It’s All Just Waves… (The Origin Of Inertia)

Where to begin…

This is the story of how I grappled with the concept of Wave/Particle Duality, and how I arrived at my final conclusion, which ended up having a profound impact on the way that I view inertia.

During my Undergraduate studies, I spent quite a lot of time trying to understand what a ‘Particle’ was. I couldn’t get past the fact that in most discussions about Wave/Particle Duality, the classification of ‘Particle’ made by Newton in the “Corpuscular Theory of Light”, or by Einstein in his description of the PhotoElectric Effect, were based more on a colloquial understanding, than a precise definition. This lack of precision has lead to a gradual shift in meaning of the term ‘Particle’ over the last century, to the point that a modern particle physicist almost always uses the term to identify a mode of wave propagation – pretty much exactly opposite from what the word meant a hundred years prior.

I came to believe that we had inherited this imprecise understanding from Democritus’ ancient definition of Particle. Initially, this definition was given in terms of a process of a sequence of division, ultimately reaching indivisibility. This idea of indivisibility is captured in the word Atom. This system of induction based on a sequence of dividing up actual physical objects ends up (sometimes mistakenly) endowing the Particle with many of the properties of the original undivided objects.

According to Democritus’ prescription, here’s how I might categorize the fundamental properties of a particle:

Massive. Cut an object in two, and the two halves have respective masses which ideally add up to the mass of the original. Although the mass of each half is smaller, we presume that they still have mass.
Small. Cut an object in two, and either of the two halves are smaller than the original. Particles must be the smallest of all.
Kinematic Behavior. The way that the two halves of an object move, or respond to forces is similar to the original. We expect particles to move in a straight line.
Rigid. If the original is rigid, then the two halves are as well. The term Rigid means that the particle moves “as one thing”. This onethingness is also related to its indivisibility, as well as the implication that the bounding edge of the particle is very distinct.
Occupying Space. Obviously, if a physical object exclusively occupies a volume of space, such that other objects cannot occupy the same space, then the two halves of that object must occupy space as well.

Basically, Particles are just really small, indivisible billiard balls.

Newton took the behavior of light, traveling in straight(ish) lines (Kinematic Behavior), as an indication that light must be composed of particles. Things that travel in straight lines are particles, right? Meanwhile Huygens was able to successfully describe light in terms of waves. And this was the beginning of the Wave/Particle debate.

Waves have distinctly different properties, which I will classify in a conjugate manner, relative to the particle:

Massless. This wasn’t ever a stated property of waves, but it was implied. Waves are just a disturbance moving through a medium, and any mass involved is just the mass of the medium. If we are talking about things with mass, it is assumed we are talking about particles, not waves.
Sizeless. The size of the wave disturbance is generally bounded in reality, but the mathematics used to describe waves are capable of describing waves that are entirely unbounded.
Boundary Conditions. Wave behavior is primarily determined by what is going on at the boundary. Some boundary conditions, like fixing the ends of a guitar string, produce waves that have quantized harmonics.
Fluid. You can’t describe the entire wave by simply describing what is going on with one part of it. There is no such onethingness here. You’ve got to capture a lot more detail that varies spatial and temporally.
Overlapping. Waves aren’t divided, or cut up, in the spatial sense that particles are. Decomposition of waves is usually done in terms of Mode (or frequency), and multiple distinct Modes are allowed to spatially overlap.

The False Dilemma…

When phrased the way that I have just done, it might appear that Waves and Particles have properties that are diametrically opposed to each other, such that they must be entirely independent things. However, with the advent of Quantum Mechanics, we have begun to see some shared behavior between both sets of classifications. In fact several particle properties could be described as an idealization of waves that travel in tight groups or packets.

Wave Packets can be thought of as being spatially localized, so you can talk about their “position”
Wave Packets in isolation tend to travel in straight lines
Wave Packets can represent relatively uniform oscillation, such that you can describe it as approximating a single thing
Zoom out far enough, and Wave Packets might be considered to have a sharp boundary
The Pauli Exclusion principle states that certain Quantum Wave Packets cannot occupy the same place at the same time

As I continued in my undergraduate studies, it seemed like every fuzzy notion that we ever had of a particle, was gradually being replaced by what we might more clearly interpret as a kind of manifestation of wave mechanics.

I started to see the whole Particle/Wave tension as a kind of False Dichotomy. I thought to myself that perhaps all of the properties that we might associate with particles are just idealizations of wave properties.

There was only one property from our original, naïve, understanding of particles, which continued to distinguish them from waves.

Particles are massive.

Perpetrated Confusion…

Unfortunately, at the beginning of the 1900’s another new behavior got thrown into the mix, which received the label of ‘particle-like’. Before moving on, we need to reconcile this new behavior. The association of this behavior as ‘particle-like’ is something that has caused confusion in myself, as well as the general physics community for more than a hundred years.

And it is all Albert Einstein’s fault.

Near the very birth of Quantum Mechanics, Einstein showed with the photo-electric effect, and to a smaller extent with Brownian motion, that light seemed to exhibit a property which he referred to as ‘particle-like’, namely, that light transferred energy in discreet units which he referred to as Photons. This discreet transfer, in his mind, was similar to the fact that particles rigidly move as a single thing, and that each photon corresponded in a way to single particles.

So what does this mean for my endeavor to understand all phenomena in terms of waves?

As I thought about what was particularly ‘particle-like’ about the quantized transfer of energy, I felt that I may have been tricked. Here’s the problem. While we might imagine that particles are quantized units, they definitely don’t transfer their energy in quantized units. A single particle might conceptually impart a continuous spectrum of energy as it undergoes elastic, or inelastic collision. In fact, it was the observation of such a continuous spectrum which led to the discovery of the neutrino. A single photon does not impart a continuum of energy. It imparts either all of its energy, or none of it.

I believe there’s actually a fair amount of harm that comes with thinking about Photons as being ‘particle-like’, mostly due to the fact that we never had a good definition of what ‘particle-like’ even meant in the first place.

Here’s one problem: Take a look at any physics book that introduces the idea of photons, and you will see a picture depicting a photon as a Wave Packet, often with an elliptical bean shaped boundary. This depiction has been reinforcing the false notion for at least a hundred years, that the thing that is particularly Photon-ey about Photons is the fact that they are localized in space.

This is not true.

This is so not true, that I actually had to spend quite a bit of time actively unlearning it.

Sure, photons can be localized in space, but photon behavior is very specifically about energy transfer, not about having a definable position. The energy of any wave is determined by its frequency, and its amplitude. When we define a ‘Photon’ for a wave with a specific frequency, what we are actually describing is discreet allowable amplitudes for that wave. Inside a reflective cavity, like those present in lasers, photons take on specific ‘modes’ which can be thought of as standing waves that entirely fill that cavity. Inside a lasing cavity, the absorption or emission of a photon is literally a discreet change in amplitude of the given mode.

The way that I eventually interpreted the photo-electric effect, was that it revealed a specific, heretofore unknown, wave-like property.

Perhaps you think that my suggestion is sort of hand wavey, that Photon behavior is actually wave-like. Well it is (for now), but consider then this. The suggestion that it is a particle-like property is just as hand wavey, if not more so.

Our classical and fuzzy understanding of what a particle is, never had anything at all to say about discreet allowable transfer of energy. Our understanding of what a wave is, has always had something to say, at least in a general way, about manifesting discreet allowable states.

And I’m going to let that be enough about that, because we’ve got to keep moving.

Red Light, Blue Light…

If my final answer to the question of Wave/Particle Duality is that, well ‘it’s all just waves’, then what wave behavior are we idealizing when we talk about mass?

Part of the answer comes with the discovery by Poynting in 1884 that ElectroMagnetic waves carry momentum. Previously, we had always attributed momentum to objects that had the property of mass. This was the first time in Physics history that we realized that objects that don’t have mass can still have momentum. If you are still stuck in the Wave/Particle Dichotomy, you would use this discovery as evidence that momentum is a wave-like, as well as a particle-like, property.

Another part of the answer comes with Einstein’s 1905 revelation about Special Relativity. The frequency (or color) of an ElectroMagnetic wave is dependent on your movement, relative to the source of that wave. If you are moving toward the source, the wave is ‘blueshifted’. If you are moving away, it is ‘redshifted’.

The final part of the answer comes in 1923, when Compton discovered that the energy of a single photon was proportional to its frequency. Correspondingly, the momentum of a photon is determined by its wavelength. The result is that blueshifted photons can transfer more momentum than redshifted photons.

We shall see that Mass can be described as being caused by the interplay of the momentum imbalances that occur between blushifted and redshifted photons.

Light in a Box…

Albert Einstein’s discovery that $E = mc^2$ can be facilitated by contemplating the center of mass of an empty mirror box, where light is bouncing back and forth inside. The light should impart momentum to the box as it strikes and reflects off its inner walls, first moving the box one way, then another, jiggling it back and forth.

Einstein posited that while the center of mass of the box itself, which is shifting slightly to the left and then to the right, might be changing, the center of mass of the ‘box + photon’ system shouldn’t be changing at all. Analyzing this center of mass change, he arrived at the conclusion that a photon with energy $E$ had a mass $m = \frac{E}{c^2}$ .

This setup actually provides the simplest possible answer to the question “What is Mass?”

Mass is light in a box.

Using a setup similar to Einstein, we can show much more explicitly that mass really is light in a box.

Let’s formulate a mental experiment where we have such a box. The box is empty and coated inside with perfect mirrors. We now stick a photon inside this box, and the photon bounces back and forth, imparting momentum to the box with each reflection, bumping the box first one way, then another.

Lacking any other influence, the momentum transfer from the photon to each side of the box is always balanced, averaging out to zero. Keep in mind, that if the momentum transfer doesn’t for some reason average to zero, then we will detect a force acting on the box, due entirely to the photon bouncing around inside. I intend to show, that in situations where such a force can arise, it produces a behavior that is identical to what has always been meant by ‘mass’.

Note: while I’ve definitely made these next two thought experiments my own, they don’t originate with me. I was introduced to them by my crazy Russian professor. Priviet, Alexander Panin.

Inertial Mass…

Imagine applying a force to a heavy, massive object. What happens? The object will accelerate, but it only does so with a corresponding pushback in the opposite direction. This opposing push, resistive to motion, is what has always been meant by the term Inertia. Claiming that mass is merely light in a box means that we should expect the light to push back, if we push on the box.

So, let’s take our light box and push on one side.

What happens? Because the box is now accelerating, and because it takes time for the photon to traverse the box length, the velocity of the box will have changed while the photon is mid-flight. Due to this change in velocity between bounces, the photon will be perceived as blueshifted if it is impinging upon the wall you are pushing against, and redshifted if it is impinging upon the opposite wall. Blueshifted light has more momentum than redshifted light.

Pushing the box throws off the balance of momentum, and causes an actual force (Blue arrow minus Red arrow), which pushes back against you.

Gravitational Mass…

In addition to resisting when you push on it, we also know that mass will fall when it is in a gravitational field. This falling is actually a force that pushes the mass toward the gravitational source. If light in a box causes mass, then we would expect the light to exert an overall force on the box in the direction of gravity.

This one is more sophisticated to describe formally, but is still easy to set up. According to General Relativity, light is blueshifted when it travels in the direction of a gravitational field, as the photon converts potential energy into momentum. Light is redshifted when it moves opposite that direction, converting momentum to potential energy.

Gravitational blue/redshift will create an imbalance in the momentum transfer, causing a force (Blue arrow minus Red arrow) which pushes the box toward the source of gravity.

Dispersion…

The previous examples provide a very clear picture of the mechanism behind the forces that we associate with mass. It’s also clear that those mechanisms are due to the wave nature of light. However, these thought experiments were not my first indication that mass was a wave-like property.

In my first upper division ElectroMagnetism class, I learned about many of the properties of waves, and the mathematical tools that were used to understand and manipulate those properties. One thing that I learned was that the relationship between the frequency (oscillation in time) and the wavelength (oscillation in space) of a wave wasn’t always trivial.

As an example, consider what is called a ‘waveguide’, which is a sort of tube that light flows down. A fiber optic cable is a ready example of a waveguide. Light traveling down a waveguide must satisfy specific boundary conditions, which end up quantizing the spectrum of allowable frequencies. The waveguide that I initially studied was the rectangular waveguide.

A wave traveling in such a waveguide will experience a phenomenon known as Dispersion. Dispersion affects the relationship between the frequency and the wavelength, as well as the speed that the wave travels. A wave that has non-zero dispersion will travel at speeds less than the speed of light. Within the waveguide, dispersion arises due to the light being trapped or confined. The exact measure of dispersion depends on the geometry of the waveguide. The dispersion equation for a rectangular waveguide looks like this:

$\omega^2 = \left(kc\right)^2 + \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2$

Upon seeing this equation for the first time, a sort of lightning bolt hit me. The form of this equation was very similar to another, rather important equation that I had been contemplating. This other equation could be considered to be one of the defining equations for mass. It is an extended version of $E = mc^2$ .

$E^2 = \left(pc\right)^2 + \left(mc^2\right)^2$

I realized, that for a photon, the only distinguishing difference between these two equations was the presence of a factor of $\hbar^2$ . You see, for a photon $\hbar\omega = E$ , and $\hbar k = p$ . If you multiply the first equation by $\hbar^2$ you get

$E^2 = \left(pc\right)^2 + \hbar^2\left[\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2\right]$

This means that the photon traveling in the waveguide must actually have a mass given by

$m = \left(\hbar / c^2\right)\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2}$

This mass is incredibly small, thanks to the factor of $\hbar / c^2$ , but it is still there. If dispersion is the wave-like mechanism for mass, then all forms of dispersion, either via waveguide or from material interaction, will induce mass in a photon.

According to our previous two thought experiments, and according to the equation in this section, a single photon inside a fiber optic cable will induce on that cable an immeasurably small amount of resistance to pushing (inertial mass), as well as an induced push toward a gravitational source (gravitational mass), in the transverse direction of the cable.

Dispersion is the wave-like mechanism which causes mass. While this Dispersion can be understood by a wave that is traveling in a confined space, like a waveguide, it can also be understood as a wave traveling through a medium that interacts with the wave, and ‘slows it down’. This kind of interaction can be thought of and modeled as a kind of confinement that happens on a very small scale.

The most common dispersive phenomena which we can ovserve in our everyday lives is the refraction of light traveling through a medium, such as a prism or a raindrop.

So what now?

At this point in my history of this development, I felt that there were several open questions:

If it really is all just waves, what then causes the waves? (This is the subject of my Master’s Thesis)
We see mass arise in places where there doesn’t appear to be any confinement, for instance in a free Electron. Is there actually a mechanism for confinement for an Electron? Like some kind of self-confinement?
Is it possible for there to be dispersion without any confinement at all? For instance, if there are natural and intrinsic non-linearities in the ElectroMagnetic field?
What happens when a photon travels through a region where its dispersion, i.e. the conditions of its confinement, changes?

Aside from my Masters Thesis, these questions still remain open for the most part. One day, when I’m all done making video games, I’ll go figure out the rest of this puzzle. If you happen to know any of the answers, don’t tell them to me until I’ve had a good chance to figure it out for myself.