Maxwell’s Equations in Wonderland
The conflict between intuition and mathematical rigor is best expressed through the relationship between the British physics of Maxwell, Kelvin, and Heaviside and the French physics of Poincare, Laplace, and Fourier.
The French mathematical physicists looked down their rigorous noses at the analogies of British physics, all while intensely studying the British ideas and trying to translate the broad strokes into the narrow confines of their strict formality. Without access to the broader perspective, they derived mathematical results without understanding their application.
With the certainty of a mathematician who wishes to supervise the physicists, Poincare wrote:
“The first time a French reader opens the book of Maxwell, a feeling of discomfort, and even often of suspicion, is mixed with his admiration. It is only with a long duration of commerce and at the expense of many efforts that this feeling dissipates.
Why do the ideas of the English scientists have so much difficulty to be acclimated by us? It is probably because the education received by the French prepares them to appreciate precision and logic before any other quality.
…
So, opening Maxwell, a Frenchman expects to find a theoretical whole as logical and as precise as the physical optics founded upon the ether; this way he’s preparing a deception I’d save the reader in showing him what he must look for in Maxwell which he cannot find.
Maxwell doesn’t give any mechanical explanation for magnetism and gravity. It is enough for him to show that such an explanation is possible.”
“Science and Hypothesis”, Chapter 12, entitled “Optics and Electricity “.
It was only late in Poincare’s life, after seeing the experimental work of the Curies, that he began to understand that those who do not think solely in mathematical frameworks can see things which he could not.
“True physicists like Curie, look neither inside themselves, nor to the surface of things, they can see under things.
Mathematics are a trouble, or even a danger, when, by the precision of their language, they lead us to affirm more than we really know.”
Poincare quoted in The Universe of Maxwell
I’n this post, I’m going to take a close look at what we really know about Maxwell’s equations. I’m going to try to look under them.
In an earlier post, I wrote about magnetic monopoles and the existence of two sets of Maxwell’s equations: Galilean and Lorentzian. Since this is not a familiar topic to most physics students, I’ll spend some time unpacking the details. Since details are tedious to 99% of people, I’ll try to keep this light and whimsical.
Jam tomorrow, jam yesterday, but never ever jam today.
Through the Looking Glass, Lewis Carroll
This statement about vectors and scalars always comes to mind whenever I see an equation expressing the expectation value for shear-stress.
Shear stress is not how you feel in rush-hour traffic, it is the tension between two surfaces that are slipping against one another and this is an important parameter because the amount of shear stress in a material tells you how it will behave when something interacts with it.
- If you have a lot of shear stress and someone hits you with a hammer, you might shatter into a million pieces.
- If you have a lot of shear stress and someone hugs you, you might melt into a little puddle as the stress leaves your body.
In this context, a particle usually acts like a hammer and a wave usually acts like a warm hug. Of course, it is possible to deliver a violent hug which can be as damaging as a hammer — as when a soprano hits a high note and shatters glass. It is also possible to gently caress someone with a hammer. This may be the reason that so many people get confused about what is a particle and what is a wave. They often act in similar ways, even though they are very different things.
The reason that everyone believes that light is both a particle and a wave is because of Einstein’s photoelectric effect interpretation. This interpretation also seems to follow from the fact that both electrons and photons can be used to do the double-slit interference experiment.
But, philosophically, this is a very literal interpretation and both experiments could also be explained by quantized properties that only reside in the electron and not in light itself. If the detector of a light wave is always a particle, it shouldn’t surprise anyone that the light wave sometimes acts like a particle. Likewise, if the particles ride on light waves, it shouldn’t surprise anyone that the particles sometimes act like light waves.
What I am trying to do here is disconnect you from the quantum weirdness that is taught in many classrooms and reconnect you with a concrete picture of particles and waves that are fundamentally different things. In this picture, light and sound propagate through a sea of particles, moving those particles around like so many aetheric grains of sand.
This picture was nicely formalized by Dirac’s sea of negative energy particles, but what is a negative energy particle? Is it nothing more than a particle which spins in the opposite direction from the particles surrounding it? I think so.
The grains of sand which spin in a clockwise direction have a positive charge and those that spin in a counterclockwise direction have a negative charge. The grains which vibrate in synch will attract one another and those which are out of synch will repel one another.
Physics couldn’t be simpler. At least it was until the invention of the Lorentzian Maxwell’s equations.
In the Lorentzian Maxwell’s equations, light is Lorentz invariant. That means that it doesn’t lose energy when it travels in a straight line. When light loses energy, one can either say that the wavelength got longer or that the speed of the light decreased. In the Lorentzian framework, instead of allowing a light wave to lose energy, the space in which the light wave exists expands while the speed of the light wave is held constant.
In the Galilean Maxwell’s equations, light is not Lorentz invariant. It can lose energy and one keeps the wavelength constant while the speed of the light changes when it, for example enters a gravitational field or a piece of glass. Why did physicists abandon this set of equations which made the physics of light in the Cosmos and on Earth so wonderfully consistent?
The reason is that the Galilean Maxwell’s equations were hard to solve for some systems, whereas the Lorentzian Maxwell’s equations made an approximation which removed a degree of freedom by eliminating the concept of the magnetic monopole, thereby making them easier to solve for some systems. But this came at a great cost to the aetheric intuition and heuristics which had been developed over the previous centuries.
In short, when Einstein popularized the Lorentzian framework and abandoned Cartesian space in favor of Minkowski’s Riemannian mathematics, many people got very confused and forgot the utility of the Galilean Maxwell’s equations. Some even forgot that they existed.
Einstein had just learned of the Lorentz transformations and when he saw the Michelson–Morley experiment, he concluded that over short distances, light does not lose energy when it propagates in a straight line. He inferred that this demonstrated the absence of aether and of the magnetic monopoles with which aether is composed.
We now know that other experiments contradict this myopic inference:
- The Sagnac loop showed that light which travels in a circle will lose energy to the aether.
- Materials science showed that light travels in tiny circles as it propagates, losing energy to the aether.
- Michelson-Morley experiments with electrons show electrons lose energy to the aether.
But what is this aether? Today it might be referred to as a quantum spin foam or a Dirac sea of negative energy particles or a quantum vacuum. It would only rarely be referred to as a sea of magnetic monopoles, but whatever you call it, it appears to be a substance that conducts electricity. If it couldn’t, an electromagnetic wave (light) could not propagate through it.
All school children are taught that an electromagnetic wave is made up of electric waves which are perpendicular to copropagating magnetic waves and that the Lorentzian Maxwell’s equations assure us that this is the truth. University students are taught that these waves are Lorentz invariant when they co-propagate in free space, but not Lorentz invariant when they do not co-propagate in a medium which refracts light. Very few university students wonder why this is the case because they have a lot of stress and a lot of homework to do.
If they did wonder about this, they would ask themselves what sort of shear stress could prevent the co-propagation of an electric and magnetic field in free space? Then they would remember that light bends in a gravitational or electric field when it has to swirl around something. In free space, that something would fit the description of a magnetic monopole. If light propagating through free space is really propagating through a sea of magnetic monopoles which have a density or shear stress determined by the gravitational field, this cleanly and simply unifies the physics of the microscale and the macroscale.
Now I’ll break the process of light propagation down into its sub-components.
The electric part of the light couples to the charge swirling around the magnetic monopole and the magnetic part of light is produced by the swirling charge around the magnetic monopole. The more magnetic monopoles a light wave interacts with, the more shear stress it encounters. When there is no shear stress, the coupling and the production have no delay and no loss, but when there is shear stress, there will be a small amount of energy lost.
On a larger length scale, the electric part of the wave couples to the electrons swirling around the nucleus and the magnetic part is produced by the electrons swirling around the nucleus.
On an even larger length scale, the electric part of a wave couples to the moons swirling around the planet and the magnetic part is produced by the moons swirling around the planet.
On an even larger length scale, the electric part of a wave couples to the planets swirling around the sun and the magnetic part is produced by the planets swirling around the sun.
On an even larger length scale, the electric part of a wave couples to the stars swirling around the galaxy and the magnetic part is produced by the stars swirling around the galaxy.
This recursive picture is just another way to describe a steady-state, gravitomagnetic, tired light, electric, plasma universe.
These systems emit a light wave when the phase relationship between the light waves propagating between the center and the orbiting spheres produces constructive interference. Likewise, these systems can only absorb energy when the phase relationship shifts in the opposite direction. The amount of energy absorbed or emitted is quantized because each orbital has a defined amount of energy and the orbitals only interact with one another when the plane of one wave discretely rotates as when the spin or angular momentum of a particle flips upside down.
I regret that I’ve let this get so complicated. I wanted it to stay light and whimsical, but here we are stuck in the details.
Summarizing: the picture of Lorentzian Maxwell’s Equations’ electromagnetic wave propagation in a vacuum is wrong because it approximates space as being empty when it is really full of magnetic monopoles.
The underlying reality is that a magnetic field only appears in the vicinity of particles or as a wake behind a purely electric pulse. Since all of our light detectors use electrons in some way, and charge swirling around a particle always gives rise to a magnetic field, we have assumed that the magnetic field must always be present — even in free space. But this was a Lorentzian approximation of a Galilean, Cartesian free-space which is full of magnetic monopoles that have a density determined by the strength of the gravitational field.
Awareness of this approximation requires the recognition that the Lorentzian Maxwell’s equations are only valid in the presence of matter and the description of light propagation in free-space is an approximation. The Galilean Maxwell’s equations are required for an accurate description of the propagation of light in free space because it is only when the energy of a wave is transformed into a swirling motion in the presence of a particle that the magnetic field is created.
To prove that light in free space is electric and not electromagnetic, you would need to have a purely electric wave which gives rise to the appearance of an electromagnetic wave within a detector. That is actually exactly what is observed in particle accelerators.
Wrapping up: To make the math of physics match up with sensible heuristics, we need to recognize that, light is an electric wave in free space, and an electromagnetic wave near matter. An electromagnetic wave in free-space is a Lorentzian approximation and in the Galilean Maxwell’s equations, free space is filled with magnetic monopoles that have a density determined by the gravitational field.
The Michelson-Morley experiment couldn’t distinguish between these frameworks because it was interpreted with the heuristic of electromagnetic waves while using a Lorentzian mathematical formalism that described electric waves. The transit time of the electric wave in one arm was actually changed, but the particles used to detect the wave had a slowed clock which perfectly compensated for this path length change. It can be explained with special relativity in terms of length-contraction of the particles in the mirrors.
Returning to jam tomorrow, jam yesterday, but never ever jam today. When people focus on a single moment without connecting it to a larger narrative, they sometimes forget that they are making an approximation of a more dynamic system. The Lorentzian Maxwell’s equations are an approximation in which magnetic monopoles are defined to be non-existent — the quantum vacuum is defined to not exist. This is all well and good when you are working with systems in which the quantum vacuum can be approximated with displacement current and virtual particles, but it removes the vital intuition required to unify the physics of systems large and small.
When you flood your students with approximations that lack any narrative connection, they get confused.
I think that video was the source of my childhood math trauma (I got over it):
The photo in the header is by Elena Kalis.
