Quaternions.com

Euler, Gauss, Hamilton, Sweetser, Maxwell: The Forgotten Path to New Physics

Written by Daniel Weidele in 2024.
Please find the archived work of Doug Sweetser below.

Disclosure

Daniel (that's me, Hi! 👋) is a computer scientist (not formally trained in college-level physics!), who has enjoyed the work of Doug, and thinks that he's been up to something.

However, there's a gap that's been bugging me that Doug has not fully closed, which led me to buy this domain sadly after Doug's passing.

Therefore I have set two goals for this site:

provide an accessible introduction leading most directly to the open gap, and
archive Doug's work for future generations.

I. Quaternion Field Theory

While today quaternions are used computationally to calculate orbits and trajectories in spacefaring or computer graphics, the following shall provide brief intuition on how they can further generate an alternative view on fields in theoretical physics, towards the quest in the understanding of our universe. Such link has already been drawn by their inventor W.R. Hamilton in the 19th century, but has been abandoned in the early 20th century, in favor of vector calculus after J.C. Maxwell and O. Heaviside.

1. What are quaternions?

Have you ever wondered how complex numbers could work in 3-D? No? Well, not too many have, so don't feel ashamed. Quaternions can do in 3-D, what complex numbers do in 2-D. So first of all, it's an algebra. And it's especially well suited for calculating rotations. But really, quaternions are defined in 4-D.

So a quaternion consists of 1 × real (a) and 3 × imaginary (b,c,d) units:

Q = a + bi + cj + dk

The real (a) is also called the scalar, and the imaginaries (b,c,d) are often referred to as the vector component.

You can refer to an introductory lecture on computer graphics to learn how to calculate rotations and orbits with this algebra.

Throughout the following, however, we're on a slightly different quest: we want to understand the universe.

Yet, it's somewhat reassuring that this algebra does work very well within our universe, especially where other forms of calculation are not sufficient (Gimbal Lock).

Would a vector-only calculus become any better when trying to describe even more complex phenomena, like fields? Or is it only practical as long as we don't run in a lock situation? What if we used both vector and scalar components in the analysis of our universe?

To do so, let's exploit the handy fact that quaternions are 4-dimensional, just like spacetime. So we could map time to the real/scalar, and space to the imaginary/vector. That yields a spacetime number χ, where the phenomena spacetime has been projected right onto the quaternion. As a spacetime number, the quaternion now has meaning.

And yes, it's okay to keep wondering why the naming scheme appears flipped overall: surely time feels more imaginary, and space more real...

Let's denote time by w in the scalar (rather than t, to not clash with the symbol for the differential of time later on). And let X be (x, y, z) our spatial vector component.

Then, instead of writing χ = w + xi + yj + zk, I prefer the more handy dense notation χ = (w, X) for spacetime numbers.

2. Quaternion multiplication rule

When multiplying quaternions in dense notation, follow this rule:

(a, b) × (c, d) = (ac - b⋅d, ad + bc + b×d)

Note that while quaternions do not commute, they are associative.

3. Spacetime differential analysis

Next let's try to derive equations that characterize the electromagnetic field (at least!). We do this by applying differential analysis, which means the following.

We let time flow a tiny little bit, and also we move the quaternion a tiny little bit in each spatial direction, only to then check how the dynamics of the system behave.

To do so, we define the differential operator ∇ for quaternions, and then apply it to our spacetime quaternion χ.

Let

∇ = (∂/∂t, ∇) "nabla" operator

and

χ = (w, X) "spacetime" number

Then

∇χ = (∂/∂t, ∇) × (w, X)
= (∂w/∂t - ∇⋅X, ∂X/∂t + ∇w + ∇×X)

The dynamics of the scalar part are described by (∂w/∂t - ∇⋅X)

The dynamics of the vector part are described by (∂X/∂t + ∇w + ∇×X)

∂/∂t is the time differential: when applied to a vector (scalar) field, it yields a vector (scalar) field representing the rate of change of the vectors (scalars) with respect to time.
∇ is the gradient: when applied to a vector (scalar) field, it yields a tensor (vector) field indicating the rate of change of the vectors (scalars).
∇⋅ is divergence: when applied to a vector field, it yields a scalar field representing the magnitude of the source or sink (think of it like a hose or drain).
∇× is the curl: when applied to a vector field, it yields a vector field measuring the tendency to rotate or circulate around a point.
(Not used here yet, but ∇² = ∇⋅∇ would be the Laplacian: when applied to a scalar field, it yields a scalar field representing the magnitude of the rate of change of the scalars.)

TODO: Build a little web tool to visualize these operators.

To sanity check our equations, let's make sure the types of the field are well-defined and compatible.

For

∇χ = (∂w/∂t - ∇⋅X, ∂X/∂t + ∇w + ∇×X)

check

type(∇χ) = (scalar - scalar, vector + vector + vector) ✔

4. Interpretation

Looking at our result from the differential operation:

∇χ = (∂w/∂t - ∇⋅X, ∂X/∂t + ∇w + ∇×X)

We can identify familiar terms from classical electromagnetism in the vector part:

electric field E = ∂X/∂t (changing vector potential creates electric field)
magnetic field B = ∇×X (curl of vector potential gives magnetic field)

But we also get a third vector term:

∇w (gradient of the scalar potential)

And in the scalar part, we find two terms:

∂w/∂t (time rate of change of scalar potential)
∇⋅X (divergence of vector potential)

In classical electromagnetic theory, these scalar terms are typically used as "gauge conditions"—mathematical constraints we impose to make the equations easier to solve. The standard approach is to set certain combinations to zero.

But here's an interesting observation: Our quaternion formulation naturally produces these terms without imposing them. They emerge from the mathematics itself. This suggests they might not just be arbitrary choices—they might represent something physical.

We can reinterpret our spacetime quaternion χ = (w, X) as a potential field, where:

w is a scalar potential
X is a vector potential

The quaternion derivative ∇χ then gives us the field strengths derived from these potentials.

The model suggests some intriguing relationships:

The ∇⋅X term (electromagnetic gauge) appears coupled with ∂w/∂t in the scalar part
The ∇w term appears alongside E and B in the vector part
Electric and magnetic fields relate through Faraday's law (changing electric field induces magnetic field, and vice versa)
The divergence and curl of the vector potential (∇⋅X and ∇×X) are connected through Helmholtz decomposition

What's notable is that all these terms coexist in the same quaternion equation. Nothing is arbitrarily set to zero yet.

5. The Critical Question: What if we DON'T set the gauge to zero?

When deriving the Maxwell homogeneous equations, Doug (similar to Lorenz) sets w = 0 as a gauge field, but leaves it up to the reader to explore other cases:

"One enormous subject I have not looked into is what happens if one keeps this gauge term. The resulting physics must describe things that do not travel at the speed of light. It is the subject of particles with a mass."

Key Insight: While gauging is fine when only interested in the EM-fields, gauging seems to be also what has caused us to not think harder about the meaning of these terms. When I first looked at the above equations, I asked myself:

Could not the gauge term be the gravitational field itself? I'd rather just keep it.

That's when I decided to buy this domain and write about it.

Let's explore what happens when we keep the scalar component w. To understand this, we need a physical picture.

6. A Physical Picture: The Universe as an Elastic Medium

Imagine for a moment that empty space isn't really empty, but rather behaves like an incredibly fine-grained elastic solid—something like an ultra-dense crystal made of the tiniest possible building blocks (think Planck-scale). This isn't a new idea; Maxwell himself considered this when developing his equations!

In such a medium, waves can propagate in two fundamentally different ways:

Transverse waves: Like shaking a rope side-to-side. The medium twists or rotates. This is what the vector component X represents—a twist in three spatial directions.
Longitudinal waves: Like pushing a slinky back and forth. The medium compresses and expands. This is what the scalar component w represents—compression energy.

Physical Interpretation:

w (scalar) = compression/expansion of the medium = gravitational potential
X (vector) = twist/rotation of the medium = electromagnetic potential

Now here's the fascinating part: In an elastic medium, you can't change one without affecting the other. They're coupled. Compress the medium (change w), and it will create rotational stress (affecting X). Twist it (change X), and you create compression (affecting w).

This coupling is exactly what we see between gravity and electromagnetism in the quaternion formulation!

7. Deriving the Wave Equations

Now let's derive what wave equations govern these two types of motion. We'll use the Laplacian (∇²) which measures how much a field differs from its surroundings.

Starting Point: The Quaternion Wave

We have our quaternion potential χ = (w, X). We want to find how this evolves in time and space. The fundamental wave equation connects second time derivatives to second space derivatives (the Laplacian).

For any wave traveling at speed c, we expect something like:

∂²χ/∂t² = c² ∇²χ

But remember, χ has both scalar and vector parts, and they behave differently in our elastic medium model.

Step 1: Separating the Components

Using the Helmholtz decomposition theorem (any vector field can be split into a curl-free part and a divergence-free part), we can write:

X = X₀ + X_rot

where:

X₀ has no curl (∇×X₀ = 0) — this couples to compression
X_rot has no divergence (∇⋅X_rot = 0) — this is pure rotation/twist

Step 2: Apply Divergence and Curl Operators

When we apply the divergence operator (∇⋅) to our wave equation, we isolate the compression part:

∂²(∇⋅X)/∂t² = c² ∇²(∇⋅X)

But ∇⋅X is related to our scalar w through the gauge condition! If we track the compression through both the scalar w and the divergence of X, we find they propagate together at a different speed. In an elastic solid with Poisson ratio ν = 0.25 (which corresponds to an ideal face-centered cubic crystal), longitudinal waves travel at speed √3 c.

A. Longitudinal Wave (Compression):
∂²w/∂t² = 3c² ∇²w

When we apply the curl operator (∇×) to our wave equation, we isolate the rotational part:

∂²(∇×X)/∂t² = c² ∇²(∇×X)

This is the pure twist/rotation in the medium, traveling at the base wave speed:

B. Transverse Wave (Twist):
∂²X_rot/∂t² = c² ∇²X_rot
where ∇⋅X_rot = 0

Step 3: The Coupling Equation

Here's where it gets interesting. These two wave types don't exist independently—they're coupled through the strain energy in the medium. The compression creates a "pressure" that affects the twist, described by a Poisson-type equation:

C. Coupling (Gravity Equation):
∇²w = -4πGρ/c²

where ρ is the mass-energy density. This is exactly the Poisson equation for gravity!

Summary of Wave Equations:

Transverse (EM): ∂²X_rot/∂t² = c² ∇²X_rot with ∇⋅X_rot = 0
Longitudinal (Gravity): ∂²w/∂t² = 3c² ∇²w
Coupling: ∇²w = -4πGρ/c²

8. What This Means

The scalar component w is not just a gauge term that can be set to zero—it's the gravitational potential itself!

By keeping it, we see that:

Electromagnetic waves (transverse) and gravitational waves (longitudinal) are two aspects of the same quaternion wave
They travel at different speeds (c vs √3c for compression waves)
They're coupled through the mass-energy density of particles
Gravity emerges naturally from the same formalism that gives us electromagnetism

9. Addressing the "Scalar Wave" Question

Now we can answer the question about longitudinal electromagnetic waves (sometimes called "scalar waves" in fringe literature):

Important Distinction: The longitudinal waves in this formulation are NOT electromagnetic—they're gravitational. They propagate through the compression of the medium (w), not through the twist (X).

However, because they're coupled to the electromagnetic potential X, manipulating one CAN affect the other. This coupling happens through the mass-energy density term in the Poisson equation.

10. Tesla, Meyl, and Longitudinal Wave Experiments

Can you create "scalar waves" by oscillating electromagnetic fields?

In principle, yes—but they would be gravitational waves, not EM waves. Rapidly oscillating EM fields (changes in X) create changes in the local energy density, which couples to the compression field w through the gravity equation. However, this coupling is extraordinarily weak, which is why we don't see strong gravitational effects from everyday EM fields.

But here's where it gets interesting: Could there be practical setups that enhance this coupling?

The Tesla-Meyl Configuration

Both Nikola Tesla and Konstantin Meyl conducted experiments with a specific setup:

Two spherical antennas (ball-shaped capacitors)
Each connected to a secondary coil (Tesla coil configuration)
Each secondary inductively coupled to a primary coil
Critically: Ground connection used as return path

In their experiments, they observed what appeared to be longitudinal wave transmission—energy transfer that didn't behave like standard electromagnetic radiation.

Could Our Framework Explain This?

Let's analyze this through our quaternion elastic medium model:

Key Observations:

1. The Spherical Geometry

Spherical antennas create a radially symmetric field pattern. In our model, this means the vector potential X has a strong radial gradient component (∇w term). The sphere "breathes" - alternately charging and discharging - which creates a time-varying scalar potential ∂w/∂t.

2. The Ground Connection

The Earth itself acts as a massive conductor with enormous capacitance. When Tesla used ground as a return path, he wasn't just completing an electrical circuit—he was using the entire Earth as part of the resonant system. In our elastic medium picture, this could create large-scale compression waves (changes in w) through the medium.

3. The Resonant Coupling

The primary-secondary coil configuration creates extremely high voltages at specific resonant frequencies. This isn't just about electromagnetic oscillation—the high energy density (ρ term in our coupling equation ∇²w = -4πGρ/c²) could enhance the coupling between the twist field (X) and compression field (w).

4. The Longitudinal Character

If you're pumping energy into the scalar potential w at resonance, you're exciting the longitudinal mode (∂²w/∂t² = 3c² ∇²w). This propagates differently than standard EM waves—it's a compression wave through the medium itself.

A Possible Mechanism

Here's a speculative but physically motivated explanation:

Charging the sphere creates a strong radial electric field and builds up charge (affecting both X and w)
Rapid discharge through the ground connection creates a sudden change in ∂w/∂t
At resonant frequencies, this creates standing waves in the scalar potential w
The ground provides a reference—a large reservoir that allows compression waves to propagate between the two spherical antennas
The receiving sphere responds to changes in both X (standard EM) and w (longitudinal/gravitational)

Why This Might Work:

The spherical geometry + ground connection + resonance might create conditions where:

The energy density is high enough that EM-to-gravity coupling becomes observable
The scalar potential w can build up coherently rather than dispersing
The ground acts as a waveguide for compression modes
The longitudinal wave (traveling at √3c ≈ 1.73c in theory) might explain seemingly faster-than-light energy transfer

Challenges and Open Questions

Important Caveats:

The coupling is still extremely weak. The gravitational constant G ≈ 6.67×10⁻¹¹ means the coupling between EM fields and compression waves is tiny. For this to work, you'd need:

Extremely high energy densities (Tesla coils provide this)
Precise resonance conditions
Possibly coherent buildup over many cycles

Alternative explanations exist. Many of the observed Tesla/Meyl effects could be explained by:

Near-field induction effects
Ground current propagation
Atmospheric plasma effects at high voltages
Resonant cavity modes using Earth's surface

How to Test This Framework

If the quaternion model is correct, we should be able to predict:

Frequency dependence: Optimal coupling at specific resonant frequencies determined by the geometry
Distance scaling: Different power law than 1/r² for pure EM radiation
Phase relationships: Specific phase differences between voltage, current, and transmitted power
Ground dependence: Transmission efficiency should depend on ground conductivity and dielectric properties
Directional characteristics: Longitudinal waves should show different directional patterns than transverse EM

Practical Implications

If this mechanism is real:

Wireless power transmission might work differently than standard EM radiation—not by beaming energy through space, but by exciting compression modes in the medium with the Earth as a resonant element
Communication through the ground using scalar potential modulation rather than EM wave propagation
Energy storage in standing waves of the scalar field w

However, extraordinary claims require extraordinary evidence. The key is designing experiments that can distinguish between:

Standard EM effects (well understood)
Near-field coupling (also well understood)
True longitudinal compression waves (predicted by quaternion framework but not standard EM theory)

Bottom Line: The quaternion elastic medium framework does provide a theoretical mechanism for longitudinal wave propagation through the scalar field w, and the Tesla/Meyl configuration with spherical antennas, resonant coupling, and ground connection might be exactly the right setup to excite these modes. Whether this actually explains their experimental observations remains an open experimental question.

II. Douglas Balch Sweetser (*//1962 - †12/10/2022)

Doug Sweetser, who used to own this domain, has conducted extensive work on quaternions in physics, dedicating a good part of his life. I am doing my best to preserve his results. If you are missing some of his work, please let me know by email, and I will post here timely.

To the best of my knowledge, however, the interpretation of the above model (particularly the elastic medium picture and the coupling between scalar and vector components) draws from recent work in quaternion quantum mechanics.

Doing Physics with Quaternions
PDF-File; 30.14 MB; Nov 29, 2024; MD5 8cb9078df79a3e88e3d12d7a73685795

Archived works
ZIP-Archive; 3.88 GB; Oct 6, 2023; MD5 b643b0d9deeb8c99e92fd0f27390ea71

Website on Github "Q"

Historical Website "The World"

YouTube channel

Questions and answers "Greenspun"

III. Related materials & Further Reading

Maxwell's original quaternion theory was a unified field theory of electromagnetics and gravitation. T. Bearden, 1988
PDF-File; 6.43 MB; Nov 28, 2024; MD5 1d5f0f2ae7742ec4108e4ee26a55081f

The study of electromagnetic processes in the experiments of Tesla. B. Sacco, A.K. Tomilin, 2012
PDF-File; 2.45 MB; Nov 28, 2024; MD5 05c8bc6d5d7e8abf29b628702f14

Wireless power transmission. S. Jackson, 2011
PDF-File; 23.10 MB; Nov 29, 2024; MD5 186b957117bdfa153d9d41a9ad00fdbc

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