
Newton’s second principle of dynamics is thus expressed classically as a vector in which force is equal to the product of mass times acceleration: F = m a.
If we examine this law in greater detail, we encounter a strange difficulty. Whereas the force acting on the body is determined objectively in terms of the factor that exerts it, the value of the acceleration depends on the way it is measured, i.e. by the body that is considered to be at rest.
Newton’s second principle should therefore be modified as follows: “force is equal to the product of the inertial mass times the absolute acceleration”. Bodies on which no force is acting will have no absolute acceleration and such bodies are said to constitute an “ inertial reference system” as accelerations measured in terms of it are considered absolute accelerations.
General Relativity has since demonstrated that there can be no absolute accelerations, however, just as there can be no absolute velocities. Acceleration too is relative to an outside observer or reference system.
The importance of the observer’s role increases as we examine the most fundamental properties of nature, and this supports the relativistic wave model of matter that we wish to illustrate. If our wave model truly possesses explanatory validity, we should be able also and above all to explain inertia of all the most fundamental properties of matter.
Dennis W. Sciama tells us that inertia is the fundamental property of matter about which there have been the smallest number of attempts to find an explanation up to now.
It is not a simple problem. Few physicists have speculated about inertia as an attribute of mass and if we try to understand why this is, we discover the banal fact that it is not a very spectacular topic.
When Einstein had firmly established the concept in physics that there is no absolute space, Newton’s thoughts about absolute rotation as equivalent to rotation with respect to the fixed stars became so much waste paper.
Once inertia could no longer depend on the centrifugal force associated with rotation in terms of absolute space, the only thing that remained was to posit that the existence of fixed stars was in some mysterious way responsible.
It is on this very basis that physics formulates its only and very fragile attempt to explain inertia.
Mach was the most representative advocate of positivist thought and established a method of analyzing the foundations of mechanics that quantum mechanics followed through to its ultimate conclusions. According to Mach, it makes sense to think that the inertia of a body is the result of a series of interactions that, “ in some way “, occur between each individual body and all the matter in the Universe.
At the cosmic level, the matter closest to any body would make only a very small contribution to the total inertia. At the same time, the enormous quantity of matter that makes up the distant masses of the stars and galaxies that populate the universe was supposed to influence each body and provide it with the specific properties of inertia.
For a brief period, Mach’s interpretation exerted some influence on Einstein, who called it the Mach principle.
Underlying the Mach principle is the idea that we are all linked to the distant regions of the universe by a mechanical bond consisting of a field. As a result, the universe as a whole exerts a powerful influence on the motion of the matter all around us. Mach argued that this influence could be seen in matter’s inertial properties in particular.
The Mach principle lacks characteristics indispensable for consideration as a serious scientific hypothesis that could lead to the formulation of a physical theory, as it is not very likely to be proven experimentally.
Dennis W. Sciama, the Indian physicist, who attempted to prove the Mach principle theoretically, thought inertia could be attributed to the sum of all the gravitational interactions exerted by the entire universe on each individual body.
From Sciama’s perspective, Mach was right and the arguments he presented in the Mach principle seemed logical and plausible.
As many more bodies are distant than nearby in the universe, their total gravitational influence was thought to bind the motion of the individual bodies we experience in a web of gravitational attractions. This would condition the motion or rest state of every body in the particular form characteristic of a reaction to any variation in motion.
According to Sciama, we call this reaction “inertia”.
This explanation appears not to affect theoretical consistency, but has the problem of being an end in itself. It in practice provides us with no further theoretical or experimental links that might lead us toward new understanding. It is also based on the idea that gravitational force is valid and dominant at “any distance” and this is a hypothesis rather than a proven fact.
In this context, A. Pais writes about inertia:
I must say that Mach’s principle as far as I know has not allowed physics to make any decisive steps forward. It should be noted that the origins of inertia are and remain the most obscure issue in particle and field theory.
In an attempt to make physics take such a decisive step, let us now use only waves to address the issue of inertia and seek to derive the inertial properties of masses from the wave properties of fields of mass.
Using the wave model of mass, we can provide a new possible interpretation of inertia that argues along the same lines as Sciama but does not constitute an end in itself. This new wave interpretation would in fact allow us to derive a rational and “comprehensible” explanation of the quantum interaction of gravity.
To construct our wave model of inertia, we will be forced to introduce a few mathematical models (some of which we will attempt to describe using geometrical models) that are already well known in the study of waves.
These models will allow us to explain inertia as a purely local and relative effect.
It is standard practice in science to investigate a particular aspect of nature by creating mathematical models that describe its behavior in various physical conditions and take the form of organic models containing simpler ones that are already familiar.
In the same way, let us create a partial model of reality and describe the waves of a particle wave source. The field these waves normally form must be considered to possess spherical symmetry. Let us now consider instead one part of the field made up of wave surfaces that propagate in an ideal tube placed on the wave source’s direction of motion.
Let us then examine only those portions of the wave surfaces that can be considered to be flat and parallel to each other as they propagate along a straight line passing through the center of the field.
These waves, which are temporal perturbations in Schild’s space-time, could be described in classical terms as non-continuous sinusoidal waves and thus as discrete sinusoidal waves.
We know they are perturbations in the geometry of space-time, but we do not yet know with certainty whether even a discontinuous sinusoidal function will be able to describe them correctly. As the source emits the waves continuously, we also know that they can be described statically and the frequency of the source at rest describes the elementary energy of the waves.
When the source is in motion in the tube, we have two different frequencies by the Doppler effect, the one in the direction of motion n1 and the other in the opposite direction n2.
Picture 40. Source of waves with evidence of the wave surfaces surrounding the trajectory of motion.
As Claude Elbaz has shown (C. R. Acad. Sc. Paris no. 13 – 1984), it is possible, according to de Broglie, to take the particle’s wave state into consideration and describe it as a stationary composition of flat waves superimposed on each other at frequency ν1 and ν2.
We can see how this would be possible if we think of geometrical optics, in which the phase of the wave satisfies Hamilton-Jacobi’s famous equation describing the trajectories of rays of light.
The Hamilton-Jacobi equation, which is linked to the abstract concept of light rays, has had only an abstract and purely ideal significance up to now. In the Wave Theory of the Field, on the other hand, it assumes a precise physical significance in terms of waves. We must see it as a description of the location of the points at which the wave surfaces are parallel to one another.
Taken together, the points the equation describes identify the trajectory of the parallel wave train which describe the energy of the waves more fully than other nearby points.
In the area of diffraction, the wave of phase satisfies another famous wave equation, this one by Klein-Gordon, to which we have already referred, which describes in relativistic terms the existence of wave packets propagating in space.
Such wave packets possess the properties common to all waves and the properties of de Broglie’s material waves can also be attributed to them.
This equation presents many interesting characteristics that can be used to describe the wave state, but it is intrinsically valuable because it is compatible with Special Relativity. It in fact describes the characteristics of de Broglie’s material wave trains in relativistic terms.
The Klein-Gordon equation would be even more significant, however, if it were not burdened with the serious inconvenience that one would necessarily be forced to reach a double solution if one applied it rigorously to de Broglie’s material waves.
As it is a quadratic equation, there are in fact two mathematical solutions to the Klein-Gordon equation that should both correspond to something physical. The one solution is positive and describes a normal amount of movement and the positive energy charge in the particle. And the other would compel us to consider the possibility that negative momenta and frequencies exist in addition to positive.
It is clear that the current view of physics cannot associate a negative physical significance with energy.
It remains true, however, that the mathematical significance of the Klein-Gordon equation, like that of any other mathematical equation to which one might wish to associate physical significance, has less explanatory value if its solutions cannot be accepted fully.
We can make a clear analogy with Dirac’s successful interpretation of an equation in which he predicted the existence of positrons.
Picture 41. The Hamilton-Jacobi formula identifies the trajectory of a photon wave train along which the wave fronts are parallel to one another. Both of the possible solutions of the Klein-Gordon formula can become meaningful if they describe de Broglie’s waves and are interpreted in terms of the elementary waves of mass.
In this case, however, no one has yet found a solution revealing the possibility that a negative energy might exist. People have therefore thought up to now that only the positive solution had physical significance and have rather arbitrarily discarded one of the two solutions to the equation and dismissed it as lacking significance.
In the light of the Wave Theory of the Field, we can now confirm the physical existence of both positive and negative energy variations by observing variations in the condition of waves within an accelerated particle wave source.
An observer who takes a position on the trajectory of a particle that has been accelerated to await its arrival can confirm, before being struck by the particle, the arrival of “a positive variation” in the energy of the particle’s wave field traveling at the speed of light.
Another observer, on the other hand, located in the path already traversed by the particle can confirm “a negative variation” in the energy of the waves originating in the particle.
Picture 42. Accelerated wave source with two observers occupying positions on the trajectory who observe two different wave variations.
In keeping with the Doppler effect, there can thus be a variation involving either an increase or a decrease in the wave energy within a particle wave source in accelerated motion.
Both cases clearly involve “energy wave variations” and it is consistent on the basis of our discussion of the Principle of Relative Symmetry to call variations involving increases “positive energy” and those involving decreases “negative energy”.
On the basis of the Principle of Relative Symmetry, both must be able to influence whether the particle wave source is in motion or at rest.
According to the Principle of Relative Symmetry, both states are in fact involved reestablishing equilibrium in the variation in momentum when they are present within the particle wave source.
The significant fact is that a wave variation, however it occurs, must always trigger the Principle of Relative Symmetry equally, regardless of whether the variation is negative or positive.
On this basis, we must seek out an entity in physics that could be associated with the wave-like nature of matter and radiation we could call “negative energy”.
Inertia is negative energy.
Before we become dissuaded by the apparent ease with which we have reached such an important statement (my heart in fact remained caught in my throat for months when I understood it), let us perform an ideal wave experiment to confirm that this statement could be falsified.
In his study of scientific methods, Popper introduced the concept of falsification of a scientific theory or hypothesis. He argued that it must be possible to prove each theory wrong, by which he meant that it must be possible to subject it to experiments that can demonstrate that it is false.
According to Popper, theories can not be “proven” by any great experiment, as another more extensive or more accurate experiment could some day prove the opposite. And this has occurred frequently in the history of science.
The offer of an experiment that could result in proving a proposed theory false suggests that it has foundations that one thinks it might be possible to link to physical reality. These foundations may prove false, but the fact that something is offered that can be used to test the theory at all gives it a certificate of correctness that in some way guarantees that the theory or hypothesis that supports it will be called “scientific”.
In any case, a theory survives by passing tests designed to prove it wrong and the more conditions it presents that could be proven false that it can pass, the more dependable it becomes.
In the spirit of Popper’s philosophy, let us suggest a key experiment that could prove the Wave Theory of the Field wrong.
- Subject a mass to a sudden acceleration and observe the variation in the momentum of a test body placed in the vicinity of the mass on the side opposite the direction of the acceleration of the mass.
- Make the tests without regard to the gravitational forces in effect by subtracting effects due to the force of gravity between the two bodies from the calculations.
Picture 43. Experiment confirming the local nature of inertia. The Theory could theoretically be refuted if this happened not to be confirmed by the experiment. On the other hand, if it were confirmed, this would prove the wave-like and purely local nature of the inertia of mass.
- Observe that a positive variation has been produced in the zone in which the wave energy variation is present in front of the accelerated mass.
- On the other hand, a negative variation of the same number as the Doppler wave has been produced at the same time behind the accelerated mass.
- This confirms the creation of a kind of “gap” in the wave energy in the zone situated behind the mass.
- As the distribution of the energy variation around the accelerated mass is asymmetrical, masses in the vicinity will be impelled toward the “gap” by the Principle of Relative Symmetry.
The phenomenon is similar to what happens in an environment in which the atmosphere is maintained at a constant pressure, when a localized pressure loss occurs in a particular zone.
All bodies are impelled toward that zone by the general redistribution of pressure. Bodies closest to the zone of the depression are subjected to the most energetic suction action and the most significant effects of the acceleration in the direction of the depression.
Picture 44. A broken window in an airplane flying at high altitude provides a typical example of objects subjected to a sudden localized variation in atmospheric pressure.
- The accelerated mass is the body closest to the zone of the energy depression and the mass that is forced to accelerate is thus necessarily also subjected to the attraction of the “gap” of negative energy.
- Each time there is an acceleration of the mass, an asymmetrical energy condition arises in the zone of space in which there is an energy depression.
- As this is located directly behind the mass in acceleration, the mass also tends to fall into the “gap” that has opened up behind it“as if a force were opposing the mass’s acceleration.”
This experiment is feasible. A mass can be accelerated in a vacuum and a small mass can be placed behind it to react to the opening of the gap in the wave energy by moving in the direction of the gap.
Picture 45. In an actual experiment to ascertain the wave-like nature of inertia, a projectile strikes the mobile arm holding the mass b and accelerates it causing it to move away from the mirror a (which constitutes the test body) suddenly. The movement of the mass away from the mirror is observed by measuring any shift in the laser beam c.
The greater the acceleration, the deeper the “gap” that opens behind the accelerating mass.
Where the acceleration of two masses is equal, the mass that emits a greater frequency at rest because it has greater mass will create a deeper gap behind itself as the energy variation will be greater.
This wave effect, which we could incorrectly attribute to a force, is what we referred to as the “force of inertia” in the past and what we now call “the inertia of the mass”.
The issue is not unimportant.
If what we have just said is true, the wave model has explained inertia in strict causal terms as an effect due to local conditions.
But we do not want to raise the issue of the “truth” of such a claim at this time.
We are nonetheless ready to ask the reader to observe that the explanation of inertia presented in the wave model is consistent with the theory and interpretation of all the phenomena examined up to now.
We would further ask the reader to observe that the Principle of Relative Symmetry has demonstrated the effectiveness of its explanatory powers in all the diverse key situations encountered up to now.
These have included everything from quantum mechanics to quantum physics and from optics to macrophysics and we have always observed the dictates of Relativity and linked diverse realms together as has never been done before.
I believe simplicity to be one of the most notable qualities of the wave interpretation of inertia.
For some time, simplicity has not been shown the respect in physics that Einstein attributed to it. Scientific currents in the past sixty years have denied that a simple scientific explanation should by rights receive more attention than another of equal significance that is not simple.
But once again we must admit that the grand old man was right: a simple scientific explanation bears a fascination from which it is difficult to turn away.
Let us now examine an effective proof of the validity of the wave explanation of inertia and observe how the wave model of inertia is able to provide a consistent wave model of centrifugal force.
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